Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I+II

Table of contents :
Contents......Page 2
Chapter 1. Introduction......Page 10
Chapter 2. Review: Floer cohomology......Page 48
Chapter 3. The A_∞ algebra associated to a Lagrangian submanifold......Page 86
Chapter 4. Homotopy equivalence of A_∞ algebras......Page 200
Chapter 5. Homotopy equivalence of A_∞ bimodules......Page 276
Chapter 6. Spectral sequences......Page 364
Chapter 7. Transversality......Page 406
Chapter 8. Orientation......Page 684
Appendices......Page 762
Bibliography......Page 800
Index......Page 810

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Lagrangian Intersection Floer Theory

Contents Volume I Preface

xi

Chapter 1. Introduction 1.1 What is Floer (co)homology 1.2 General theory of Lagrangian Floer cohomology 1.3 Applications to symplectic geometry 1.4 Relation to mirror symmetry 1.5 Chapter-wise outline of the main results 1.6 Acknowledgments 1.7 Conventions

1 1 5 13 16 25 35 36

Chapter 2. Review: Floer cohomology 2.1 Bordered stable maps and the Maslov index 2.1.1 The Maslov index: the relative first Chern number 2.1.2 The moduli space of bordered stable maps 2.2 The Novikov covering and the action functional 2.2.1 The Γ-equivalence 2.2.2 The action functional and the Maslov-Morse index 2.3 Review of Floer cohomology I: without anomaly 2.3.1 The L2 -gradient equation of A 2.3.2 Floer’s definition: Z2 -coefficients. 2.3.3 Bott-Morse Floer cohomology 2.4 Review of Floer cohomology II: anomaly appearance 2.4.1 The Floer cochain module 2.4.2 The Floer moduli space 2.4.3 The Novikov ring ΛR (L) 2.4.4 Monotone Lagrangian submanifolds 2.4.5 Appearance of the primary obstruction

39 39 39 43 49 50 51 53 53 57 59 60 61 62 66 69 71

Chapter 3. The A∞ algebra associated to a Lagrangian submanifold 3.1 Outline of Chapter 3 3.2 Algebraic framework on filtered A∞ algebras 3.2.1 A∞ algebras and homomorphisms 3.2.2 Filtered A∞ algebras and homomorphisms 3.3 Algebraic framework on the homotopy unit 3.3.1 Definition of the homotopy unit 3.3.2 Unital (resp. homotopy unital) A∞ homomorphisms 3.4 A∞ deformation of the cup product

77 77 86 86 89 94 94 97 97

v

vi

CONTENTS

3.5 The filtered A∞ algebra associated to a Lagrangian submanifold 3.6 Bounding cochains and the A∞ Maurer-Cartan equation. 3.6.1 Bounding cochains and deformations 3.6.2 Obstruction for the existence of bounding cochain 3.6.3 Weak unobstructedness and existence of Floer cohomology  3.6.4 The superpotential and M(C) 3.7 A∞ bimodules and Floer cohomology 3.7.1 Algebraic framework 3.7.2 A∞ bimodule homomorphisms 3.7.3 Weak unobstructedness and deformations 3.7.4 The filtered A∞ bimodule C(L(1) , L(0) ; Λ0,nov ) 3.7.5 The Bott-Morse case 3.7.6 Examples 3.7.7 The multiplicative structure on Floer cohomology 3.8 Inserting marked points in the interior 3.8.1 The operator p 3.8.2 Applications to vanishing of the obstruction classes ok (L) 3.8.3 Outline of the construction of the operator p 3.8.4 The operator q 3.8.5 Bulk deformation of filtered A∞ structures 3.8.6 Outline of the construction of the operator q 3.8.7 The operator r and the A∞ bimodule 3.8.8 Construction of the operator r 3.8.9 Generalization of the operator p 3.8.10 Proof of parts of Theorems B, C and G

102 107 108 111 114 117 120 120 123 125 126 137 151 155 156 156 159 161 165 168 175 178 181 182 188

Chapter 4. Homotopy equivalence of A∞ algebras 4.1 Outline of Chapters 4 and 5 4.2 Homotopy equivalence of A∞ algebras: the algebraic framework 4.2.1 Models of [0, 1] × C 4.2.2 Homotopies between A∞ homomorphisms 4.2.3 The unital or homotopy-unital cases 4.3 Gauge equivalence of bounding cochains 4.3.1 Basic properties and the category HA∞ 4.3.2 Mweak (C) and its homotopy invariance 4.3.3 Mweak,def (L) and its homotopy invariance 4.4 Uniqueness of the model of [0, 1] × C 4.4.1 Induction on the number filtration I 4.4.2 AK structures and homomorphisms 4.4.3 Induction on the number filtration II 4.4.4 Unital case I: the unfiltered version 4.4.5 Coderivation and Hochschild cohomology 4.4.6 Induction on the energy filtration 4.4.7 Unital case II: the filtered version 4.5 Whitehead theorem in A∞ algebras 4.5.1 Extending AK homomorphisms to AK+1 homomorphisms 4.5.2 Proof of Theorem 4.2.45 I: the number filtration 4.5.3 Unital case: the unfiltered version

191 191 197 197 205 208 211 211 215 216 217 218 219 220 223 226 230 232 233 234 236 237

CONTENTS

4.5.4 Extending filtered A∞ homomorphism modulo T λi to modulo T λi+1 4.5.5 Proof of Theorem 4.2.45 II: the energy filtration 4.6 Homotopy equivalence of A∞ algebras: the geometric realization 4.6.1 Construction of A∞ homomorphisms 4.6.2 Homotopies between A∞ homomorphisms 4.6.3 Compositions 4.6.4 Homotopy equivalence and the operator q I: changing the cycle in the interior 4.6.5 Homotopy equivalence and the operator q II: invariance of symplectic diffeomorphisms 1 4.6.6 Homotopy equivalence and the operator q III: invariance of symplectic diffeomorphisms 2 Chapter 5. Homotopy equivalence of A∞ bimodules

vii

239 241 242 242 249 257 259 261 264 267

5.1 Novikov rings 5.1.1 Reduction to universal Novikov ring 5.1.2 Hamiltonian independence of the Novikov ring 5.1.3 Floer cohomologies over Λ(L(0) , L(1) ; 0 ) and Λnov

267 267 270 272

5.2 Homotopy equivalences of A∞ bimodules: the algebraic framework 5.2.1 Weakly filtered A∞ bimodule homomorphisms 5.2.2 Deformations of A∞ bimodule homomorphisms 5.2.3 Homotopies between A∞ bimodule homomorphisms 5.2.4 Gauge invariance and the category HA∞ (C1 , C0 ) 5.2.5 Obstructions to defining A∞ bimodule homomorphisms I 5.2.6 Whitehead theorem for A∞ bimodule homomorphisms 5.2.7 Obstructions to defining A∞ bimodule homomorphisms II

275 275 276 282 288 291 292 294

5.3 Homotopy equivalences of A∞ bimodules: the geometric realization 5.3.1 Construction of filtered A∞ bimodule homomorphisms 5.3.2 Moving Lagrangian submanifolds by Hamiltonian isotopies 5.3.3 Homotopies between bimodule homomorphisms 5.3.4 Compositions of Hamiltonian isotopies and of bimodule homomorphisms 5.3.5 An energy estimate. 5.3.6 The operators q, r and homotopy equivalence 5.3.7 Wrap-up of the proof of invariance of Floer cohomologies 5.4 Canonical models, formal super schemes and Kuranishi maps 5.4.1 Canonical models, Kuranishi maps and bounding cochains 5.4.2 The canonical models of filtered A∞ bimodules 5.4.3 Filtered A∞ bimodules and complex of coherent sheaves 5.4.4 Construction of the canonical model 5.4.5 Including the operator q 5.4.6 Wrap-up of the proofs of Theorems F, G, M, N and Corollaries O, P Chapter 6. Spectral sequences 6.1 Statement of the results in Chapter 6

296 296 306 313 319 321 326 327 330 330 336 337 339 347 349 355 355

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CONTENTS

6.1.1 The spectral sequence 6.1.2 Non-vanishing theorem and a Maslov class conjecture 6.1.3 Applications to Lagrangian intersections

355 357 360

6.2 A toy model: rational Lagrangian submanifolds 6.3 The algebraic construction of the spectral sequence 6.3.1 c.f.z. 6.3.2 d.g.c.f.z. (differential graded c.f.z.) 6.3.3 Construction and convergence

362 366 367 369 371

6.4 The spectral sequence associated to a Lagrangian submanifold 6.4.1 Construction 6.4.2 A condition for degeneration: proof of (D.3) 6.4.3 Non-vanishing theorem: proof of Theorem 6.1.9. 6.4.4 Application to the Maslov class conjecture: proofs of Theorems 6.1.15 and 6.1.17 6.4.5 Compatibility with the product structure

375 375 375 377

6.5 Applications to Lagrangian intersections 6.5.1 Proof of Theorem H 6.5.2 Proof of Theorem I 6.5.3 Torsion of the Floer cohomology and Hofer distance: Proof of Theorem J 6.5.4 Floer cohomologies of Lagrangian submanifolds that do not intersect cleanly 6.5.5 Unobstructedness modulo T E

385 385 385

381 382

388 393 395

Volume II Chapter 7. Transversality 7.1 Construction of the Kuranishi structure 7.1.1 Statement of the results in Section 7.1 7.1.2 Kuranishi charts on Mmain,reg (β): Fredholm theory k+1 7.1.3 Kuranishi charts in the complement of Mmain,reg (β): gluing k+1 7.1.4 Wrap-up of the proof of Propositions 7.1.1 and 7.1.2   7.1.5 The Kuranishi structure of Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)): A∞ map analog of Stasheff cells 7.2 Multisections and choice of a countable set of chains 7.2.1 Transversality at the diagonal 7.2.2 Inductive construction of compatible system of multisections in the Bott-Morse case 7.2.3 Perturbed moduli space running out of the Kuranishi neighborhood I 7.2.4 Statement of results 7.2.5 Proof of Proposition 7.2.35 7.2.6 Filtered An,k structures 7.2.7 Construction of filtered An,K structures 7.2.8 Perturbed moduli space running out of the Kuranishi neigborhood II 7.2.9 Construction of filtered An,K homomorphisms

397 398 398 401 404 418 425 435 436 437 444 445 449 458 461 466 468

CONTENTS

7.2.10 Constructions of filtered A n,K homotopies 7.2.11 Constructions of filtered A∞ homotopies I: a short cut 7.2.12 Constructions of filtered A∞ homotopies II: the algebraic framework on homotopy of homotopies 7.2.13 Constructions of filtered A ∞ homotopies III: the geometric realization of homotopy of homotopies 7.2.14 Bifurcation vs cobordism method: an alternative proof 7.3 Construction of homotopy unit 7.3.1 Statement of the result and the idea of its proof 7.3.2 Proof of Theorem 7.3.1 7.3.3 Proof of (3.8.36) 7.4 Details of the construction of the operators p, q and r 7.4.1 Details of the construction of p 7.4.2 Construction of q I: the An,K version 7.4.3 Construction of q II: q is an L∞ homomorphism  7.4.4 Construction of q III: the homotopy invariance of Der(B(C[1]),  B(C[1])) 7.4.5 Construction of q IV: wrap-up of the proof of Theorem 3.8.32 7.4.6 Proof of Theorem Y 7.4.7 Algebraic formulation of r I: Der B(C1 , C0 ; D) and its homotopy invariance 7.4.8 Algebraic formulation of r II: via bifurcation argument 7.4.9 Algebraic formulation of r III: via cobordism argument 7.4.10 Algebraic formulation of p I: the cyclic bar complex is an L∞ module 7.4.11 Algebraic formulation of p II: p induces an L∞ module homomorphism 7.5 Compatibility with rational homotopy theory 7.5.1 Statement of results 7.5.2 Virtual fundamental chain in de Rham theory 7.5.3 The Kuranishi structure of Mmain k+1 (β0 ) 7.5.4 Construction of the AK homomorphism I 7.5.5 Construction of the AK homomorphism II  7.5.6 The A∞ map to a topological monoid and Nk+1 Chapter 8. Orientation 8.1 Orientation of the moduli space of unmarked discs 8.1.1 The case of holomorphic discs 8.1.2 The example of non-orientable family index 8.1.3 The case of connecting orbits in Floer theory 8.1.4 Change of relatively spin structure and orientation 8.2 Convention and preliminaries 8.3 Orientation of the moduli space of marked discs and of the singular strata of the moduli space 8.4 Orientation of M+1 (β; P1 , . . . , P ). 8.4.1 Definition of the orientation of M+1 (β; P1 , . . . , P ) 8.4.2 Cyclic symmetry and orientation 8.5 The filtered A∞ algebra case 8.6 Orientation of the moduli space of constant maps

ix

483 502 505 534 569 574 574 576 587 589 589 595 596 601 621 625 631 637 640 644 647 650 650 652 654 655 663 669 675 675 675 684 686 690 691 698 703 703 705 708 713

x

CONTENTS

8.7 Orientation of the moduli space of connecting orbits 8.8 The Bott-Morse case 8.9 Orientations of the top-moduli spaces and the twp-moduli spaces   8.9.1 Orientation of Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) main 8.9.2 Orientation of Mk+1 ({Jρ }ρ : β; twp(ρ); P1 , . . . , Pk ) 8.10 Homotopy units, the operators p,q, continuous families of perturbations, etc. 8.10.1 Homotopy unit 8.10.2 Operators p, q 8.10.3 Continuous families of perturbations Appendices A1 Kuranishi structures A1.1 Review of the definition of the Kuranishi structure and multisections A1.2 Fiber products A1.3 Finite group actions and the quotient space A1.4 A remark on smoothness of coordinate transforms A1.5 Some counter examples A1.6 Some errors in the earlier versions and corrections thereof A2 Singular chains with local coefficients A3 Filtered L∞ algebras and symmetrization of filtered A∞ algebras A4 The differential graded Lie algebra homomorphism in Theorem 7.4.132 Bibliography Index

716 719 731 731 735 738 738 738 749 753 753 754 764 766 768 778 779 780 782 787 791 801

Preface With the advent of the method of pseudo-holomorphic curves developed by Gromov in the 80’s and the subsequent Floer’s invention of elliptic Morse theory resulted in Floer cohomology, the landscape of symplectic geometry has changed drastically. Many previously intractable problems in symplectic geometry were solved by the techniques of pseudo-holomorphic curves, and the concept of symplectic topology gradually began to take shape. This progress was accompanied by parallel developments in physics most notably in closed string theory. In 1993, partially motivated by Donaldson’s pants product construction in Floer cohomology, the first named author introduced the structure of an A∞ -category in symplectic geometry whose objects are Lagrangian submanifolds and whose morphisms are the Floer cohomologies (or complexes). Based on this algebraic framework, Kontsevich proposed the celebrated homological mirror symmetry between the derived category of coherent sheaves and the Fukaya category of Lagrangian submanifolds in his 1994 ICM talk in Z¨ urich. Enhanced by the later development in open string theory of D-branes, this homological mirror symmetry has been a source of many new insights and progresses in both algebraic geometry and symplectic geometry as well as in physics. However the rigorous formulation of homological mirror symmetry has not been made, largely due to lack of understanding the Floer theory of Lagrangian submanifolds itself. In this book, we explain how the obstruction to and anomaly in the construction of Floer cohomology arise, provide a precise formulation of the obstructions and then carry out detailed algebraic and analytic study of deformation theory of Floer cohomology. It turns out that even a description of such an obstruction (in a mathematically precise way) requires new homological algebra of filtered A∞ algebras. In addition, verification of existence of such an algebraic structure in the geometric context of Lagrangian submanifolds requires non-trivial analytic study of the corresponding moduli space of pseudo-holomorphic discs. We also provide various immediate applications of the so constructed Floer cohomology to problems in symplectic topology. Many of these improve the previously known results obtained via Floer theory and some firsthand applications to homological mirror symmetry are new. We expect more nontrivial applications of the theory will soon follow as its true potential is unveiled and then realized. While we have been preparing this book, there have been several important developments in symplectic geometry and in related fields. The relationship between topological strings, D-branes and pseudo-holomorphic curves and symplectic Floer theory is now more clearly understood. The usage of higher algebraic structures in Floer theory, which we have been promoting while writing this book, has now become a popular and essential area of research. Furthermore advances of the techniques handling various moduli spaces of solutions to nonlinear PDE’s, xi

xii

PREFACE

intertwined with the formalism of higher algebraic structures, has now made the geometric picture more transparent. This will help facilitate the further progression of the geometric theory. In this book we take full advantage of these developments and provide the Floer theory of Lagrangian submanifolds in the most general form available at this time. We hope that this book will be a stepping stone for future advancements in symplectic geometry and homological mirror symmetry. Our collaboration which has culminated in completion of this book started during the 1996 (8 July–12 July) conference held in Ascona, Switzerland. We hardly imagined then that our project would continue to span more than 10 years. We have greatly enjoyed this collaboration and hope to continue it into the coming decades. In fact our second journey into newly landscaped field of symplectic topology and mirror symmetry has already begun, and we hope to garner more fruits of collaboration: The scene in front looks very different and much more exciting than the one we left behind 13 years ago! June 9 2009. Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono.

https://doi.org/10.1090/amsip/046.1/01

CHAPTER 1

Introduction 1.1. What is the Floer (co)homology Floer homology is a semi-infinite (∞/2) dimensional homology theory on an infinite (∞) dimensional space in general. It has been defined in various contexts and proven to be an extremely deep and useful tool to study many problems arising in the various areas including symplectic geometry, gauge theory and the low dimensional topology. Floer’s main motivation to introduce Floer homology in symplectic geometry [Flo88IV] was to prove Arnold’s conjecture [Arn65] for the fixed points of Hamiltonian diffeomorphisms (or exact symplectic diffeomorphisms). One version of Arnold’s conjecture is that any Hamiltonian diffeomorphism φ : (M, ω) → (M, ω) on a compact M satisfies  (1.1) # Fix φ ≥ rank Hk (M ) k

provided all of its fixed points are nondegenerate. A Hamiltonian diffeomorphism (or exact symplectic diffeomorphism) is a symplectic diffeomorphism obtained as the time one map of a (time-dependent) Hamiltonian flow. By considering the diagonal Δ ⊂ (M × M, ω ⊕ −ω), which is a Lagrangian submanifold in the product, one can instead ask its intersection theoretic version, which is indeed the original approach taken by Floer in [Flo88IV]. Let L1 , L0 ⊂ (M, ω) be two compact Lagrangian submanifolds. The ideal statement one might expect for Floer homology is as follows: (1.2.1) We can assign a (graded) abelian group (or Z2 vector space), denoted by HF (L1 , L0 ), to each pair (L1 , L0 ) of Lagrangian submanifolds. We call HF (L1 , L0 ) the Floer homology. (1.2.2) Floer homology is invariant under the Hamiltonian diffeomorphisms. Namely if φi , i = 0, 1 are two Hamiltonian diffeomorphisms, then HF∗ (L1 , L0 )  HF∗ (φ1 L1 , φ0 L0 ). (1.2.3) If L0 = L1 = L, then the Floer homology group coincides with the standard homology group of Lagrangian submanifold L. Namely HF∗ (L, L)  H∗ (L; Z2 ). (1.2.4) When L0 is transversal to L1 , Floer homology HF∗ (L1 , L0 ) is a homology group of the chain complex that is given by  (1.3) CF∗ (L1 , L0 ) = Z2 [p] p∈L1 ∩L0

as a group. 1

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2

INTRODUCTION

Actually Floer defined HF (L1 , L0 ) as the homology of the chain complex (1.3) where the matrix element ∂p, q of the boundary operator ∂ is defined by counting the number of pseudo-holomorphic (cornered) discs with each half of its boundary lying on L0 and L1 respectively, and with its asymptotic limits given by p, q ∈ L1 ∩ L0 (see Section 2.3 for more details). The main task to be carried out is to justify this definition and to prove the boundary property ∂∂ = 0 of the operator ∂. In [Flo88IV], Floer carried out this analysis, defined the Floer homology group HF (L1 , L0 ), and proved the properties (1.2.1)-(1.2.4) above for the case where L0 = φ(L),

L1 = L

for L satisfying π2 (M, L) = 0 and φ : M → M a Hamiltonian diffeomorphism. After Floer, the second named author generalized the construction of Floer homology to the arbitrary pairs (L1 , L0 ) of monotone Lagrangian submanifolds with minimal Maslov number > 2 in [Oh93,95I] (See Subsection 2.4.5 and Theorem 2.4.42 for its definition). He also proved (1.2.2),(1.2,4) but observed that (1.2.3) does not hold in general. Both authors used Z2 -coefficients. At first sight, these conditions look rather restrictive and technical especially when compared to the development of the other version of Floer homology, Floer homology for Hamiltonian diffeomorphisms, and of Arnold’s conjecture (1.1). Namely, for the case of diffeomorphisms, the following result has been already proved for arbitrary compact symplectic manifold M : For any Hamiltonian diffeomorphism φ : M → M , one can associate a graded abelian group, Floer homology HF (M ; φ) with suitable coefficients, such that 

HF (M ; φ) ∼ = H(M ) rank HFk (M ; φ) ≤ # Fix(φ).

k

Here Fix(φ) is the fixed point set of φ. In [Flo89I], Floer gave the definition of HF (M ; φ) for Hamiltonian diffeomorphisms and proved that HF (M ; φ) ∼ = H(M ; Z), when (M, ω) is monotone. As a consequence, he proved Arnold’s conjecture for the monotone case. (For this matter, we should mention the earlier result by Conley-Zehnder [CoZe83] for the case of symplectic tori and by Floer himself for the case where π2 (M ) = 0 as a consequence of [Flo88IV]). This was generalized by Hofer-Salamon [HoSa95] and the fourth named author [Ono95] to semi-positive symplectic manifolds. Construction of the Floer homology and proof of Arnold’s conjecture for the general case have been carried out in [FuOn99II, LiuTi98, Rua99] over Q, and by the first and fourth named authors in [FuOn01] over Z. (The detail of [FuOn01] is still to be written.) Since HF (M ; φ) is isomorphic to the homology group H(M ) of M , the group itself does not give rise to a new invariant of the symplectic manifold (M, ω). However defining Floer homology itself has application to Arnold’s conjecture in various settings. On the other hand, we can not expect the theory of Lagrangian intersection Floer homology would completely resemble that of Hamiltonian diffeomorphisms. Namely we can not expect all of the properties (1.2.1)-(1.2.4) hold in general.

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1.1. WHAT IS THE FLOER (CO)HOMOLOGY

3

Firstly, Lagrangian intersection Floer homology concerns pairs (L1 , L0 ) of Lagrangian submanifolds, but cannot be defined for completely arbitrary pairs. Secondly, unlike the case of diffeomorphisms, the relevant moduli spaces are neither (stably almost) complex spaces nor deformable to them and so do not carry canonical orientations in general. For the case of cotangent bundles, orientation problem in the Lagrangian intersection Floer homology was previously studied by the second named author in [Oh97II] where he improved Floer’s result [Flo88IV] to define the Floer homology over Z coefficients. In general, providing an orientation on the moduli space of pseudo-holomorphic discs (or bordered Riemann surfaces) attached to a Lagrangian submanifold is not always possible, and is a rather nontrivial family index problem of the Cauchy-Riemann operator over the bordered Riemann surface. This leads to the condition on the spin structure on the Lagrangian submanifold (see Definition 1.6 below). Thirdly, even when HF (L1 , L0 ) is defined, the (symplectic) geometric meaning of the invariants HF (L1 , L0 ) becomes less clear. For example, let us consider a pair (L1 , L0 ) of compact Lagrangian submanifolds of Cn (Cn is noncompact but is tame in the sense of Gromov [Grom85] which still allows one to do the Floer theory for compact Lagrangian submanifolds in the same way). Then obviously there exists a Hamiltonian isotopy φt of compact support such that φ1 (L1 ) ∩ L0 = ∅. This fact and (1.2.2), (1.2.4) would mean that if one could define Floer homology of the pair (L1 , L0 ) of Lagrangian submanifolds invariant under the Hamiltonian isotopy, then it should become necessarily trivial. Hence it appears at the first sight that the Floer theory would give nothing interesting for the important case of Lagrangian submanifolds of Cn . However there are various results on Lagrangian submanifolds of Cn which one can prove using the Floer theory, just knowing that it is defined. For example, using the Floer theory, one can prove Gromov’s theorem of nonexistence of exact Lagrangian submanifold [Grom85] and some topological obstruction to the Maslov class of compact Lagrangian embedding in Cn as illustrated in [Oh96I,96II]. This line of ideas has been further developed by others, most notably by Biran and Cieliebak [BiCi01,02]. Therefore it is an important task to understand when the Floer homology can be defined and when not. As pointed out in [Oh93], if we define the operator ∂ just by counting the number of pseudo-holomorphic discs as Floer did in [Flo88IV], then ∂ will not satisfy the boundary property, i.e., ∂∂ = 0 in general. This fact was applied to a variant of Floer homology in [Ono96]. In physical terms, ∂∂ = 0 might be regarded as the breaking of BRST symmetry by soliton effects (i.e., by the presence of pseudo-holomorphic discs). The word “anomaly” in the sub-title of the present book indicates this (BRST) symmetry breaking. This means that the mathematical framework for the Floer homology of Lagrangian submanifolds will be more involved (and interesting) than that of Hamiltonian diffeomorphisms. The main result that we establish in this book is an obstruction theory to defining the Floer homology of Lagrangian intersections. We also describe how the homology of L is related to the Floer homology HF (L, L) and provide various applications of Floer homology to the study of geometry of Lagrangian submanifolds. The result of this book is also fundamental to make precise formulation of the celebrated homological mirror symmetry conjecture proposed by Kontsevich [Kon95I]: This is based on the framework of Floer cohomology, or more precisely of the A∞ category that the first named author introduced in [Fuk93]. The most important

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4

INTRODUCTION

steps towards a rigorous construction of the A∞ category on a general symplectic manifold are given in this book. Recently there are many works both by physicists and by mathematicians on the calculation of the number of (pseudo)holomorphic discs attached to a given Lagrangian submanifold. (See for example, [KatLi01, LiSo01, AAMV05].) As in the closed case of Gromov-Witten invariants, they mainly use the fixed point localization under the torus action in their computations. As is widely known by now, the number itself of holomorphic discs is not well defined in that the number depends on the various choices involved, such as perturbations to make the moduli space transversal. Therefore, it is a highly nontrivial problem to extract an object, out of the numbers calculated by localization and others, that is mathematically well defined. In this book, we have solved this problem by constructing an algebraic structure, the filtered A∞ algebra, whose structure constants are the numbers of pseudo-holomorphic discs and satisfy various relations. We prove that the homotopy equivalence class of this filtered A∞ algebra associated to a given Lagrangian submanifold is independent of various choices involved. It is not yet very clear to the authors precisely how the numbers obtained by the localization and others are related to the structure studied in this book in general, although it is quite apparent that they must be related. The algebraic basis of the present work is the theory of A∞ algebras, which we will explain more in the next section. The notion of A∞ algebra was introduced by Stasheff [Sta63] in the homotopy theory using the idea of Sugawara [Sug57] in 1960’s. Witten [Wit86] (the end of 1980’s) and Zwiebach [Zwi91,97] (the beginning of 1990’s) observed that A∞ structure is a natural frame work for the open string theory. Just after that, inspired by Donaldson’s lecture, the first named author independently introduced the A∞ structure to the Floer theory. In those days the relationship between these works was well understood by neither the researchers in mathematics nor in physics. We are quite happy to see that the developments in both sides in the last decade have completely changed this landscape which made the relationship between the (topological) open string with D-brane and the Floer cohomology much clearer. Now appearance of the A∞ structures in both fields is not an alien but a natural phenomenon. In this midst the relationship of this A∞ structure to that of homotopy theory has also become clearer. In the beginning of 1990’s when the idea of using the A∞ structure or homological algebra of that kind in the Floer theory was first incepted, it was so alien that it was not well-received among the symplectic geometers. Only recently the study of various algebraic structures in Floer theory became natural and unavoidable among the researchers. On the other hand, Floer cohomology, especially the one arising in symplectic geometry, is still rarely understood in detail by theoretical physicists working on the string theory. (After the work of Vafa [Vaf99], Douglas [Dou02] etc., it becomes more familiar of some physicists.) We sincerely hope that publication of this book will improve the situation even more. We would also like to mention that the basic technology developed in the second half of 1990’s ([FuOn99II, LiTi98, Rua99, Sie96]) for the purpose of handling the moduli spaces defined by nonlinear elliptic partial differential equations has made it possible to work out the mathematical details needed for the systematic construction of various algebraic structures coming out of the moduli space. Especially the method of using Kuranishi structure and multi-sections (or something

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1.2. GENERAL THEORY OF LAGRANGIAN FLOER COHOMOLOGY

5

equivalent to them) is crucial for such constructions. The machinery of Kuranishi structure and multi-sections was developed by the first and the fourth named authors in 1996 for the purpose of handling many transversality problems needed in the construction of appropriate (virtual) fundamental cycles in various contexts. In this book, we employ them in an even more nontrivial way of defining (virtual) fundamental chains. This book is a research monograph. All the results are new unless otherwise mentioned. 1.2. General theory of Lagrangian Floer cohomology From now on we use the cohomological notation for the Floer theory. Therefore we will exclusively use cohomology, not homology, unless otherwise stated. To obtain the whole picture of Floer cohomology of Lagrangian submanifolds, it turns out necessary to use the language of filtered A∞ algebras and filtered A∞ modules. We will describe them systematically in Chapters 3-5. Here we give only the definitions that are necessary in stating Theorem A below. Firstly we introduce our coefficient ring Λ0,nov .  ∞    λi ni  ai T e  ai ∈ Q, λi ∈ R≥0 , ni ∈ Z and lim λi = ∞ . Λ0,nov = i→∞  i=0

Here T and e are formal generators of deg e = 2 and deg T = 0. We call Λ0,nov the universal Novikov ring. We may assume that λ0 = 0. Consider a graded vector space C (over Q) and its shifted one C[1] defined by C[1]k = C k+1 . We consider a series of operations mk,i : C[1] ⊗ · · · ⊗ C[1] → C[1] 

 k

of degree 1 − μi . Here μi are even numbers and μ0 = 0. For each k = 0, 1, 2, · · · , we put ∞  mk = T λi eμi /2 mk,i . i=0

Here λi are nonnegative real numbers such that λ0 = 0, λi > 0 for i > 0, limi→∞ λi = ∞. We assume m0,0 = 0. The operations mk define homomorphisms ˆ 0,nov )⊗ · · · ⊗(C[1]⊗Λ ˆ 0,nov ) → (C[1] ⊗ ˆ Λ0,nov ). mk : (C[1]⊗Λ

  k

ˆ stands for the completion of algebraic tensor product with Here and hereafter ⊗ respect to appropriate filtration. (See Sections 3.1 and 3.2.) We say that the operations mk define the structure of filtered A∞ algebra on ˆ 0,nov if they satisfy the quadratic relations C[1]⊗Λ (1.4)



k1 

(−1)∗ mk1 (x1 , · · · , mk2 (xj , · · · , xj+k2 −1 ), · · · , xk ) = 0

k1 +k2 =k+1 j=1

for each k = 0, 1, 2, · · · , where ∗ = deg x1 + · · · + deg xj−1 + j − 1. We call (1.4) the A∞ relation or A∞ formula. (See Definition 3.2.20 and Remark 3.2.21.)

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6

INTRODUCTION

Remark 1.5. The ring Λ0,nov is a version of the one introduced by Novikov [Nov81] in his Morse theory of closed one forms, which is closely related to Floer cohomology. Therefore we call it the (universal) Novikov ring in this book. (A similar ring have been appeared in some different contexts before Novikov.) It was Floer [Flo89I] who first observed that Novikov ring is the correct coefficient ring to be used for the Floer cohomology. The fourth named author [Ono95] and HoferSalamon [HoSa95] independently used Novikov ring in their proofs of Arnold’s conjecture (1.1) for semi-positive symplectic manifolds. To study the general Floer theory over the rational coefficients, we need to assume that Lagrangian submanifolds Li , i = 0, 1 are relatively spin. Definition 1.6. We call a submanifold L ⊂ M relatively spin if it is orientable and there exists a class st ∈ H 2 (M ; Z2 ) that restricts to the second Stiefel-Whitney class w2 (L) of L. We call a pair of submanifolds (L1 , L0 ) of M relatively spin if they are orientable and there exists a class st ∈ H 2 (M ; Z2 ) that restricts to the second Stiefel-Whitney classes w2 (Li ) of both of Li , i = 0, 1. We put H ∗ (L; Λ0,nov ) = H ∗ (L; Q) ⊗Q Λ0,nov . Theorem A. To each relatively spin Lagrangian submanifold L we can associate a structure of filtered A∞ algebra mk on H ∗ (L; Λ0,nov ), which is well-defined up to isomorphism. If ψ : (M, L) → (M  , L ) is a symplectic diffeomorphism, then we can associate to it an isomorphism ψ∗ := (ψ −1 )∗ : H ∗ (L; Λ0,nov ) → H ∗ (L ; Λ0,nov ) of filtered A∞ algebras whose homotopy class depends only of isotopy class of symplectic diffeomorphism ψ : (M, L) → (M  , L ). The Poincar´e dual P D([L]) ∈ H 0 (L; Λ0,nov ) of the fundamental class [L] is the unit of our filtered A∞ algebra. The homomorphism ψ∗ is unital. Here, an isomorphism of a filtered A∞ algebra is by definition a filtered A∞ homomorphism that has an inverse. The notion of filtered A∞ homomorphism is defined in Definition 3.2.29. A homotopy between them is defined in Definition 4.2.35. See Definition 3.2.20 and 3.3.11 for the definition of a unit and the unitality of filtered A∞ homomorphisms. The proof of Theorem A is completed in Section 5.4 (Corollary 5.4.6) based on the results of Chapters 7 and 8. Much of the rigorous construction in this whole book is based on the results in Chapters 7 and 8, which deal with the problems of transversality and of orientation respectively. Therefore we will not repeat this kind of the phrase in italic from now on. The filtered A∞ algebra in Theorem A will be homotopy equivalent to the de Rham complex when we forget the contribution of nontrivial pseudoholomorphic discs. (Theorem X.) We remark that the filtered A∞ structure appearing in Theorem A is defined for an arbitrary relatively spin Lagrangian submanifold L. It induces a coderivation dˆ on the free (formal) coalgebra BH(L; Λ0,nov )[1] =

k

H(L; Λ0,nov )[1]⊗ · · · ⊗H(L; Λ0,nov )[1]

 

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k

1.2. GENERAL THEORY OF LAGRANGIAN FLOER COHOMOLOGY

7

(see Section 3.2 formula (3.2.1)) and the A∞ relation (1.4) is equivalent to dˆdˆ = 0. ˆ This is an A∞ Thus we obtain a differential graded coalgebra (BH(L; Λ0,nov )[1], d). analog to the classical bar resolution of differential graded algebra (see [Mac63] Chapter 3 Section 3.5). We call it the bar complex. The bar complex of A∞ algebra is studied in [GeJo90], for example. Compared to the notation of [GeJo90], our notation BH(L; Λ0,nov )[1] corresponds to BH(L; Λ0,nov ) in [GeJo90]. The dual of a bar complex is a differential graded algebra in general. A similar differential graded algebra was defined by Chekanov [Chek02] for a Legendrian knots in R3 . (The idea of using this kind of bar complex in Floer theory is independently discovered by Chekanov and by the present authors in 1990’s. See also [EGH00].) The second half of Theorem A implies that cohomology of the ˆ is an invariant of L. However cohomology of bar complex (BH(L; Λ0,nov )[1], d) ˆ (BH(L; Λ0,nov )[1], d) is isomorphic to Λ0,nov , since it is an A∞ analogue of the bar complex of a differential graded algebra with unit. (See for example in [Mac63] p 306.) Some reduced and/or symmetrized versions of the bar complex (for example the reduced or the cyclic and reduced ones) have nontrivial cohomology. (See [Fuk05II] for its relation to the loop space.) Theorem A also implies that the cohomology of them is an invariant of the relatively spin Lagrangian submanifold. However rather than studying these reduced versions, we study another cohomology also induced naturally from the A∞ algebra, whose definition we will explain below. We call this cohomology the Floer cohomology. There are several reasons for doing this. Firstly cohomology of (versions of) the bar complex is usually of infinite dimension when it is nontrivial. This makes it harder to apply: For example its rank is not a good invariant. Secondly, Floer cohomology is more directly related to various geometric properties of the Lagrangian submanifold, such as the classical cohomology of Lagrangian or the number of intersection points between them. Thirdly, it is Floer cohomology (and not the cohomology of bar complex), that corresponds to sheaf cohomology (or Ext group) under the homological mirror symmetry correspondence. (See Conjecture R.) To obtain Floer cohomology of L out of Theorem A, we proceed in the following way. Let (C, mk ) be a filtered A∞ algebra. Note that we assume m0,0 = 0 but do not assume m0,i : Q → C[1] is zero for i > 0. This is related to the phenomenon that δδ = 0 in general Floer theory: Namely (1.4) for k = 2 implies m1 ◦ m1 (x) = (−1)deg x+1 m2 (x, m0 (1)) + m2 (m0 (1), x). There are three different levels on how one can arrive at a coboundary operator δ : C → C, i.e, the map satisfying δδ = 0: (1) The case where we have m0 (1) = 0 (2) The case where m0 (1) is a multiple of the fundamental class P D[L] (3) The cases where we can deform m to m(b1 ,b0 ) for which the corresponding (b ,b ) m0 1 0 satisfies (1) or (2). For the first case, it is obvious that the operator δ : C → C defined by δ = m1 satisfies δδ = 0 and so defines a coboundary map. For the second case, one can check that the two terms of the right hand side of the above equation cancel each other and again δ = m1 satisfies δδ = 0. (See Addendum [Oh93], [ChOh03] and Subsection 3.6.3 for this cancellation argument.) In the case (3), we call the A∞

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8

INTRODUCTION

algebra (C, m) unobstructed (respectively weakly unobstructed) if it can be deformed to the case (1) (respectively to (2)). Now we explain how we deform δ for the unobstructed case. In this case we consider the deformed operator δb1 ,b2 given by (1.7)

δb1 ,b0 (x) =

∞ ∞   k=0 =0

mk++1 (b1 , · · · , b1 , x, b0 , · · · , b0 ).     k



Then it can be shown (see Section 3.8) that δb1 ,b0 ◦ δb1 ,b0 = 0 if (1.8)

∞  k=0

mk (bj , · · · , bj ) = 0,  

for j = 0, 1.

k

We call b a bounding cochain if b satisfies (1.8). One can also deform the A∞ algebra (C, m) by a cohomology class coming from the ambient symplectic manifold (See Section 3.8). We call this deformation a bulk deformation. By amplifying the above deformation argument to the case of bulk deformations, we obtain the following Theorem B. Here  ∞     + λi ni Λ0,nov = ai T e ∈ Λ0,nov  λi > 0 .  i=0

See (Conv.4) at the end of Chapter 1. Theorem B. To each relatively spin Lagrangian submanifold L ⊂ M , we can associate a set Mweak,def (L), and the maps πamb : Mweak,def (L) → H 2 (M ; Λ+ 0,nov ),

PO : Mweak,def (L) → Λ+ 0,nov ,

with the following properties: Hereafter we put (1.9)

Mweak,def (L) ×(πamb ,PO) Mweak,def (L) = {(b1 , b0 ) | πamb (b1 ) = πamb (b0 ), PO(b1 ) = PO(b0 )}.

(B.1) There is a Floer cohomology HF ((L, b1 ), (L, b0 ); Λ0,nov ) parameterized by (b1 , b0 ) ∈ Mweak,def (L) ×(πamb ,PO) Mweak,def (L). (B.2)

There exists a product structure m2 : HF ((L, b2 ), (L, b1 ); Λ0,nov ) ⊗ HF ((L, b1 ), (L, b0 ); Λ0,nov ) → HF ((L, b2 ), (L, b0 ); Λ0,nov )

if (b1 , b0 ), (b2 , b1 ) ∈ Mweak,def (L) ×(πamb ,PO) Mweak,def (L). m2 is associative. In particular, HF ((L, b), (L, b); Λ0,nov ) has a ring structure for b ∈ Mweak,def (L). (B.3) If ψ : (M, L) → (M  , L ) is a symplectic diffeomorphism, then it induces a bijection ψ∗ : Mweak,def (L) → Mweak,def (L ) such that πamb ◦ ψ∗ = ψ∗ ◦ πamb ,

PO ◦ ψ∗ = PO.

The map ψ∗ depends only on the isotopy class of symplectic diffeomorphism ψ. Here ψ∗ = (ψ −1 )∗ on the right hand side. (B.4) In the situation of (B.3), we have an isomorphism ψ∗ : HF ((L, b1 ), (L, b0 ); Λ0,nov ) ∼ = HF ((L , ψ∗ (b1 )), (L , ψ∗ (b0 )); Λ0,nov ).

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1.2. GENERAL THEORY OF LAGRANGIAN FLOER COHOMOLOGY

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And ψ∗ commutes with m2 . (B.5) The isomorphism ψ∗ in (B.4) depends only on the isotopy class of symplectic diffeomorphism ψ : (M, L) → (M  , L ). Moreover (ψ ◦ ψ  )∗ = ψ∗ ◦ ψ∗ . See (4.3.27) for Πamb and Lemma 4.3.28 for PO. Construction of the Floer cohomology and the proof of (B.2) of Theorem B are completed in Subsection 3.8.10. Statement (B.1), especially the statement that Floer cohomology depends only of the pair (b1 , b0 ), follows from Proposition 4.3.29 which is a consequence of Corollary 5.2.40 in Subsection 5.2.4. The proofs of (B.3)-(B.5) are completed in Subsection 4.6.5. Theorem B follows from Theorem A (and its ‘def’ version) by a purely algebraic argument. (In this book we give a proof of Theorem B before the proof of Theorem A is completed, however.) We also consider the following subsets of Mweak,def (L): Mweak (L) = Zero(PO), Mdef (L) = Zero(πamb ), M(L) = Zero(πamb ) ∩ Zero(PO). Then M(L) will be identified with the set of gauge equivalence classes of the bounding cochains b, i.e., solutions of (1.8). (See Section 4.3 Definition 4.3.1 for the definition of gauge equivalence.) Each of these sets will be regarded as a quotient of the zero set of a formal power series as we will describe in Theorem M. The Floer cohomology, denoted by HF ((L, b1 ), (L, b0 ); Λ0,nov ), for b1 , b0 ∈ M(L), is the cohomology group of the boundary operator δb1 ,b0 defined by (1.7). The version HF ((L, b1 ), (L, b0 ); Λ0,nov ) for (b1 , b0 ) ∈ Mweak,def (L) ×(πamb ,PO) Mweak,def (L) is its generalization. More precisely speaking, for the generalization of the definition of Floer cohomology from M(L) to Mweak (L), we need to study the unit of our filtered A∞ algebra (See Sections 3.3,7.3 and Subsection 3.7.3.), and for the generalization from M(L) to Mdef (L), we need to study bulk deformations of the filtered A∞ algebra in Theorem A. (See Section 3.8.) Theorem B implies that Floer cohomology is defined if Mweak,def (L) = ∅. We would like to note that the set Mweak,def (L) could be empty, however. A sufficient condition for the non-emptiness of Mweak,def (L) can be described in terms of the cohomology group of L. Theorem C. There exists a series of positive integers mk < dim L/2 and classes k [o2m (L; weak, def)] ∈ k

Im

(i∗

:

H 2mk (L; Q) → H 2mk (L; Q))

H 2mk (M ; Q)

k k = 1, 2, · · · , such that, if the obstruction classes [o2m (L; weak, def)] are all zero, k then Mweak,def (L) is nonempty. The number 2 − 2mk is a sum of the Maslov indices of a finite collection of the homotopy classes in π2 (M, L) realized by pseudoholomorphic discs (with respect to a given almost complex structure on M ).

Theorem C is a consequence of Theorems 3.8.41, 3.8.50 and 3.8.96, which is proved in Subsections 3.8.5 and 3.8.9. The proof of Theorem C is completed in Subsection 3.8.10. k Remark 1.10. The classes [o2m (L; weak, def)] are not invariants of L itself k because they will depend on various choices in general, let alone the given almost complex structure. A more functorial (or invariant) way of describing the set Mweak (L) is Theorem M.

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10

INTRODUCTION

We next describe how we need to modify (1.2.3) in the general situation. Hereafter we denote the Poincar´e duality by P D. Theorem D. For each (b1 , b0 ) ∈ Mweak,def (L) ×(πamb ,PO) Mweak,def (L), there exists a spectral sequence with the following properties:

(D.1) E2p,q = k H k (L; Q) ⊗ (T qλ Λ0,nov /T (q+1)λ Λ0,nov )(p−k) . Here λ > 0. (D.2) There exists a filtration F∗ HF ((L, b1 ), (L, b0 ); Λ0,nov ) on the Floer cohomology HF ((L, b1 ), (L, b0 ); Λ0,nov ) such that

(D.3)

q p p,q ∼ F HF ((L, b1 ), (L, b0 ); Λ0,nov ) . E∞ = q+1 F HF p ((L, b1 ), (L, b0 ); Λ0,nov )

Consider the subgroup Kr ⊂ Er = p,q Erp,q defined by K2p,q = P D(Ker(Hn−k (L; Q) → Hn−k (M ; Q))) k

⊗ (T qλ Λ0,nov /T (q+1)λ Λ0,nov )(p−k) , Kr+1 =

Kr ∩ Ker δr ⊂ Er+1 . Kr ∩ Im δr

Then we have Im δr ⊆ Kr for every r, under the additional assumption that b0 = b1 . In particular, the spectral sequence collapses at the E2 level, if the inclusioninduced homomorphism i∗ : H∗ (L; Q) → H∗ (M ; Q) is injective. (D.4) The spectral sequence is compatible with the ring structure in (B.2). In other words we have the following. Each of Er has a ring structure m2 which satisfies: δr (m2 (x, y)) = −m2 (δr (x), y) + (−1)deg x m2 (x, δr (y)). The filtration F is compatible with the ring structure in (B.2). The isomorphisms in (D.1) and (D.2) are ring isomorphisms. The proof of Theorem D is completed in Subsection 6.4.5. For various applications in practice, it is important to know when Floer cohomology does not vanish, which will then give rise to an existence theorem of a suitable form of pseudo-holomorphic curves. For this purpose, one may often use either the point class P D[pt] ∈ H n (L; Q) or the fundamental class P D[L] ∈ H 0 (L; Q), that we know are not zero at least in the classical cohomology: Theorem E. In the situation of Theorem D we assume b0 = b1 . Then, there exists a cohomology class P D[L] ∈ H n (L; Λ0,nov ) with P D[L] ≡ P D[L] mod Λ+ 0,nov which has the following properties. / Im(δr ), where δr (E.1) For each r ∈ Z≥0 we have δr (P D[L] ) = 0 and P D[pt] ∈ is the differential of the spectral sequence in Theorem D. (E.2) If the Maslov index of all the pseudo-holomorphic discs bounding L are non-positive, then δr (P D[pt]) = 0 and P D[L] ∈ / Im(δr ). Theorem E is Theorem 6.1.9 which is proved in Chapter 6 Subsection 6.4.3. Theorem E holds over Λnov coefficient. (Λnov is defined in (Conv.3) at the end of Chapter 1.)

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1.2. GENERAL THEORY OF LAGRANGIAN FLOER COHOMOLOGY

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Remark 1.11. The properties of P D[L] in Theorem E is closely related to the fact that the filtered A∞ algebra in Theorem A has a unit P D[L]. (See Section 3.2 Condition 3.2.18 and Definition 3.2.20, for the definition of unit.) The statement on P D[pt] is related to the fact that the map H0 (L; Q) → H0 (M ; Q) is injective, and also to the operator p discussed in Section 3.8. We next discuss Floer cohomology of a pair of Lagrangian submanifolds. Let (L1 , L0 ) be a relatively spin pair of Lagrangian submanifolds. We assume that they have clean intersection. Namely we assume that the dimension of Tx L1 ∩ Tx L0 is a locally constant function of x ∈ L1  ∩ L0 . (This is the Bott-Morse theory ([Bot59]) version of the Floer theory.) Let h Rh = L1 ∩ L0 be the decomposition to the connected components. In Subsection 3.7.5 and Section 8.8, we will define a local system Θ− Rh on Rh defined by a homomorphism π1 (Rh ) → {±1} = Aut(Z) ⊂ Aut(Q). We then put H(Rh ; Θ− C(L1 , L0 ; Λ0,nov ) = Rh )[μL (Rh )] ⊗ Λ0,nov . h

Here μL (Rh ) is the Bott-Morse version of Maslov index which is defined in Proposition 3.7.59, (3.7.60.1). (Note μL ([h, w]) = μL (Rh ), where the left hand side is defined in Proposition 3.7.59.) Actually we need to fix an extra data, (denoted by w there) to define the Bott-Morse version of the Maslov index. [·] denotes the degree shift. (Namely C[k]d = C k+d .) See Section 8.8 for the reason why this degree shift occurs. Theorem F. C(L1 , L0 ; Λ0,nov ) has the structure of a unital filtered A∞ bimodule over the pair ((H(L1 ; Λ0,nov ), m∗ ), (H(L0 ; Λ0,nov ), m∗ )). This is well-defined up to a unital filtered A∞ bimodule isomorphism. The definition of filtered A∞ bimodules is given in Section 3.7. An isomorphism of A∞ bimodules is a filtered A∞ bimodule homomorphism that has an inverse. The filtered A∞ bimodule homomorphism is defined in Subsection 3.7.2. The proof of Theorem F is completed in Section 5.4 (Corollary 5.4.21). Using Theorem F, we can define Floer cohomology of a pair of Lagrangian submanifolds and obtain the following Theorem G. Theorem G. Let (L1 , L0 ) be a relatively spin pair of Lagrangian submanifolds of M and (b1 , b0 ) ∈ Mweak,def (L1 ) ×(πamb ,PO) Mweak,def (L0 ), (see (1.9) for the definition of right hand side). Then, we can associate a Floer cohomology HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) with properties (G.1) - (G.4). Let HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) ⊗Λ0,nov Λnov = HF ((L1 , b1 ), (L0 , b0 ); Λnov ). (G.1) If L0 = L1 = L, then HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) coincides with the one in Theorem B. (G.2)

If L1 and L0 have clean intersection, then we have  rankQ H k (Rh ; Θ− rankΛnov HF ((L1 , b1 ), (L0 , b0 ); Λnov ) ≤ Rh ), h,k

where each Rh is a connected component of L0 ∩ L1 and Θ− Rh is a local system on it associated with certain representation π1 (Rh ) → Aut(R) = {±1}.

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12

INTRODUCTION

(G.3) If ψ : M → M  is a symplectic diffeomorphism with ψ(Li ) = Li , (i = 0, 1), then we have a canonical isomorphism ψ∗ : HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) ∼ = HF ((L1 , ψ∗ b1 ), (L0 , ψ∗ b0 ); Λ0,nov ) where ψ∗ : Mweak,def (Li ) → Mweak,def (Li ) is as in Theorem B. The isomorphism ψ∗ depends only on the isotopy class of symplectic diffeomorphism ψ with ψ(Li ) = Li . We also have (ψ ◦ ψ  )∗ = ψ∗ ◦ ψ∗ . (G.4) If ψis : M → M (i = 0, 1, s ∈ [0, 1]) are Hamiltonian isotopies with 0 ψi = identity, ψi1 (Li ) = Li , then it induces an isomorphism (ψ1s , ψ0s )∗ : HF ((L1 , b1 ), (L0 , b0 ); Λnov ) ∼ HF ((L , ψ 1 b1 ), (L , ψ 1 b0 ); Λnov ). = 1

1∗

0

0∗

(ψ1s , ψ0s )∗

depends only on the isotopy class of the Hamiltonian The isomorphism isotopies ψis : M → M (i = 0, 1, s ∈ [0, 1]) with ψi0 = identity, ψi1 (Li ) = Li . The isomorphism (ψ1s , ψ0s )∗ is functorial with respect to composition of the Hamiltonian isotopies. (G.5)

HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) is a bimodule over the ring pair (HF ((L1 , b1 ), (L1 , b1 ); Λ0,nov ), HF ((L0 , b0 ), (L0 , b0 ); Λ0,nov )) .

The isomorphisms (G.3), (G.4) are bimodule isomorphisms. Remark 1.12. (G.3) above asserts that the Floer cohomology remains to be isomorphic when we move L1 , L0 by the same symplectic diffeomorphism. (G.4) above asserts that Floer cohomology does not change even when we move L1 , L0 independently by different Hamiltonian isotopies. The latter invariance property holds only over the Λnov coefficient but not over the Λ0,nov coefficient. This point will be further explored in Theorem J. Construction of the Floer cohomology and the proofs of (G.1), (G.5) of Theorem G are completed in Subsection 3.8.10. (G.3) is a consequence of Theorems 4.1.4 and 5.3.64, which are proved in Subsections 5.3.1 and 5.3.6, respectively. (G.4) is a consequence of Theorems 4.1.5 and 5.3.65, which are proved in Subsections 5.3.2-5.3.6. The proofs of (G.3), (G.4) are completed in Subsection 5.3.7. (G.2) is proved in Subsection 5.4.6. It also follows from Theorem 6.1.4 which is proved in Chapter 6. For the case of non-clean intersection, we need some more arguments to prove Theorem G. It is given in Subsection 6.5.4 (Proposition 6.5.38, Definition 6.5.39). Theorem G is also a consequence of the functoriality of the filtered A∞ bimodule given in Theorem F. (See Theorems 5.3.1 and 5.3.14.) (But, in this book, we give the proof of Theorem G before we complete the proof of Theorem F.) We remark that Theorem F implies that we can construct various kinds of bar complex for an arbitrary relatively spin pair (L1 , L0 ) whose cohomology is independent of the choices. We do not need to assume M(L1 ) = ∅, M(L0 ) = ∅ to obtain such a bar complex. (See Definition 3.7.5.) We use elements of M(Lj ) to cut down the bar complex to a more reasonable and applicable size. We also have a spectral sequence for a pair of Lagrangian submanifolds (L1 , L0 ) of clean intersection (see Theorem 6.1.4).

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1.3. APPLICATIONS TO SYMPLECTIC GEOMETRY

13

We can amplify this to a product structure involving three or more Lagrangian submanifolds. Namely we obtain a filtered A∞ category. Existence of such a category was observed by the first named author around 1993 inspired by Donaldson’s lecture at University of Warwick 1992. The details of its construction are provided in [Fuk02II], where the contents of present book are assumed. 1.3. Applications to symplectic geometry In this section, we describe applications of the Floer cohomology to symplectic topology of Lagrangian submanifolds. We first state the following consequence of Theorems B,C,D and G combined. Theorem H. Let L be a relatively spin Lagrangian submanifold and assume that the natural map H∗ (L; Q) → H∗ (M ; Q) is injective. Then, for any Hamiltonian diffeomorphism φ : M → M such that L  φ(L), we have  #(L ∩ φ(L)) ≥ rank Hk (L; Q). k

The proof of Theorem H is given in Subsection 6.5.1. We remark that the diagonal Δ ⊂ M × M satisfies the assumption in Theorem H and so the version of Arnold’s conjecture established in [FuOn99II, LiuTi98, Rua99] for Hamiltonian diffeomorphisms (over Q-coefficients) follows from Theorem H. The following is a more precise version of Theorem H. Theorem I. LetL be relatively spin and  assume that Mweak,def (L) is nonempty. Denote A = ∗ rank H∗ (L; Q), B = ∗ rank Ker(H∗ (L; Q) → H∗ (M ; Q)). Then we have #(L ∩ φ(L)) ≥ A − 2B for any Hamiltonian diffeomorphism φ : M → M such that L  φ(L). The proof of Theorem I is completed in Subsection 6.5.2. As we mentioned before, the Floer cohomology over Λ0,nov is not independent of the Hamiltonian isotopies of the pair of Lagrangian submanifolds L1 , L0 , while Floer cohomology over Λnov is so. We study this phenomenon more closely and relate the ‘torsion part’ of the Floer cohomology to the Hofer distance between Hamiltonian diffeomorphisms. (See [HoZe94] for the definition of Hofer distance.) We first remark that Floer cohomology HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) has the decomposition (1.13)

HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) ∼ = (Λ0,nov )a ⊕

b i=1

Λ0,nov T λi Λ0,nov

in general (Theorem 6.1.20). We call a ∈ Z≥0 the Betti number and λi ∈ (0, ∞) torsion exponents of the Floer cohomology. (The integer b is finite if L1 has clean intersection with L0 but may be infinite in general. (Example 6.5.40).) Theorem J. Let (L1 , L0 ) be a relatively spin pair of Lagrangian submanifolds of M and (b1 , b0 ) ∈ Mweak,def (L1 ) ×(πamb ,PO) Mweak,def (L0 ). Let φ : M → M be a Hamiltonian diffeomorphism and μ its Hofer norm μ = φ. Assume that φ(L1 ) is transversal to L0 and denote b(μ) = #{i | λi ≥ μ}

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14

INTRODUCTION

where λi are the torsion exponents in (1.13). Then we have #(φ(L1 ) ∩ L0 ) ≥ a + 2b(μ). The proof of Theorem J is completed in Subsection 6.5.3. Theorem J is closely related to Chekanov’s work [Chek96,98] and we use Chekanov’s idea in the proof of Theorem J. Actually a slight improvement of Theorem J (Theorem 6.5.47 in Subsection 6.5.5) implies Chekanov’s theorem. In Subsection 6.5.3 we prove a stronger result than Theorem J. Namely we prove that the torsion exponents λi are Lipschitz function with Lipschitz constant 1 with respect to Hofer’s distance, when we move Li in the Hamilton isotopy class (Theorem 6.1.25). Next application concerns the Maslov class μL ∈ H 1 (L; Z) of a Lagrangian embedding L ⊂ Cn . The general folklore conjecture says that this Maslov class is non-trivial for any compact Lagrangian embedding in Cn . There have been several partial results towards this conjecture (see [Vit90, Pol91I, Oh96I]). Here we add another: Theorem K. Let L be a compact spin Lagrangian submanifold of Cn that satisfies H 2 (L; Q) = 0. Then its Maslov class μL ∈ H 1 (L; Z) is nonzero. Theorem K follows from Corollary 6.1.16 which is proved in Subsection 6.4.4. Theorem K can be generalized to the case of more general (M, L). See Theorem L and Corollary 6.1.16. There is another application (Theorem 6.4.35) using the compatibility of the spectral sequence with the multiplicative structure (Theorem D (D.4)). Theorem L. Let L ⊂ (M, ω) be a relatively spin Lagrangian submanifold such that the Maslov index homomorphism Iμ : π2 (M, L) → Z is trivial. We assume H 2 (L; Q) = 0. Then for any Hamiltonian diffeomorphism φ : M → M , we have L ∩ φ(L) = ∅. Moreover if L is transversal to φ(L), there exists p ∈ L ∩ φ(L) whose Maslov index is 0 . Theorem L is proved in Subsection 6.4.3. A special class of pairs (M, L) for which the Maslov index homomorphism Iμ : π2 (M, L) → Z is trivial consists of those for which (M, ω, J) is a Calabi-Yau manifold and L ⊂ M is a Lagrangian submanifold whose Maslov class is trivial: In this case, the Maslov index μL (β) depends only on ∂β ∈ π1 (L) and so induces a well-defined cohomology class in H 1 (L, Z), which is called the Maslov class of L. (See [Mor81, Daz81, HaLa82] for more explanations.) Theorem L is then used by Thomas and Yau [ThYa02] in their proof of a uniqueness theorem of special Lagrangian homology 3-sphere (i.e., L with H 1 (L, Z) = 0) in its Hamiltonian isotopy class in Calabi-Yau 3-folds. As we mentioned already, Floer cohomology HF ((L, b), (L, b); Λ0,nov ) is not necessarily isomorphic to the cohomology of L. The difference is determined by the differentials δr of the spectral sequence in Theorem D. The differentials are in turn determined by the fundamental chains of appropriate moduli spaces of pseudo-holomorphic discs. Therefore it is in general very hard to analyze this spectral sequence in the actual computation of HF ((L, b), (L, b); Λ0,nov ), because it requires a careful study of the counting problem of pseudo-holomorphic discs. This counting problem is even more delicate and non-trivial than the counting problem of pseudo-holomorphic spheres in the usual Gromov-Witten theory. On the other hand, Theorem A implies that there are various product structures on

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1.3. APPLICATIONS TO SYMPLECTIC GEOMETRY

15

the ordinary cohomology and Floer cohomology of a Lagrangian submanifold which may be exploited in calculations. However computing these products in examples is also a non-trivial problem in general. Here is a list of some calculations. (1.14.0) If π2 (M, L) = π1 (L) = 0, then Theorem X implies that the filtered A∞ algebra in Theorem A is nothing but the real homotopy type of L. (1.14.1) There are some calculations carried out by C.-H. Cho and the second named author for the case of Lagrangian torus orbits of toric Fano manifolds [Cho04I, ChOh03]. Their works are partially motivated by the works of Hori and Vafa [HoVa00, Hor01]. In particular, a variant of the function PO : Mweak,def (L) → Λ+ 0,nov in Theorem B is calculated in [ChOh03], which corresponds to the Landau-Ginzburg superpotential in physics [HoVa00]. Roughly speaking, they proved that the torus orbits that have non-trivial Floer cohomology correspond to the critical points of this superpotential via the mirror map. Cho and the second named author call those orbits balanced torus fibers in [ChOh03]. It is also proved in [ChOh03] that for the balanced torus fibers whose Floer cohomology HF (L, L) is isomorphic to the classical cohomology group H ∗ (T n ; Λnov ). In [Cho04II], Cho proved that the natural ring structure on HF (L, L) arising from the A∞ structure becomes isomorphic to the Clifford algebra associated to the quadratic form of the Hessian of the Landau-Ginzburg superpotential after the LandauGinzburg mirror map is applied. This was earlier predicted by Hori [Hor01] and Kapustin-Li [KapLi04]. In particular, this computation provides an example in which the Floer cohomology is isomorphic to the classical one but its ring structure is deformed from the classical cup product. Recall that the cohomology ring of the torus is isomorphic to the exterior algebra generated by the degree one cohomology thereof. (1.14.2) The result of [ChOh03] is used and generalized in [FOOO08I] and [FOOO08II], where the present authors apply many of the results of this book to study Floer cohomology of Lagrangian submanifolds which are T n orbits in a compact toric manifold (which is not necessary Fano). (1.14.3) When L ⊆ Cn+1 is a Lagrangian submanifold homotopy equivalent to S 1 × S n , the spectral sequence is studied in detail in the paper [FOOO09I]. (1.14.4) Some calculation is carried out in [FOOO09II] for the case of L obtained by the Lagrangian surgery from flat Lagrangian sub-tori in T 2n . (See also [Fuk02III].) Theta functions appear in the calculation. We obtain various other examples in [FOOO09II]. (1.14.5) Homological mirror symmetry conjecture, if proved, will become a powerful tool for the computation of Floer cohomology and the filtered A∞ algebras of Lagrangian submanifolds. For example, P. Seidel’s proof of homological mirror symmetry of quartic surface [Sei03II] should give rise to some consequences on the calculation of these structures. (1.14.6) There are numerous results on the counting problem of holomorphic discs attached to a given Lagrangian submanifold in thephysics literature, especially when the ambient symplectic manifold is a noncompact Calabi-Yau manifold in relation to the local mirror principle (see [AAMV05, DiFl05, LLLZ04] and the references therein.) It is, however, not yet clear how one can use these studies in actual calculation of the structures we have constructed in this book.

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16

INTRODUCTION

1.4. Relation to mirror symmetry So far we have explained various applications of the Floer theory to the problems in symplectic topology. Another direction of applications are towards the proof of the celebrated homological mirror symmetry conjecture proposed by Kontsevich [Kon93,95I] around 1994, in which the A∞ category introduced by the first named author [Fuk93] plays a key role in its formulation. (Vafa [Vaf99] made an interpretation of this proposal into the physics language around 2000.) There remain many more things to be carried out towards this goal. We would like to explain the relationship of our work with the homological mirror symmetry briefly here (See also [Fuk02I,02III,03I,03II] for other explanations). In this respect, we recall that the set M(L) = (Zero(πamb ) ∩ Zero(PO))/ ∼ which was introduced in the paragraph right after Theorem B is consisting of the gauge equivalence classes of bounding cochains. (Here and hereafter in this section, L is assumed to be a relatively spin Lagrangian submanifold of M .) Because this set plays a central role in the application to homological mirror symmetry, we first state a theorem in which the set is described as a formal scheme. Let C0 , C1 be two finitely generated free Λ0,nov modules. We choose bases {v1 , · · · , vm } of C0 and {w1 , · · · , w } of C1 respectively. A formal map P : C0 → C1 is defined to be  a linear combination P = j Pj wj where each Pj a formal power series  (1.15)

m 

Pj

 si vi

=

i=1

∞ 

···

k1 =0

∞ 

ajk1 ··· ,km sk11 · · · skmm

km =0

with the coefficients ajk1 ··· ,km ∈ Λ0,nov . We note that although the formula (1.15) will not make sense in general if we substitute elements of Λ0,nov in sj , it does if we substitute elements from Λ+ 0,nov in sj . This is because the right hand side of (1.15) will converge in the Λ+ -adic topology for the latter case. Therefore we may put 0,nov  (P

−1

(0))(Λ+ 0,nov )

:=



sj vj ∈

Λ+ 0,nov C0

 m      si vi = 0 P  i=1

and regard this set as the set of Λ+ 0,nov -valued points (or rigid points) of an affine formal scheme Spf(R(P; Λ0,nov )) associated to the (commutative non-Noetherian) complete valuational ring R(P; Λ0,nov ) =

Λ0,nov [[s1 , · · · , sm ]] . (P1 , · · · , P )

Here (P1 , · · · , P ) is the ideal generated by the elements P1 , · · · , P of the formal power series ring Λ0,nov [[s1 , · · · , sm ]]. Namely we have the identification (P−1 (0))(Λ+ 0,nov ) = Hom(R(P; Λ0,nov ), Λ0,nov ) where the right hand side is the set of continuous Λ0,nov algebra homomorphisms. (See Proposition 5.4.15 in Subsection 5.4.1.)

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1.4. RELATION TO MIRROR SYMMETRY

17

−1 For each element b ∈ H 2 (M ; Λ+ 0,nov ), we consider the inverse image πamb (b) of + the map πamb : Mweak,def (L) → H 2 (M ; Λ0,nov ) and define the sets Mbweak (L) and Mb (L) by −1 Mbweak (L) = πamb (b),

−1 Mb (L) = πamb (b) ∩ Zero(PO).

Theorem M. There exist a map + 2 PbL,weak : H 1 (L; Λ+ 0,nov ) → H (L; Λ0,nov )

and an equivalence relation ∼ on ((PbL,weak )−1 (0))(Λ+ 0,nov ) so that we have Mbweak (L) = ((PbL,weak )−1 (0)(Λ+ 0,nov ))/ ∼ . The continuous isomorphism class of R(PbL,weak ; Λ0,nov ) as the Λ0,nov algebra depends only on the pair (M, L) up to a symplectic diffeomorphism. A similar + 2 statement holds for Mb (L) if we consider PbL : H 1 (L; Λ+ 0,nov ) → H (L; Λ0,nov ) instead. Theorem M is a consequence of Theorem 5.4.12 and Proposition 5.4.15 and is proved in Subsection 5.4.6. Now we consider a relatively spin pair (L0 , L1 ) of Lagrangian submanifolds. b For a given b ∈ H 2 (M ; Λ+ 0,nov ) and bi ∈ Mweak (Li ), we denote by ϕbi : R(PbLi ; Λ0,nov ) −→ Λ0,nov a homomorphism corresponding to a representative of bi via the identification given in Theorem M. For the case of the pair (L1 , L0 ) that is relatively spin, we obtain the following: Theorem N. There exists a complex (D(L1 , L0 ; Λ0,nov ), d) where the Λ0,nov module D(L1 , L0 ; Λ0,nov ) is a finitely generated complete bimodule over the ring pair (R(PbL1 ; Λ0,nov ), R(PbL0 ; Λ0,nov )) and the coboundary map d : D(L1 , L0 ; Λ0,nov ) −→ D(L1 , L0 ; Λ0,nov ) is continuous with respect to the topology on D(L1 , L0 ; Λ0,nov ) induced by the filtration. The complex is independent of the various choices made up to isomorphism and has the following properties: (N.1) Consider bi ∈ Mb (Li ) (i = 0, 1) a pair of bounding cochains and let ϕbi : b R(PLi ; Λ0,nov ) −→ Λ0,nov be their representatives, respectively. Then cohomology of the complex of the Λ0,nov bimodule (Λ0,nov ϕb1 ⊗R(PbL

1

;Λ0,nov )

D(L1 , L0 ; Λ0,nov )

R(Pb L ;Λ0,nov ) 0

⊗ϕb0 Λ0,nov , 1 ⊗ d ⊗ 1)

is isomorphic to the Floer cohomology HF ((L1 , b1 ), (L0 , b0 ); Λ0,nov ) as a Λ0,nov bimodule. (Here bi = (b, bi ), (i = 0, 1).) (N.2) If φi : (M, Li ) −→ (M, Li ) (i = 0, 1) are Hamiltonian isotopies, we have a chain homotopy equivalence (φ1 , φ0 )∗ : (D(L1 , L0 ; Λ0,nov ), d) ⊗Λ0,nov Λnov −→ (D(L1 , L0 ; Λ0,nov ), d) ⊗Λ0,nov Λnov

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18

INTRODUCTION

over the ring isomorphisms φi∗ : R(PbLi ; Λ0,nov ) → R(PbL ; Λ0,nov ). Under the i isomorphism given in (N.2), (φ1 , φ0 )∗ induces the isomorphism of Floer cohomology provided in (G.5). Statement (N.1) above can be reinterpreted as follows: By the general theory of formal schemes, the complex (D(L1 , L0 ; Λ0,nov ), d) defines an object of the derived category of coherent sheaves on the affine formal scheme Spf(R(PbL1 ; Λ0,nov )) × Spf(R(PbL0 ; Λ0,nov )). Then the fiber at the (rigid) point (b1 , b0 ) is the Floer cohomology. Theorem N is proved in Subsection 5.4.6. Remark 1.16. In Theorem N we considered Mb (L). We can use Mbweak (L) as well to obtain similar conclusions. Namely using the morphism PObL : Mbweak (L) → Λ+ 0,nov b (which is a continuous ring homomorphism Λ+ 0,nov → R(PL,weak ; Λ0,nov )), we can define the fiber product

ˆ Λ+ R(PbL1 ,weak ; Λ0,nov )⊗

0,nov

R(PbL0 ,weak ; Λ0,nov ).

Then D(L1 , L0 ; Λ0,nov ) can be extended to a chain complex of modules over ˆ Λ+ R(PbL1 ,weak ; Λ0,nov )⊗

0,nov

R(PbL0 ,weak ; Λ0,nov ).

Its fibers at rigid points define a family of Floer cohomology parameterized by Mbweak (L1 ) ×Λ+

0,nov

Mbweak (L0 ),

where the fiber product is taken with respect to PObLi . The rest of this section discusses the relationship of the result obtained in this book with the mirror symmetry. This discussion will not be used for the rest of the book except for the purpose of making similar remarks in the book from time to time. For the application to mirror symmetry, the most relevant class of Lagrangian submanifolds is the one consisting of those L ⊂ M where M is a Calabi-Yau manifold and L has vanishing Maslov class. Therefore we will restrict ourselves to this class of Lagrangian submanifolds for the discussion below. (However the results of this book apply to the Fano case as well which also plays an important role in mirror symmetry.) We first remark that we do not need to use the formal parameter e, which encodes the Maslov index of the discs, in Λ0,nov for such Lagrangian submanifolds. (See Subsection 5.4.6.) Therefore we consider the smaller rings Λ0 , Λ, Λ+ instead of Λ0,nov ,Λnov , Λ+ 0,nov , respectively. (See (Conv.5) at the end of Chapter 1.) We note that the ring Λ is a field in this case. Then we can define the structure of a filtered A∞ algebra on H(L; Λ0 ). And we can define a formal scheme as before for this smaller ring in the same way as for Λ+ 0,nov . To make this change explicit, we denote the corresponding formal scheme by M(L; Λ+ ) to highlight the dependence on Λ+ . The following version of Theorems G and M is useful to describe the relationship of the Floer cohomology studied in this book with mirror symmetry.

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1.4. RELATION TO MIRROR SYMMETRY

19

Corollary O. Let (M, ω) satisfy c1 (M ) = 0. Let L be a relatively spin Lagrangian submanifold of M with vanishing Maslov class and b ∈ H 2 (M ; Λ+ ). Then there exists a map PbL : H 1 (L; Λ+ ) → H 2 (L; Λ+ ) such that the set Mb (L; Λ+ ) = (PbL )−1 (0)(Λ+ ) = {b ∈ H 1 (L; Λ+ ) | PbL (b) = 0} has the following properties: Let (L0 , L1 ) be a relatively spin pair of such Lagrangian submanifolds of M . (O.1) For each bi ∈ Mb (Li ; Λ+ ), we put bi = (b, bi ). Then we have a Floer cohomology HF (((L1 , b1 ), (L0 , b0 ); Λ0 ). (O.2) For the case (L1 , b1 ) = (L0 , b0 ) = (L, b), the Floer cohomology group HF ((L, b), ((L, b); Λ0 ) has a ring structure. (O.3) The Floer cohomology HF ((L1 , b1 ), (L0 , b0 ); Λ0 ) has a structure of a left HF (((L1 , b1 ), (L1 , b1 ); Λ0 ) and right HF (((L0 , b0 ), (L0 , b0 ); Λ0 ) filtered bimodule. (O.4) If m[ω] ∈ H 2 (M, L; Z) for some integer m, then we can reduce the coefficient ring Λ0 to the ring Q[[T 1/m ]] ⊂ Λ0 of formal Puiseux series in the above statements. Remark 1.17. (1) Contrary to the case of Theorem M, for the case of vanishing Maslov class, we do not need to divide the zero set of PbL by an equivalence relation to obtain Mb (L; Λ+ ). See Subsection 5.4.6 Lemma 5.4.71. (2) In fact we can extend the statements in this corollary to those in the context of the filtered A∞ category by considering arbitrary finite chains (L0 , L1 , · · · , Lk ) of Lagrangian submanifolds. In this book, we restrict ourselves to those statements up to the definitions of objects (one Lagrangian) and morphisms (two Lagrangians) referring to [Fuk02II] for the discussion in this general context of A∞ category. Corollary O is a minor modification of Theorems G and M. Only the point mentioned in Remark 1.17 (1) and the statement (O.4) do not immediately come from Theorems G, M. We will discuss these points and complete the proof of Corollary O in Subsection 5.4.6. Now we rewrite Corollary O with the language of formal scheme. Consider the situation of Corollary O. We put m = dim H 1 (L; Q),

 = dim H 2 (L; Q)

and fix a basis of H 1 (L, Q). Denote by s1 , · · · , sm the associated coordinates on H 1 (L, Q) and Λ0 [[s1 , · · · , sm ]] . R(PbL ; Λ0 ) = ((PbL )1 , · · · , (PbL ) ) Here PbL = ((PbL )1 , · · · , (PbL ) ) is regarded as a formal power series of s1 , · · · , sm . Corollary P. Let L, M be as in Corollary O. The ring R(PbL ; Λ0 ) is an invariant of L. Moreover, to each relatively spin pair (L0 , L1 ), we can associate a cochain complex (D(L1 , L0 ; Λ0 ), d) of finitely generated continuous complete bimodules over the ring pair (R(PbL1 ; Λ0 ), R(PbL0 ; Λ0 )) such that if bi ∈ Mb (Li ; Λ+ ) and ϕbi : R(PbLi ; Λ0 ) → Λ0 is its representative, then cohomology of the complex (Λ0

ϕb1

⊗R(PbL

1

;Λ0 )

D(L1 , L0 ; Λ0 )

R(Pb L ;Λ0 )

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0

⊗ϕb0 Λ0 , 1 ⊗ d ⊗ 1)

20

INTRODUCTION

is isomorphic to Floer cohomology HF ((L1 , b1 ), (L0 , b0 ); Λ0 ), where bi = (b, bi ). Up to chain homotopy equivalence, the bimodule (D(L1 , L0 ; Λ0 ), d) is independent of various choices made in its construction. (O.4) still applies. Corollary P is a minor modification of Theorem N and is proved in Subsection 5.4.6. Now we describe the conjectural mirror picture of the structures in Corollaries O and P. To describe this mirror in the algebraic (rather than transcendental) context, we need some restrictions on M and L, in addition to M being Calabi-Yau and L having the vanishing Maslov class. Condition 1.18. We assume that L is rational. Namely (mω) ∩ β ∈ Z for all β ∈ H2 (M, L; Z) for a nonnegative integer m. We remark that it automatically implies m[ω] ∈ H 2 (M, L; Z). We consider the ring of the Peiseux series Q[[T 1/m ]] ⊂ Λ0 . (See [Fuk03I] more about this assumption.) Conjecture Q. Let (M, ω) be a Calabi-Yau manifold such that for some integer m, m[ω] ∈ H 2 (M ; Z). Consider an element b ∈ H 2 (M ; Q[[T 1/m ]]) ⊂ H 2 (M ; Λ0 ) and suppose the formal power series b has its radius of convergence strictly positive at T = 0. Then we can associate a holomorphic family (1.19)

π : X(M,ω,b) −→ D2 ( )

over a disc D2 ( ) ⊂ C for some sufficiently small > 0, so that its fiber π −1 (q) at each q = 0 is a Calabi-Yau manifold. The fiber π −1 (0) at q = 0 is singular in a way that the family defines a maximally degenerating family as q → 0. See [LTY05] for the definition of a maximally degenerating family. We give a brief explanation how Conjecture Q follows from the various facts that are widely believed (but not proved) among the researchers on mirror symmetry. (The description below, except how b enters, is not new. See [Fuk03II], for example.) ∞ the way k/m Let b = , where bk ∈ H 2 (M ; Q). We choose closed 2 forms k=1 bk T representing the classes bk and denote them by the same symbols. After these choices made, assume that the series has positive radius of convergence. For τ ∈ √ h = {τ ∈ C | Im τ > 0}, we put q = e2π −1τ /m . Heuristically, one may regard q = T 1/m . However T is a formal parameter while q ∈ D1 ( ) is a genuine complex number. For each such τ , we define a complex valued closed two form by   2π  (1.20) ωτ (b) = √ τω + bk q k . −1 By the convergence assumption on b, it follows from this expression that ωτ (b) varies holomorphically over τ ∈ h. It is also easy to see that Re(ωτ (b)) is nondegenerate and so defines a symplectic form on M , as long as Im(τ ) is sufficiently large. We regard the family (1.20) as a holomorphic family of complexified symplectic structures on M . The imaginary part of ωτ (b) is called a B-field. We remark that m[ω] ∈ H 2 (M ; Z) implies √ (1.21) τ − τ  ∈ mZ =⇒ ωτ (b) − ωτ  (b) ∈ 2π −1H 2 (M ; Z). Here comes a fact that is widely believed to be true:

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1.4. RELATION TO MIRROR SYMMETRY

21

To each (M, ωτ (b)) that is a Calabi-Yau manifold with a B field, we can associate its mirror Xωτ (b) that is also a Calabi-Yau manifold. Moreover if the dif√ ference between two complexified symplectic structures is in 2π −1H 2 (M ; Z), then they correspond to the same mirror manifold. Therefore it is expected from (1.21) that we have defined the family parameterized by q alluded in (1.19). We remark that for β ∈ H2 (M ; Z) we have the identity   √  (1.22) exp(−ωτ (b)[β]) = q m(ω[β]) exp 2π −1 (bk [β])q k . We can see from (1.20) and (1.22) that or the cohomology class b deforms the symplectic form ω as q → 0 in the way √ −1 ωτ (b) → ω as Im τ → ∞. 2π τ This is the reason why we use ‘def’ in the notation Mdef (L). Now we want to formulate a conjecture which describes the conjectural mirror object of the pair (L, b), where L is a spin Lagrangian submanifold that satisfies Condition 1.18 and μL = 0, and b ∈ Mb (L; Λ+ ). (We can include a flat complex bundle on L with a slight modification of our construction. We omit its discussion here. One may look at [Fuk02II] for the detailed explanation on such a modification.) ˆ (M,ω,b) of X(M,ω,b) at π −1 (0). (See [GrDi60] We consider a formal completion X ˆ i be an object of the derived category of coherent I Section 10 for its meaning.) Let F ˆ1, F ˆ 2 ) of Hom ˆ sheaves on X(M,ω,b) for i = 1, 2. The right derived functor RHom(F is defined as a C[[q]] module. (Since we take formal completion, q is now a formal parameter.) Remark 1.23. (1) There does not seem to exist a canonical choice for the ˆ· , F ˆ · ) ⊗C[[q]] C[[q]][q −1 ] in singular fiber π −1 (0). So we need to consider RHom(F 1 2 reality. (See [Fuk03I] Section 4 for a more precise formulation.) In the point of view in [Ray74], forgetting the information of the fiber at 0 means that we regard ˆ i as an object of derived category of associated rigid analytic space when we take F RHom. This point may be related to Remark 1.12. ˆ (M,ω,b) to exist, we may not even need to (2) In order for the formal scheme X assume, in Conjecture Q, that b converges. Conjecture R. To each pair (L, b) with L a spin Lagrangian submanifold satisfying Condition 1.18 and μL = 0, and b ∈ Mb (L; Λ+ ), we can associate to ˆ  (M,ω,b) so that F(L, b), an object of the derived category of coherent sheaves on X the following holds: (R.1) The formal neighborhood of b in Mb (L; Λ+ ) is isomorphic to a formal ˆ neighborhood of F(L, b) in its moduli space. (R.2) We have a canonical isomorphism HF ((L1 , b1 ), (L0 , b0 ); C[[q]][q −1 ]) ∼ ˆ 1 , b1 ), F(L ˆ 0 , b0 )) ⊗C[[q]] C[[q]][q −1 ]. = RHom(F(L

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22

INTRODUCTION

Here bi = (b, bi ). (It follows from (O.4) that the Floer cohomology of the left hand side is defined over the ring C[[q]][q −1 ].) (R.3) The ring and bimodule structures in (O.2), (O.3) are isomorphic to those in the right hand side of (R.2) respectively. One can generalize Conjecture R to the context of the filtered A∞ category by considering arbitrary finite chains of Lagrangian submanifolds as described in [Fuk02II]. Remark 1.24. We remark that the moduli space of objects in the derived category is a notion that is not well established yet in algebraic geometry. In many  (M,ω,b) , not just an important cases, the mirror object is a coherent sheaf on X object of its derived category. In these cases, the moduli space mentioned in (R.1) is well established in algebraic geometry. For the general case, one may use the notion of filtered A∞ algebras studied in this book to give a rigorous meaning to (R.1): We first observe that both cohomology groups appearing in (R.2) are δb1 ,b0 cohomology of some filtered A∞ bimodule. For the case of Floer cohomology, this follows from the results of this book. More specifically from Theorem A and Definition 3.6.9. For the case of RHom, this follows from Theorem W and the standard construction of RHom in homological algebra. We can amplify (R.3) to a conjecture that they are isomorphic as a filtered A∞ bimodule. We next recall that a formal neighborhood of the moduli space is controlled by the (filtered) A∞ structure. Namely (1.8) is the equation that determines the formal neighborhood of the moduli space. (See [Fuk03II] for more explanation.) In this sense, the amplified version of (R.3) contains (R.1) as a special case. We note that (R.1) above would mean that the formal map PbL given in Corollary O coincides with the Kuranishi map of the moduli space of holomorphic vector bundles (or coherent sheaves) of its mirror. After the substitution of T 1/m = q, the map PbL is regarded as a map from H 1 (L; qC[[q]]) to H 2 (L; qC[[q]]) (see [Fuk03I]), and has the form (1.25)

PbL (b) =

∞  k=0

mk (b, · · · , b).   k

(Strictly speaking, (1.25) is the formula in the chain level. To define PbL on the cohomology H 1 (L; C[[q]]), we need to use Theorem W below and the filtered A∞ homomorphism defined in Subsection 5.4.3.) The right hand side of (1.25) may be regarded as a generating function of the numbers of pseudo-holomorphic discs attached to L, which is an open string analogue of the Gromov-Witten potential. (The Gromov-Witten potential is the generating function of (genus zero) Gromov-Witten invariants defined by counting pseudo-holomorphic spheres). (In case dim L = 3, the map PbL is the differential of a super potential which is a more direct analog to the Gromov-Witten potential. See Subsection 3.6.4.) Therefore (R.1) is a conjecture on the relationship between the number of pseudo-holomorphic discs attached to L and the deformation theory of vector bundles on its mirror. In this sense, Conjecture R can be regarded as the open string version of the famous conjecture quantum cup product = Yukawa coupling

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1.4. RELATION TO MIRROR SYMMETRY

23

which was discovered by Candelas et al. [COGP91] and partially proved by Givental [Giv96]. (See also [LLY97].) The open string analogue of this kind was first discovered by Kontsevich [Kon95I] for the case of elliptic curve. The case of elliptic curve was further explored by Polishchuk-Zaslow [PoZa98] and was partially generalized to higher dimensions by first named author [Fuk02III]. Recently a few more cases have been studied including the cases (1.14.1) and (1.14.5) mentioned at the end of Section 1.3. Next we outline a scheme of the proof of Conjecture R by using Corollary P combined with some rigid analytic geometry. Remark 1.26. The idea of using rigid analytic geometry to prove homological mirror symmetry is due to Kontsevich and Soibelman [KoSo01]. There is an approach closely related to but different from this: This is the approach using asymptotic analysis proposed by the first named author [Fuk05I]. We mention that Gross and Siebert [GrSi03I,03II] use the log scheme in their attempt to construct mirror manifold, which is also of a similar flavor. In the rest of this section we only consider the case b = 0 for the simplicity of exposition. Motivated by the picture of Strominger-Yau-Zaslow [SYZ96], we start with a 2n-dimensional symplectic manifold M that has a singular fibration over a manifold B whose general fiber is a Lagrangian torus T n with its Maslov class being zero. We assume that ω is integral, i.e., [ω] ∈ H 2 (M ; Z). We denote by Bsm the subset of B consisting of the points y ∈ B at which its fiber Fy is smooth. Under the ˜ n (flat affine) assumptions given on the fibration, we can construct a GL(n; Z)×Z structure on Bsm . (See [Dui80, KoSo01, GrSi03I].) We also assume that the codimension of B \ Bsm is bigger than or equal to 2. Define B Q to be the set of points whose affine coordinates are rational. When π1 (M ) = 1, a point y being in B Q is equivalent to the condition that Fy is a rational Lagrangian submanifold. Remark 1.27. Let y ∈ B Q . If Fy is singular, the definition of Floer cohomology of Fy is not given in this book. In many important cases, the singular fiber Fy turns out to be an immersed Lagrangian submanifold. Therefore it would be desirable to extend our story to such cases. (See [Aka05, AkJo08].) Conjecture S. Suppose y ∈ B Q and let PFy be the formal power map introduced in (1.25). Then we have PFy ≡ 0, i.e., M(Fy ; Λ+ ) = H 2 (Fy , Λ+ ). We next consider the pair (L, b) as in Conjecture R. We obtain (1.28)

(Λ0

ˆ R(PL ;Λ0 ) D(L, Fy ; Λ0 ), ϕb ⊗

1 ⊗ d)

from Corollary P. This defines an object of the derived category of coherent sheaves on Spf(R(PFy ; Λ0 )) = Spf(Λ0 [[s1 , · · · , sn ]]). We recall that the set of rigid points of the formal scheme Spf(R(PFy ; Λ0 )) is Λn+ where s1 , · · · , sn are its coordinates. In fact, the natural coordinates are not si but their exponentials esi = xi . (See Subsection 3.6.4 for some discussion on this point.) The set exp(H 2 (Fy , Λ+ )) is an infinitesimally small (open) neighborhood of (1, · · · , 1) ∈ (Λ0 \ {0})n .

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24

INTRODUCTION

Conjecture T. The object given in (1.28) descends to an object on the open set exp(H 2 (Fy , Λ+ )) and can be extended to an object on the annular region A1− ,1+ (x1 , · · · , xn ) = {(x1 , · · · , xn ) ∈ (Λ0 \ {0})n | 1 − < xi  < 1 + } for some sufficiently small > 0. Here the norm  ·  is the non-Archimedean norm defined by   ai T λi  = e−λ1 where λ1 is the leading exponent, i.e., satisfies a1 = 0 and λi > λ1 for i > 1. To specify the base point y ∈ B in our discussion of the infinitesimal neighborhood and its coordinates, we write xyi in place of xi . Namely we define xyi = exp(si ) where si are the chosen coordinates of Spf(R(PFy ; Λ0 )). Let y  = (y1 , · · · , yn ) be a point in the neighborhood of y = (y1 , · · · , yn ) expressed in the given affine coordinates of B. By comparing symplectic areas of the discs attached to each of the Lagrangian submanifolds Fy and Fy , we observe that it is natural to define the  relation between the coordinates xyi and xyi by (1.29)





xyi = T yi −yi xyi .

Conjecture U. There is a coordinate change map Φyy (of the rigid analytic space) from (1.30)





A1− ,1+ (xy1 , · · · , xyn ) ∩ A1− ,1+ (xy1 , · · · , xyn ) ⊂ A1− ,1+ (xy1 , · · · , xyn )

to (1.31)









A1− ,1+ (xy1 , · · · , xyn ) ∩ A1− ,1+ (xy1 , · · · , xyn ) ⊂ A1− ,1+ (xy1 , · · · , xyn ).

The map Φyy is induced by an appropriate filtered A∞ homomorphism H(Fy ; Λ0 )   → H(Fy ; Λ0 ). We can glue {A1− ,1+ (xy1 , · · · , xyn )} by transition maps Φyy from ˆ (1.30) to (1.31) and obtain a rigid analytic space X. Moreover we can glue the objects obtained in Conjecture T by ΦL yy  to obtain ˆ ˆ an object F(L, b) of derived category of coherent sheaves on X. The transition maps ΦL yy  are induced by appropriate filtered A∞ bimodule homomorphisms. ˆ Conjecture U, if proved, will give a way of constructing the mirror object F(L, b) provided in Conjecture R. Then functoriality of the construction would imply that there is a filtered A∞ homomorphism from the left hand side of (R.2) to the right hand side of (R.2). More arguments are needed to show that it is an isomorphism. To carry out the scheme described above, there still remain other difficulties. One outstanding problem is the study of the singular fibers and various structures associated to them. We expect the results of [FOOO09II] will be useful for this purpose. We point out that the scheme described above uses a family version of Floer cohomology. The idea of using the family version of Floer cohomology to prove homological mirror symmetry dates back to the summer of 1997 in the discussion between the first named author and M. Kontsevich at IHES. It was partially realized for the case of abelian varieties in [Fuk02III]. In this particular case, all the operations mk appearing in [Fuk02III] indeed converge since they are reduced to multi-theta functions, and so translation of the story in the language of the formal scheme or the rigid analytic space was not needed in [Fuk02III]. See also [Fuk02I].

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25

The approach of using the family version of Floer cohomology to the homological mirror symmetry conjecture that we have described above seems to be the most promising approach at this stage, which we hope to carry through in a near future. Before closing out this section, we would like to indicate a few possible generalizations of our story. Our construction of the Floer cohomology and of the A∞ structure in this book uses the moduli space of pseudo-holomorphic maps from the disc D2 . We may also consider pseudo-holomorphic maps from bordered Riemann surface of higher genus, and consider more general flat bundles than flat line bundles on L. If we perform these constructions for the zero section of the cotangent bundle of a 3-manifold, the corresponding open string theory is expected to coincide with the Chern-Simons perturbation theory, [AxSi91I,91II, Bar91, GMM89, Kon94]. This fact was pointed out by Witten in [Wit95]. (See also [Fuk96I,97III, FuOh97].) Its mirror would be a version of the quantum KodairaSpencer theory of Bershadsky-Ceccoti-Ooguri-Vafa [BCOV94], or more precisely its analog of the moduli space of coherent sheaves. In [BCOV94], the quantum Kodaira-Spencer theory of the moduli space of complex structure on a manifold was described. Unfortunately, formulating a conjecture of this coincidence in a mathematically precise form such as in Conjecture R is still beyond our reach. One might, however, be able to say something rigorous for the case of genus one. Namely in the symplectic side, the genus one part would correspond to an invariant defined by Hutchings and Lee, [HuLe99], after including quantum contributions. This invariant is related to the number of pseudo-holomorphic annuli and the Reidemeister torsion. (See also [Fuk97I, Lee01].) In the complex side, the genus one part would correspond to the analytic torsion of ∂ operator with ˆ ˆ Hom(F(L, b), F(L, b)) as its coefficients. The relationship of the analytic torsion of ∂ operator to the Quillen metric on the moduli space of coherent sheaves and others has been discussed by various people, for example, by Bismut [Bis98]. The first step towards the study of the higher genus analog to Conjecture R would be to prove the coincidence of these invariants defined for the genus-one case. 1.5. Chapter-wise outline of the main results Lagrangian intersection Floer theory as a whole turns out to be a highly complex and non-trivial theory partly because many basic constructions are required to be carried out in the chain level. This chain level constructions are intricately intertwined with by now the well-known construction of (virtual) fundamental chains of the various moduli spaces of pseudo-holomorphic maps, via the Kuranishi structure [FuOn99II] or the equivalent kind. This partly explains why the theory requires many disciplines of modern mathematics like infinite dimensional topology, nonlinear elliptic partial differential equation, deformation theory and higher algebraic structures in addition to symplectic geometry. On top of this, the theory has been further enhanced by the physics of open string theory with D-branes. We have attempted to set the rigorous foundation of this complex theory in all aspects having applications both to symplectic topology and to homological mirror symmetry in mind. This explains why the volume of the book is unusually bulky as a research monograph. Since we expect that not only mathematicians from related fields but also physicists working on the string theory may be reading at least some part of the book depending on their interest, we would like to give a brief outline of the

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26

INTRODUCTION

contents of the book chapter-wise and assist them to decide where to look for what they are interested in. Now chapter-wise summary of the contents of the book is in order. Along the way, we will also provide brief outlines of the proofs of the main theorems already stated before. We remark that many of the chapters have their own introductions (Sections 3.1, 4.1, 6.1, and the beginning of Chapter 7) where more detailed summary of the chapters are given. Chapter 2 is mostly devoted to a review of the Floer’s original version of Floer cohomology [Flo88IV] and explains how this definition meets obstruction as soon as we leave the world of Lagrangian submanifolds L that Floer looked at in [Flo88IV], i.e., of those without quantum effects or of L satisfying π2 (M, L) = {e}. In Sections 2.1-2.2, we give the definitions of various kinds of Maslov indices and of the stable map moduli space of pseudo-holomorphic discs on which the whole constructions in the book are based on. In Section 2.3, we explain the basic idea of Floer’s construction of Lagrangian intersection Floer cohomology that is due to Floer himself [Flo88IV]. Then in Section 2.4, we first review the Floer cohomology of monotone Lagrangian submanifolds that the second named author studied in [Oh93,95I] and then explain where the direct generalization of these works fail in general and indicate how one might be able to overcome the failure. Proofs of our basic results, Theorems A and F, begin in Chapter 3. Chapters 3-5 are occupied by the basic constructions leading to the definition of our deformed Floer cohomology and then to the proofs of Theorems A and F. For this purpose, we use the (virtual) fundamental chains of the moduli spaces of pseudo-holomorphic discs of various kinds. Two essential points needed for the construction are postponed until Chapters 7 and 8: one is the transversality problem in Chapter 7 and the other is the orientation problem in Chapter 8. The way how Chapters 3-5 are written is always, first develop necessary homological (homotopical) algebra of filtered A∞ algebras and filtered A∞ modules in the purely algebraic context, and then apply the algebraic machinery to our geometric context of Lagrangian submanifolds. In this way, the parts of homological (homotopical) algebra can be read separately from the geometric theory of Floer cohomology, which may be beneficial to some algebraically minded readers. For the geometric constructions in Chapters 3,4 and most part of Chapter 5, we need to work in the (co)chain level. Our filtered A∞ structure of a given Lagrangian submanifold L ⊂ (M, ω) is constructed on a cochain complex of L whose cohomology group is isomorphic to that of L. As in the usual cohomology theory, there are several different choices of the cochain complex that realizes the cohomology group of L. We have experimented several different choices while (and before) we were writing this book. Eventually, in this version of the book, we mostly use the version of singular chain complex. More precisely speaking, we construct a filtered A∞ structure on a countably generated sub-complex of the smooth singular chain complex of L. We regard it as a cochain complex by identifying k chain as a dim L − k cochain. (This can be regarded as a kind of Poincar´e duality.) Chapter 3 gives the definition of filtered A∞ algebras and their deformation theory. Notions of obstruction cycles and of bounding cochains are introduced first in the algebraic context. Then we associate a filtered A∞ algebra to each Lagrangian submanifold L over a countably generated complex on L in Section 3.5. The algebraic arguments explaining how to proceed from the filtered A∞ algebra to the definitions of (deformed) Floer cohomology and of the moduli space M(L)

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1.5. CHAPTER-WISE OUTLINE OF THE MAIN RESULTS

27

that parameterizes the Floer cohomologies are given in Section 3.6. Our deformed Floer cohomology on a single Lagrangian submanifold L should be regarded as a version of Bott-Morse type Floer cohomology. Then to each pair of Lagrangian submanifolds we associate a filtered A∞ bimodule which is given in Section 3.7. The filtered A∞ structure on the cohomology group itself used in the statements of Theorems A and F is constructed only at the end of Chapter 5. There are two other important issues discussed in Chapter 3. One is the issue of the unit of filtered A∞ algebra. Due to the transversality problem, which is solved in Chapter 7, we need to introduce the notion of homotopy unit. Its definition is given in Section 3.3. The relationship of the unit with the moduli space Mweak (L) of weak bounding cochains is explained in Subsection 3.6.3. The other issue discussed in Section 3.8 is the deformation of the filtered A∞ algebra by the cohomology classes from the ambient symplectic manifold. We use various operators denoted by p, q and r for this purpose. To define these operators, we use the moduli space of pseudo-holomorphic discs that have marked points both in the interior and on the boundary of the disc. This is the first step towards involving the open-closed string moduli spaces in the Floer theory. We expect that more extensive penetration of the ideas from the open closed string theory into the Floer theory and symplectic geometry will occur. (See [FOOO08II].) Such an idea is already used in this book in the proof of Theorems H and I. The relationship of the operator q with the moduli space Mdef (L) is discussed in Section 3.8, and the operator p is used in Chapter 6 to study spectral sequence and prove (D.3). The filtered A∞ algebra on the countably generated sub-complex of a singular cochain complex constructed in Chapter 3 depends on the various choices we make in the construction. In Chapters 4 and 5, we discuss in what sense it is independent of the choices made, and prove that the filtered A∞ algebra is unique up to the homotopy equivalence of filtered A∞ algebras. However before stating this homotopy equivalence, we need to give the definitions of various basic concepts in the homological algebra of A∞ structures. In Chapter 4, we give the definition of homotopy equivalence between (filtered) A∞ algebras and prove its basic properties. We provide a detailed exposition of homological (homotopical) algebra of (filtered) A∞ algebras in this chapter. There are some literature in which this notion of homotopy equivalence between A∞ algebras is defined (see, for example, [Smi00, MSS02]). However we make our exposition self-contained as much as possible. There are various reason for doing this. One reason is that several different versions of the definition of homotopy equivalence are used in the literature which we feel is too confusing to directly borrow for our purpose. We give our definition of homotopy between two A∞ homomorphisms in Definition 4.2.35. Our idea is to use the notion that is an algebraic counterpart to the operation of taking the product of a space with the interval [0, 1] in geometry, and to mimic this operation in defining the homotopy of A∞ homomorphisms. Depending on the way how we algebraically realize the multiplication by the interval [0, 1], we obtain several different versions of the definition of homotopy. We, however, prove that the different choices are all equivalent in a precise sense. (This is a consequence of Theorem 4.2.34.) The other reason for making our exposition self-contained is that we also need to generalize the homological algebra of A∞ algebra in two different directions. The first one is to define the filtered version of A∞ structures and the other is to

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28

INTRODUCTION

consider the case where the coefficient ring is not Q, but Z or Z2 . Our exposition is designed so that this generalization can be easily carried out. Now we describe two basic results, among others, in the homological algebra of filtered A∞ structures. Let f : (C1 , m∗ ) → (C2 , m∗ ) be a filtered A∞ homomorphism between two filtered A∞ algebras over the ring Λ0,nov (R). We assume that Ci and f are gapped in the sense of Definitions 3.2.26 and 3.2.29. We reduce the coefficient ring Λ0,nov (R) to R to obtain an A∞ homomorphism f : (C 1 , m∗ ) → (C 2 , m∗ ) over R. For the (unfiltered) A∞ algebra (C, m∗ ), we always assume m0 = 0 and so m1 satisfies m1 ◦ m1 = 0. Namely, for the unfiltered case, we always assume that we are given a genuine cochain complex (C, m1 ). For our geometric application, this will be nothing but a version of singular chain complex of L. We now state the first main theorem in the homological algebra of filtered A∞ algebras, which is an algebraic counterpart of Whitehead’s theorem in topology. Theorem V. If f induces a chain homotopy equivalence (C 1 , m1 ) → (C 2 , m1 ), then f : (C1 , m∗ ) → (C2 , m∗ ) has a homotopy inverse (of gapped filtered A∞ algebra). Theorem V is useful for the purpose of proving that various filtered A∞ homomorphisms we define throughout the book are homotopy equivalences. We would like to point out that Theorem V does not hold in the category of differential graded algebras in general. Thanks to Theorem V, we do not need to ‘invert the quasi isomorphism’ which is an essential ingredient in the standard homological (homotopical) algebra of differential graded algebras. Because of this reason, the homotopy theory of A∞ algebras is much closer to that of spaces than the homotopy theory of differential graded algebras. Theorem V is Theorem 4.2.45 whose proof is given in Section 4.5. Then in Section 4.6 we use this algebraic machinery and prove that the filtered A∞ algebra constructed for a Lagrangian submanifold L in Chapter 3 is independent of various choices up to this homotopy equivalence. Another matter for which we need to use homological algebra of A∞ algebras concerns the notion of gauge equivalence of solutions of (1.8). We define M(L) to be the set of gauge equivalence classes of solutions of (1.8). We define the gauge equivalence between two solutions to be a version of homotopy that is again an algebraic counterpart of taking the product with [0, 1]. In this way, we can associate the set M(C) to each filtered A∞ algebra (C, m∗ ) in a functorial way (Theorem 4.3.13). Then we prove that M(C) is an invariant of the homotopy type of the filtered A∞ algebra (C, m∗ ) (Corollary 4.3.14). Chapter 5 is devoted to the filtered A∞ bimodule version of Chapter 4. In Section 5.1, we first discuss the relationships between various Novikov rings, Λ0,nov and others, and how they behave when we move Lagrangian submanifolds. In Section 5.2, we discuss the homological algebra of filtered A∞ bimodules, and especially define the notion of homotopy between them. We then prove in Section 5.3 that the filtered A∞ bimodule associated to a relative spin pair (L0 , L1 ) of Lagrangian submanifolds is independent of various choices involved up to the chain homotopy equivalence. In Section 5.4, we transfer various A∞ structures constructed in the cochain level to those defined on the cohomology. In this regard, the following is the second main theorem in the homological algebra of filtered A∞ structures.

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1.5. CHAPTER-WISE OUTLINE OF THE MAIN RESULTS

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Theorem W. Let R be a field. For any gapped filtered A∞ algebra (C, m∗ ) there exists a gapped filtered A∞ structure on H(C, m1 ) ⊗R Λ0,nov (R) that is homotopy equivalent to (C, m∗ ). Theorem W is the filtered A∞ algebra version of a classical result in homological algebra which dates back to [Kad82]. Our proof is based on the Feynman diagram technique of summing over trees which is similar to the one given by [KoSo01] and to the argument in Section A.6 of the year-2000-version [FOOO00] of this book. The purpose of Chapter 6 is to construct a spectral sequence appearing in Theorem D and to explain its various applications to symplectic geometry and topology. The universal Novikov ring Λ0,nov has a filtration defined by F λ Λ0,nov = T λ Λ0,nov . (See (Conv.6.)) However the filtration is given by real numbers λ, not by integers. It turns out that constructing a spectral sequence and proving its convergence out of such a filtered complex is a nontrivial problem. In Chapter 6, this problem is solved and Theorem E is proved. For this purpose, we also use the operator p and the existence of unit in the filtered A∞ algebra attached to the given Lagrangian submanifold L. In Chapter 7, we discuss the problem of transversality. This is one of the two main ingredients in the construction of the virtual fundamental chains of various moduli spaces of pseudo-holomorphic discs used in this book. (The other is the problem of orientation which is discussed in Chapter 8.) For this purpose, we systematically use the notion of the Kuranishi structure and its multi-sections introduced in [FuOn99II]. The Kuranishi structure and perturbations by multi-sections provide an optimal framework to study the transversality problem in the general moduli problem, where the concerned objects have finite automorphism groups. This established machinery still seems to be overlooked by many researchers in the related areas, although more than 10 years have passed since its first appearance in January 1996. Because of this, in Appendix Section A1 we provide a rather detailed account on the Kuranishi structure and on the construction of virtual fundamental chains via the Kuranishi structure. We would also like to give rather detailed account of the contents of Chapter 7 here. In Section 7.1, we give a construction of the Kuranishi structures on the moduli spaces of various type of pseudo-holomorphic discs used in the book. In our circumstances of defining the A∞ algebra and the Floer cohomology, we need to use virtual fundamental chains in place of virtual fundamental cycle. This means that the structure constants of various algebraic structures we construct depend on the choice of various perturbations, and so only their homotopy types could possibly become an invariant independent of such perturbations. Because we need to consider a whole family of moduli spaces attached to a Lagrangian submanifold L or its pair (L0 , L1 ), we need to choose perturbations of various moduli spaces so that they satisfy certain compatibility conditions, if one hopes the final outcome to be an invariant of L or (L0 , L1 ) itself. (This point is of sharp contrast to the case of usual Gromov-Witten invariants, where we can safely take perturbations individually for each moduli space, because the corresponding invariant is defined in the homology level.) We define the fiber product of Kuranishi structures in Section A1 and use this to formulate the compatibility of Kuranishi structures defined on the different strata of the compactified moduli spaces in Subsection 7.1.1. In Section 7.2, we carefully state the compatibility conditions of multi-sections (perturbations) using

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30

INTRODUCTION

this compatibility of Kuranishi structures. We emphasize that without precisely stating the compatibility condition of perturbations, any statement asserting the well-definedness of various algebraic structures defined by using the compactified moduli space would not make sense and not be well-founded. In Subsections 7.1.2-7.1.3, we discuss the analytic part of construction of Kuranishi structures of the disc moduli spaces of our concern. Many arguments there are combinations of various techniques that have already appeared in the literature. For example, the gluing technique in the Bott-Morse situation appeared in [Fuk96II], the study of pseudo-holomorphic discs together with moduli parameters in [FuOh97], and gluing construction of Kuranishi structures in [FuOn99II] respectively. Our construction of Kuranishi structures of the disc moduli spaces used for the definition of the filtered A∞ structures associated to Lagrangian submanifolds is completed in Subsection 7.1.4. In Subsection 7.1.5, we study the moduli space that we use to define a filtered A∞ homomorphism between the filtered A∞ algebras attached to Lagrangian submanifolds. To give a construction of Kuranishi structures that satisfy the above mentioned compatibility conditions, we need to study certain cell decomposition of a disc Dn , which in turn is used to define the notion of the A∞ map (Theorem 7.1.51.) Note that the Stasheff cell or rather Stasheff’s cell decomposition of a disc Dn is used to define the notion of the A∞ space ([St63]) and is related to both the moduli space of marked bordered Riemann surface of genus zero and to the moduli space of metric ribbon trees. (See [FuOh97].) Stasheff told us that the cell decomposition of the disc which leads a definition of A∞ maps was known to him. (See also [MSS02, BoVo73].) However, it seems that our geometric description of this cell decomposition is new. We emphasize that novelty of our exposition on the transversality in this book does not lie in the analytic part, which has been in principle understood since 1996, but rather lies in its topological part that we present in Section 7.2. In Section 7.2, for the purpose of constructing the filtered A∞ structure and others, we make suitable inductive strata-wise perturbations of the Kuranishi map of the given pseudo-holomorphic equation on each stratum of the compactified moduli space, and make a choice of countably generated sub-complex of the singular cochain complex. Remark 1.32. In this book, we use singular chain complex for the construction of the A∞ structure associated to Lagrangian submanifolds. This is the outcome of some deliberation of ours. We point out that there are also two other possible choices. We briefly mention the other choices and explain why we have made the current choice in this book. One such choice is to be using the de Rham complex. This choice works as explained in [Fuk05II] and worked out in [Fuk07I, FOOO08II], (where the contents of present book is assumed). It has some of its own advantage. For example, it makes easier to keep more symmetry like the cyclic symmetry in the construction of A∞ structures. However with the de Rham complex, we are unable to work over the Z2 coefficient. Usage of the Z2 coefficient is essential for several applications to symplectic geometry, especially to the study of the Arnold-Givental conjecture. We will discuss the partial generalization of our story to Z or Z2 coefficient and

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apply it to prove Arnold-Givental conjecture in some cases in [FOOO09I]. There we use the algebraic (as well as other) frame works of this book. So we did not use the de Rham complex in this book, except in Section 7.5 and Theorem X. The other choice is to use the Morse complex. In fact this is the approach that the first named author initiated in 1993. (See [Fuk93] and (3.3), Figure 8 in [Fuk97III].) The second named author also used this approach to construct a spectral sequence similar to the one in Theorem D and pointed out that this approach could be amplified to a construction of A∞ -category in [Oh96I, 96II]. We used a similar argument in pp 287 - 291 of the year-2000-version of this book for the proof of weak finiteness (Definition 6.3.27), exploiting the obvious fact that the Morse complex of a compact manifold is always finitely generated. During the preparation of the current revision, we have discovered a (purely algebraic and simpler) proof of weak finiteness of our countably generated complex, and preferred to include this simpler proof and to drop the old proof from this current version. We believe that the approach of using the Morse complex would work to some extent. However we would like to point out that there is some trouble that occurs in this approach. Recall that the product structure on the (filtered) A∞ algebra associated to L is a deformation of the cup product on the cohomology H ∗ (L). As studied by the first named author [Fuk93,97III] and Betz-Cohen [BeCo94], one needs to simultaneously use several different Morse functions on a manifold to study the product of Morse homology in the chain level. The product in Morse homology is defined by considering the moduli space of maps from the “Y-graph” to the given manifold. To define the structure constants, we want this moduli space to be transversal. Note that this moduli space will be never transversal, if one consider the same Morse function on the three edges of the Y-graph. This transversality problem is a fundamental issue in algebraic topology and seems to be extremely hard to resolve unless one uses several Morse functions simultaneously. See [FOOO08III]. One implication of the above discussion is that one can only construct a (kind of) filtered A∞ category in which morphisms (= Floer cohomology) can be composed only generically. This leads to the notion of topological A∞ category. As a consequence, it is difficult to define a unit which is nothing but the identity morphism. The problem of the unit is discussed in the context of topological A∞ category by the first named author in [Fuk97II]. However the exposition in [Fuk97II] was quite unsatisfactory. Only after we succeeded in constructing the (filtered) A∞ algebra for each L as in the way taken in this book (or by using de Rham cohomology), we have been able to properly understand the unit of (filtered) A∞ category. (See [Fuk02II].) The unit plays a very important role in applications to symplectic geometry, because the unit can be used to prove non-triviality of Floer cohomology in some favorable circumstances. (See Theorem E, for example.) The unit also plays an important role in the application to mirror symmetry. For example, the proof of Yoneda’s lemma for the A∞ category is based on the existence of a unit. (See [Fuk02II].) Therefore having a proper definition of the unit in this story is an important ingredient for the A∞ category of Lagrangian submanifolds. Because it does not seem to be easy to give a proper definition of the unit in the context of the approach using Morse functions, we do not use Morse homology either in this book.

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32

INTRODUCTION

The problem of transversality already occurs even if we do not take contributions from (pseudo-)holomorphic discs into account. The problem in classical algebraic topology is then, that of defining an A∞ structure on the singular chain complex that realizes the intersection product in the chain level, for example. The main trouble for the singular chain complex is that a chain is never transversal to itself and so pairing between the same chain is not well-defined unless the chain is the fundamental chain of the underlying manifold. Among the algebraic topologists, this trouble is well known as well as the difficulty of the problem of realizing the intersection product in the chain level (unless one use the de Rham complex). We have solved this transversality problem by carefully perturbing the usual intersection pairing of chains over Z. However, then, associativity of the resulting product inevitably breaks down. We compensate the failure of associativity by systematically introducing the higher product operations mk through which we obtain an A∞ algebra on a countably generated (co)chain complex. Once we have done this, we then continue to incorporate contributions of pseudo-holomorphic discs in a certain compatible way, and define our filtered A∞ algebra associated to the given Lagrangian submanifold L. In Section 7.3, we provide the construction of the homotopy unit. In Section 7.5, we prove the following theorem, which is close to the equivalence statement of various approaches, that was mentioned in Remark 1.32, to the construction of the filtered A∞ algebra appearing in Theorem A. We recall that the de Rham complex is a differential graded algebra which can be regarded as an A∞ algebra. (See Remark 3.2.4 (2) for the precise meaning of this statement.) Theorem X. Let (H ∗ (L; Λ0,nov ), m∗ ) be the filtered A∞ algebra given in Theorem A and (H ∗ (L; Q), m∗ ) the reduction of the coefficient ring to Q. Then the A∞ algebra (H ∗ (L; Q), m∗ ) ⊗Q R is homotopy equivalent to the de Rham complex of L as an A∞ algebra. This theorem implies that if we take the coefficient R, then the filtered A∞ algebra in Theorem A is a deformation of the real homotopy type of L. One might say that it is intuitively clear that all three approaches, singular, de-Rham, and Morse, will boil down to the same theory. However giving a rigorous proof of such an equivalence is not an easy problem. Theorem X gives a part of the proof of the equivalence between the de Rham cohomology version and the singular cohomology version. It is very likely that further refining of these arguments leads to a complete proof of this equivalence using the de Rham version constructed in [Fuk07I]. As for the equivalence between the Morse homology (or Morse homotopy) version and de Rham cohomology version in the classical topology, a relevant statement is that the Morse A∞ category of a given compact manifold, which was constructed in [FuOh97], is homotopy equivalent to the rational homotopy type of the manifold. We have no doubt that this is true as we mentioned in [FuOh97]. Unfortunately such a statement is not proved in detail in the literature yet. It appears that the discussion of Kontsevich and Soibelman in page 239 around Theorem 2 [KoSo01] is related to such a statement. Detail of the proof of that statement, however, is missing in [KoSo01]. (However their idea of the proof proposed there is convincing and very likely to be correct.)

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1.5. CHAPTER-WISE OUTLINE OF THE MAIN RESULTS

33

Remark 1.33. We conjecture that (H ∗ (L; Q), {mk }∞ k=0 ) is homotopy equivalent to the Q-de Rham complex (see [Sul78]). This does not follows from Theorem X. In fact, there is an example of a pair of differential graded algebras over Q which are homotopy equivalent to each other over R but are not homotopy equivalent to each other over Q. (See [BrSz89].) In Section 7.4 we present the details of the construction outlined in Section 3.8. Namely we deform our filtered A∞ algebra attached to the Lagrangian submanifold using the moduli space of pseudo-holomorphic discs with both interior and boundary marked points. There are two main issues we discuss in this section. One is about compactification of the moduli space of pseudo-holomorphic discs with interior marked points but without boundary marked points. One essential new phenomenon we discovered is the fact that the standard stable map compactification does not give a correct compactification in this case. Because of this phenomenon, the term containing Gromov-Witten invariant appears in the formula (3.8.10.3) which spells out the main property of the operator p. Another issue dealt with in Section 7.4 is to provide an algebraic framework for the description of the bulk deformation of (H(L; Λ0,nov ), m∗ ). The general strategy in the deformation theory (whose origin goes back to Gerstenhaber) states that deformations of an algebraic system are controlled by an appropriate Lie algebra. For the case  of filtered A∞ algebra C, we have the Hochschild chain complex CH(C, C) = k Hom(Bk C[1], C[1]) which is a differential graded Lie algebra. We regard this complex as an L∞ algebra which we denote by DerB(C[1]). (See Definition 7.4.26). (See Section A3 for the definition of L∞ algebra.) Its unital version is denoted by DerBunit (C[1]). On the other hand, the cohomology group H(M ; Λ0,nov )[1] (with degree shifted) of our ambient symplectic manifold is regarded as an L∞ algebra with all the operations trivial. The main properties of our operator q can be stated in the language of L∞ algebra as follows: Theorem Y. There exists a strict and filtered L∞ homomorphism qo : H(M ; Λ0,nov )[1] → DerBunit (H(L; Λ0,nov )[1]) such that after the reduction of the coefficient ring to Q, the map qo reduces to a linear map H(M ; Q) → Hom(Q, H(L; Q)) ∼ = H(L; Q) induced by the inclusion into the cohomology group H(L; Q). If ψ : (M, L) → (M  , L ) is a symplectic diffeomorphism we have the following diagram which commutes up to an L∞ homotopy: qo

H(M, Λ0,nov )[1] −−−−→ DerBunit (H(L; Λ0,nov )[1]) ⏐ ⏐ ⏐ψ ⏐ψ  ∗  ∗ qo

H(M  , Λ0,nov )[1] −−−−→ DerBunit (H(L ; Λ0,nov )[1]) Theorem Y is the combination of Theorems 7.4.118 and 7.4.120 which are proved in Subsections 7.4.5 and 7.4.6. We can use the notion of L∞ modules to state a similar result for the map r: This is the operator appearing in the deformation of Floer cohomology of the pair by the cohomology class of ambient symplectic manifold. See Theorems 7.4.154, 7.4.156, 7.4.162 in Sections 7.4.7-9. The operator p has an interpretation in

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34

INTRODUCTION

terms of the cyclic bar complex of H(L; Λ0,nov ) and an L∞ module structure over DerBunit (H(L; Λ0,nov )[1]) on it. See Theorems 7.4.192 and 7.4.195 in Subsection 7.4.11. The results in Section 7.4 are closely related to the loop space formulation of Lagrangian Floer theory [Fuk05II] and to the generalization of our story to the higher genus cases. Chapter 8 is devoted to the other technical heart of matter in our constructions, the problem of orientation. In Section 8.1, we study the moduli space of pseudoholomorphic discs without marked points. This problem is naturally related to the index theory for the family of elliptic operators. In our case of holomorphic discs (or the case of bordered Riemann surfaces in general), we need an index theory for the family of elliptic operators with boundary conditions. We show how the second Stiefel-Whitney class comes into this index theory, and prove that the moduli space of pseudo-holomorphic discs without marked points is orientable if the Lagrangian submanifold is relatively spin. More precisely, we prove the following Theorem Z. We denote by M(L; β) the stable map compactification of the moduli space of pseudo-holomorphic discs w : (D2 , ∂D2 ) → (M, L) in homology class β ∈ H2 (M, L). Theorem Z. The moduli space M(L; β) is orientable, if L ⊂ (M, ω) is a relatively spin Lagrangian submanifold. Furthermore the choice of relative spin structure on L canonically determines an orientation on M(L; β) for all β ∈ H2 (M, L) in a coherent way. We refer to Chapter 8 Definition 8.1.2 for the definition of relative spin structure. We would like to remark that Theorem Z is proved independently by de Silva [Sil97]. To illustrate that the index bundle of the Cauchy-Riemann operator with boundary conditions is not orientable in general, we give an explicit example for which the index bundle (of the Cauchy-Riemann operator on the disc with totally real boundary condition) is not orientable. Other sections of Chapter 8 are devoted to defining coherent orientations on various moduli spaces of pseudo-holomorphic discs with respect to the fiber product. This problem turns out to be an essential point in our construction of the A∞ algebra. The problem of giving the coherent orientations is closely tied to that of finding a good sign convention for the A∞ formulae, and also very much related to the supersymmetry. According to Getzler-Jones [GeJo90], there are basically two different kinds of the sign convention for the A∞ formulae. We now briefly recall the differences between them here: Consider the (singular) chain complex C(L) of L. Let xi ∈ C and let the degree be di = codim xi . In the first convention, we shift its degree by one and regard it as an element of degree di + 1 in the shifted complex. In this convention, the sign change we get is (−1)(d1 +1)(d2 +1) when one commutes two elements in the product. In the second convention, each element will have two different degrees, one the usual degree in the graded complex and the other the super-degree. In this convention, an element xi ∈ C(L) has the usual degree di coming from the grading and has super degree 1. The sign change we get is (−1)d1 d2 +1 when one commutes two elements in the product .  for the operator with the For this matter, Getzler-Jones used the notation m first sign convention and m for the second sign convention. They are related by

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1.6. ACKNOWLEDGMENTS

35

the map b in pp 259-260 [GeJo90]. Our convention in this book seems to coincide with the first sign convention. (Note that the second sign convention was used in [Fuk02III].) From the geometric point of view, these two sign conventions correspond to two different ways of equipping with orientation on the moduli space of pseudoholomorphic discs with marked points on the boundary. Thus, Sections 8.2-8.10 are devoted to a detailed description of the orientations of the various moduli spaces, and to the verification of the A∞ formulae with sign using the first sign convention. We include a thorough account on this orientation problem because it is a complicated and confusing matter. Without the thorough analysis, there would be no way to see the correct formulae with sign throughout the book. 1.6. Acknowledgments We owe M. Kontsevich the idea of constructing a series of obstruction classes and deforming Floer’s boundary map when the obstruction classes are trivial, in the study of Floer theory of general Lagrangian submanifolds. We have also learned many things from his papers which are used in several places of this book in an essential way. We like to thank U. Frauenfelder for pointing out one essential error in relation to the proof of the Arnold-Givental conjecture in the year-2000 version of this book. We also thank C.-H. Cho for several valuable discussions, especially the ones related to the isomorphism between quantum cohomology and Floer cohomology of the diagonal and also for a comment that led us to writing Subsections 7.4.2-11. We also thank K. Hori, J. Stasheff, N. Minami, M. Furuta, T. Gocho, A. Kono and F. Kato, for providing various useful information on potential function & Landau-Ginzburg model, A∞ algebra, rational homotopy theory, renormalization theory, relation of noncommutative symplectic geometry to A∞ algebra, homology of loop space, and rigid analytic geometry, respectively, during the preparation or revision of this book. We started the project of writing this book in 1998 and a preliminary version was completed in December of the year 2000. Since then, we have put it on the home page (http://www.math.kyoto-u.ac.jp/∼ fukaya/) of the first named author. Since the appearance of the year-2000 version of the current book, our revision process of completing this final version has taken much more time than we originally expected. In the mean time, many results out of this book have been further explained in various articles of the authors. (See [ChOh03, Fuk03II,05II, Oht01].) The contents of the book have been much modified and improved from those in the year-2000 version. Therefore some explanations on the relationship between the year-2000 version and the present final version should be in order. First, we would like to point out that there are several errors in the year-2000 version. The most conspicuous one is the extra condition of spherical positivity of (M, ω) needed for the authors to prove the Arnold-Givental conjecture at the time of writing this book. Except this added condition, all errors in the proofs of the theorems stated in the introduction of the year-2000 version are corrected in this book or in several other previous papers [FOOO09I,09II]. We emphasize that all the theorems stated in the introduction of the year-2000 version hold true as stated there, with the exception of ‘Theorem H’ of the year-2000 version, for which we need an additional assumption. (See [FOOO09I].) For the readers of

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36

INTRODUCTION

the year-2000 version or of the year-2006 version who are interested in finding out the differences between the year-2000 version and this final version, we prepared the document [FOOO09III] that describes the difference. Actually two chapters which were included in the previous year-2007 version of this book are excluded from this final version due to the publisher’s page restriction on this AMS/IP series: One chapter studies Lagrangian Floer cohomology over Z or Z2 coefficient under the additional assumption ‘spherically positivity’ of Lagrangian submanifolds. The other one studies the relationship between Lagrangian Floer cohomology and Lagrangian surgery. These two chapters will be published elsewhere as separate papers [FOOO09I,09II]. We would like to point out, though, the two chapters in the year-2007 version of this book are already in the final form. One drawback of this departure of the two chapters from this book is that not many examples of calculation of Lagrangian Floer cohomology are included in this book. We refer readers to [FOOO08I,08II,09I,09II] for more examples of such calculations. The first named author (K.F.) would like to thank IHES, where he had fruitful discussion with M. Kontsevich related to this project during his visit of IHES in 1997. The first, the third and the fourth named authors (K.F., H.O., K.O.) would also like to thank Y.-G. Oh and KIAS for their hospitality during their visits of KIAS in 1999 and 2000. They are also grateful to ETH, University of Warwick, Johns Hopkins University and Max Planck Institute f¨ ur Mathematik, Bonn for the hospitality during their visits. In these visits, they are benefited much from these institutions where they provide excellent working environment so that they could focus their attention on this project. The second named author (Y.O.) thanks K. Fukaya and RIMS for its financial support and hospitality during his stay in RIMS in the fall of 1999. The fourth named author would like to thank IHES for its hospitality during his stay in the summer of 1998. The authors would like to thank Professor S.T. Yau for suggesting to publish this book as a volume in the series, AMS/IP Studies of Mathematics. They would also like to thank Mrs. Tanaka for drawing excellent figures in this book. 1.7. Conventions (Conv.1)

The Hamiltonian vector field XH is defined by dH = ω(XH , ·).

(Conv.2) An almost complex structure is called tame to ω if the bilinear from ω(·, J·) is positive and compatible to ω if the bilinear form defines a Riemannian metric (which is also bounded at infinity if M is non-compact). (Conv.3) Let x ∈ BC[1] be an element of bar complex. We define a coassociative coproduct Δ : BC[1] → BC[1] ⊗ BC[1] by (1.34)

Δ(x1 ⊗ · · · ⊗ xn ) =

n 

(x1 ⊗ · · · ⊗ xk ) ⊗ (xk+1 ⊗ · · · ⊗ xn ) .

k=0

We consider its n − 1 iteration, Δn−1 : BC[1] → BC[1] ⊗ · · · ⊗ BC[1] . 

 n times

We write (1.35)

Δn−1 (x) =



n:n xn:1 c ⊗ · · · xc ,

c

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1.7. CONVENTIONS

37

where c runs on some index set depending n and x. In the case n = 2 we also write  (1.36) Δ(x) = xc ⊗ xc . c

(Conv.4) Let R be a commutative ring with unit. We use the following version of Novikov ring:  ∞     Λ0,nov (R) = ai T λi eni  ai ∈ R, λi ∈ R≥0 , ni ∈ Z, lim λi = ∞ . i→∞  i=0

Here T and e are formal generators of deg e = 2 and deg T = 0. We all it the universal Novikov ring. We define:  ∞    λi ni  Λnov (R) = ai T e  ai ∈ R, λi ∈ R, ni ∈ Z, lim λi = ∞ . i→∞  i=0

This is a localization of Λ0,nov (R) and is a field of fraction of in case R is a field. We denote by Λ+ 0,nov (R) an ideal of Λ0,nov (R) given by it  ∞     + λi ni ai T e ∈ Λ0,nov (R)  λi > 0 . Λ0,nov (R) =  i=0

We remark that Λ0,nov (R) ∼ = R[e, e−1 ]. Λ+ 0,nov (R)

(1.37) (d)

(d)

We denote by Λ0,nov (R) and Λnov (R) the degree d parts of them, respectively. In case R = Q we omit R and write Λ0,nov etc. (Conv.5)

We also consider the ring.  ∞    λi  ai T  ai ∈ R, λi ∈ R≥0 , and lim λi = ∞ Λ0 (R) = i→∞  i=0

and Λ+ (R) = Λ0 (R) ∩ Λ+ 0,nov and the localization Λ(R) = Λ0 (R)[T −1 ]. We denote Λ0 (Q) etc. by Λ0 etc.. (Conv.6) We define a filtration on Λ0,nov (R) by (1.38)

F λ Λ0,nov (R) = T λ Λ0,nov (R).

(λ ≥ 0.) It induces a filtrations F λ Λnov (R) of Λnov (R), where λ ∈ R. It also induces a filtration on Λ+ 0,nov (R). Those filtrations induce metrics on Λ0,nov (R), Λnov (R), Λ+ (R). These rings are complete with respect to this metric. 0,nov We define filtrations on Λ0 (R), Λ+ (R), Λ(R) in a similar way.

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https://doi.org/10.1090/amsip/046.1/02

CHAPTER 2

Review: Floer cohomology 2.1. Bordered stable maps and the Maslov index Floer’s original construction of Floer cohomology does not meet any obstruction when there occurs no bubbling phenomenon. When bubbling occurs, one needs to study how bubbling interacts with the classical Smale-Morse-Witten-Floer boundary map. By now, this analysis has been thoroughly carried out for the Floer cohomology of Hamiltonian diffeomorphisms using the virtual fundamental cycle techniques in [FuOn99II, LiTi98, Rua99]. The outcome is that there occurs no anomaly in the construction of Floer cohomology in that case. However for the Lagrangian intersection Floer cohomology, anomaly occurs even under the simplest possible bubbling contribution: the case of non-exact monotone Lagrangian submanifolds. In this chapter, we review construction of Floer cohomology for a monotone Lagrangian pair (L0 , L1 ) and explain how the anomaly occurs and can be resolved in some favorable situations just making a generic choice of almost complex structures. We refer to [Oh93,96I,96II] for more details of this construction for the monotone case. However we need to make clearer usages of the Novikov ring and the geometric set-up needed for our construction of Lagrangian intersection Floer cohomology beyond the monotone case. These points were not addressed in [Oh93,96I,96II] or in other previous literature. (Except them, the contents of this chapter is not new.) To carry out a self-contained explanation of these points, we provide a review on the Maslov index and the Novikov ring of Lagrangian submanifolds. 2.1.1. The Maslov index: the relative first Chern number. We consider the standard symplectic vector space (R2n , ω0 ) with the canonical coordinates n (x1 , · · · , xn , y1 , · · · , yn ) with ω0 = i=1 dxi ∧ dyi . The Lagrangian Grassmanian Λ(n) in (R2n , ω0 ) is defined to be Λ(n) = {V | V is a Lagrangian subspace of (R2n , ω0 )}. When we equip R2n with the standard √ complex multiplication and identify it with Cn by the map (xi , yi ) → zi = xi + −1yi any Lagrangian subspace V ⊂ Cn can be written as V = A·Rn for a complex matrix A ∈ U (n). Obviously A·Rn = Rn if and only if the matrix A is a real matrix. These show that Λ(n) is a homogeneous space Λ(n) ∼ = U (n)/O(n). It is shown in [Arn67] that H 1 (Λ(n), Z) ∼ = Z and Λ(n) carries the well-known characteristic class μ ∈ H 1 (Λ(n), Z), the Maslov class [Arn67]. This assigns an integer to each given loop γ : S 1 → Λ(n) given by μ(γ) = deg(det2 ◦ γ). Furthermore two loops γ1 , γ2 are homotopic if and only if μ(γ1 ) = μ(γ2 ). 39

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40

2.

REVIEW: FLOER COHOMOLOGY

In fact, there is a purely symplectic description of the Maslov class which we describe now. Let (S, Ω) be a symplectic vector space and let Λ(S, Ω) be the set of Lagrangian subspaces. Let V0 be a given Lagrangian subspace and consider the stratification of Λ(S, Ω) Λ(S, Ω) ⊃ Λ1 (S, Ω; V0 ) ⊃ · · · ⊃ Λn (S, Ω; V0 ) = {V0 } where Λk (S, Ω; V0 ) is the subset of Λ(S, Ω) defined by Λk (S, Ω; V0 ) = {V ∈ Λ(S, Ω) | dim(V ∩ V0 ) ≥ k}

for k = 0, · · ·, n.

It is proven in [Arn67] that Λk (S, Ω; V0 ) is compact and Λ1 (S, Ω; V0 ) is co-oriented and so defines a cycle whose Poincar´e dual is precisely the Maslov class μ ∈ H 1 (Λ(S, Ω), Z). More precisely, Arnold proved the following Proposition 2.1.3. This proposition will be used in our construction of Novikov covering space in the next section. Lemma 2.1.1. There exists a neighborhood U of V0 ∈ Λ(S, Ω), the set U \ Λ1 (S, Ω; V0 ) has exactly n + 1 connected components each of which contains V0 in its closure. Proof. We identify S = T ∗ V0 as a symplectic vector space. Then any V ∈ U is uniquely identified to a graph Graph(dfV ) for a function fV : V0 → R such that fV (0) = 0. fV is a quadratic form. If V ∈ / Λ1 (S, Ω; V0 ) then fV is nondegenerate. We put (2.1.2)

Uk (V0 ) = {Graph(dfV ) | Index fV = k} \ Λ1 (S, Ω; V0 ).

It is easy to see that Uk (V0 ) is connected (and is nonempty if k = 0, · · · , n). The lemma follows.  Proposition 2.1.3. Let (S, Ω) be a symplectic vector space and V0 ∈ (S, Ω) be a given Lagrangian subspace of (S, Ω). Let V1 ∈ Λ(S, Ω) \ Λ1 (S, Ω; V0 ) i.e., be a Lagrangian subspace with V0 ∩V1 = {0}. Consider smooth paths α : [0, 1] → Λ(S, Ω) satisfying (2.1.4.1) (2.1.4.2) (2.1.4.3)

α(0) = V0 , α(1) = V1 . α(t) ∈ Λ(S, Ω) \ Λ1 (S, Ω; V0 ) for all 0 < t ≤ 1. α(t) ∈ U0 (V0 ) for small t, where U0 (V0 ) is as in (2.1.2).

Then any two such paths α1 , α2 are homotopic to each other via a homotopy s ∈ [0, 1] → αs such that each αs also satisfies the condition (2.1.4). See [Arn67] or [GuiSt77] for the proof of Proposition 2.1.3. In relation to the study of the relative version of the Riemann-Roch formula, it is useful to generalize the Maslov index to a loop of (maximally) totally real √ subspace. A real subspace V ⊂ Cn is called totally real if V ∩ −1V = {0} and dimR V = n. We denote the set of totally real subspaces by R(n). Any totally real subspace V in Cn can be written as V = A · Rn for some A ∈ GL(n, C) and A1 · Rn = A2 · Rn if and only if A−1 2 A1 ∈ GL(n, R). Therefore the set R(n) of totally real subspaces is a homogeneous space R(n) = GL(n, C)/GL(n, R). The following lemma from [Oh95II] is a useful fact for the study of index problem.

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2.1. BORDERED STABLE MAPS AND THE MASLOV INDEX

41

Lemma 2.1.5. Consider the subset  R(n) = {D ∈ GL(n, C) | DD = In } where In is the identity matrix. Then the map  B : R(n) ∼ = GL(n, C)/GL(n, R) → R(n);

A · Rn → A−1 A

is a diffeomorphism with respect to the obvious smooth structures on R(n) and  R(n). Next we specialize Lemma 2.1.5 to the subset of Lagrangian subspaces. Not−1 ing that when A is a unitary matrix we have A = At , At the transpose of A. Therefore we have the following corollary of Lemma 2.1.5.   Corollary 2.1.6. Denote by Λ(n) ⊂ R(n) the image of B restricted to Λ(n) ⊂ R(n), i.e., (2.1.7)

 Λ(n) := {D ∈ U (n) | D = D t }

the set of symmetric unitary matrices. Then B restricts to a diffeomorphism on Λ(n). The following generalizes the above Maslov index to loops of totally real subspaces. Definition 2.1.8. (Generalized Maslov index) Let γ : S 1 → R(n) be a loop. The generalized Maslov index μ(γ) is defined to be the winding number of det ◦B ◦ γ : S 1 → C \ {0}. Now we adopt the terminology of bundle pairs used by de Silva [Sil97] and Katz-Liu [KatLi01]. We will treat two cases separately, one the complex case and the other the symplectic case. Let Σ be an oriented compact surface with boundary ∂Σ. We denote by g the genus of Σ and h be the number of connected components of ∂Σ. We start with the case of complex vector bundles V → Σ. Note that if ∂Σ = ∅, then any complex vector bundle V → Σ is topologically trivial. Definition 2.1.9. A complex bundle pair (V, λ) is a complex vector bundle V → Σ with a (maximally) totally real bundle λ → ∂Σ and with an isomorphism (2.1.10)

V|∂Σ ∼ = λ ⊗ C.

We fix a trivialization Φ : V → Σ × Cn and let ∂1 Σ, · · · , ∂h Σ be the connected components of ∂Σ with boundary orientation induced from Σ. Then due to the i given isomorphism V|∂Σ ∼ : S 1 → R(n). = λ⊗C, Φ(λ⊗R|∂i Σ ) gives rise to a loop γΦ,λ i Setting μ(Φ, ∂i Σ) = μ(γΦ,λ ), we have Proposition 2.1.11. Let (V, λ) be a complex vector bundle pair over (Σ, ∂Σ).  Then the sum hi=1 μ(Φ, ∂i Σ) is independent of the choice of trivialization Φ : V → Σ × Cn depending only on the pair (V, λ). We refer to, for example, [KatLi01] for the details of the proof of this proposition or see the proof below for an analogous proof for the case of symplectic bundle pairs. Proposition 2.1.11 says that the following definition is well-defined.

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42

2.

REVIEW: FLOER COHOMOLOGY

Definition 2.1.12. The Maslov index of the complex bundle pair (V, λ) is defined by h  μ(V, λ) = μ(Φ, ∂i Σ) i=1

where Φ : V → Σ × Cn is a (and so any) trivialization. Next we consider the case of symplectic vector bundles V → Σ: each fiber Vx carries a symplectic inner product ωx depending smoothly on x ∈ Σ. We first note that the symplectic Lie group Sp(2n) = {A ∈ GL(2n, R) | A∗ ω0 = ω0 } has the maximal compact subgroup consisting of the matrices     √ X Y   ∈ GL(2n, R)  X + −1 Y ∈ U (n) . U (n) := −Y X In particular, we have the polar decomposition of A = U P of A ∈ Sp(2n) into  (n). We denote by U = A  and call it the angular part of A. U ∈U It follows from this observation that if ∂Σ = ∅ any symplectic vector bundle on Σ is symplectically trivial as in the case of complex vector bundle. Definition 2.1.13. A symplectic bundle pair is a pair (V, λ) where V → Σ is a symplectic vector bundle and λ → ∂Σ is a Lagrangian subbundle of V|∂Σ . We fix a trivialization Ψ : V → Σ × (R2n , ω0 ). Similarly as in the complex i case, the restriction Ψ(λ|∂i Σ ) gives rise to a loop γΨ,λ : S 1 → Λ(n). We denote i μ(Ψ, ∂i Σ) = μ(γΨ,λ ). Then we have the following: Proposition 2.1.14. Let (V, λ) be a symplectic bundle pair over (Σ, ∂Σ). h Then the sum i=1 μ(Ψ, ∂i Σ) is independent of the choice of symplectic trivialization Ψ : Σ → Σ × Cn . Proof. Let Ψ1 , Ψ2 : V → (R2n , ω0 ) be two trivializations. The map 2n 2n Ψ2 ◦ Ψ−1 1 : (R , ω0 ) → (R , ω0 )

is given by the assignment (x, v) → (x, g(x)v) for a map g : Σ → Sp(2n, R). We write the natural action of Sp(2n, R) on Λ(n) by g · V . Then we have the identity i i g(x)γΨ (x) = γΨ (x) 1 ,λ 2 ,λ

for x ∈ ∂i Σ.

This identity gives rise to μ(Ψ2 , ∂i Σ) = μ(Ψ1 , ∂i Σ) + 2 ind(g|∂i Σ ). Here ind(g|∂i Σ ) is defined to be the degree of the loop det( g |∂i Σ ) : S 1 → U (1)

 where g|∂i Σ is the angular part of g|∂i Σ . Since g|∂Σ = hi=1 g|∂i Σ and g extends to h Σ and so does g, we have i=1 deg( g |∂i Σ ) = 0 by the cobordism invariance of the degree map. This finishes the proof.  This proposition allows us to define the Maslov index μ(V, λ) for the symplectic bundle pair (V, λ).

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Definition 2.1.15. The Maslov index of the symplectic bundle pair (V, λ) is defined by h  μ(V, λ) = μ(Ψ, ∂i Σ) i=1

where Ψ : V → Σ × Cn is a (and so any) trivialization. Now when we are given a smooth map f : (Σ, ∂Σ) → (M, L) with L ⊂ (M, ω) being Lagrangian, we can associate a symplectic bundle pair (f ∗ T M, f |∗∂Σ T L). We define the Maslov index, denoted by μL (f ), of a smooth map f : (Σ, ∂Σ) → (M, L) by μL (f ) = μ(f ∗ T M, f |∗∂Σ T L). We will just denote μ(f ) = μL (f ) when there is no danger of confusion. It is easy to check that μL (f ) is invariant under the homotopy of f . 2.1.2. The moduli space of bordered stable maps. We first recall that a marked semi-stable curve (S, z ) is a pair of (complex) one dimensional variety S and z = (z1 , · · · , zk ), zi ∈ S and zi = zj for i = j where S has at worst ordinary double points as singular points and all zi ’s are away from double points. A marked semi-stable curve (S, z) is stable if and only if the automorphism group of (S, z) is finite (See [KoMa94] for example). We first review several basic facts on bordered Riemann surfaces (cf. [Sep84,91, SeSi89]). Definition 2.1.16. A real marked semi-stable curve is a system (S, τ, z, z ± ) = (S, τ, (z1 , . . . , zk ), ({z1+ , z1− }, . . . , {z+ , z− })) where S is a semi-stable curve (without boundary) and τ : S → S is an antiholomorphic involution such that τ (zi ) = zi τ (zi+ )

=

zi−

for i = 1, . . . , k for i = 1, . . . , 

and that z1 , . . . , zk , z1+ , . . . , z+ , z1− , · · · , z− are all distinct and are nonsingular. Note that (S, τ, z, z ± ) is unchanged when we interchange zi+ and zi− since the pairs {zi+ , zi− } are assumed to be unordered. We define the genus of a real marked semi-stable curve (S, τ, z, z ± ) to be the genus of S. We denote by MR g,k, the set of all isomorphism classes of real marked stable curve of genus g and with (k, ) marked points. For each (S, τ, z, z ± ) ∈ MR g,k, , we consider the set of fixed points of τ , S τ = {z ∈ S | τ z = z}. We note that if S τ is non empty, it is a union of finitely many circles glued at finitely many points. Definition 2.1.17. A marked semi-stable bordered Riemann surface of genus g = 0 and with (k, ) marked points is a real marked semi-stable curve of genus 0 and with (k, ) marked points together with an orientation ori of S τ , assuming that S τ is non empty. We call a marked bordered semi-stable Riemann surface (S, τ, z, z ± , ori), stable if the corresponding semi-stable curve (S, z, z ± ) is stable.

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In case g ≥ 1 (which will not be needed in this paper though), we need two additional assumptions on ori which we will be included in the description below. For each marked semi-stable bordered Riemann surface (S, τ, z, z ± , ori), we select a closed subset Σ ⊂ S with ∂Σ = S τ in the following way: Let Si ⊂ S be an irreducible component with Si ∩ S τ = ∅. The set Si \S τ will have two connected components. We select one of the components, denoted by Σi , so that the induced orientation on ∂Σi = S τ ∩ Si coincides with the given orientation, ori. In case when S τ ∩ Si is connected (which is always the case when g = 0), the existence of Σi will be automatic. Otherwise, we impose the existence of Σi as a part of the definition of bordered Riemann surface.  We take Σ0 , a union of components of S\ i:Si ∩S τ =∅ Σi , so that the set Σ0 ∪  i:Si ∩S τ =∅ Σi is connected, Σ0 ∩ τ (Σ0 ) = ∅ and that the set ⎛ ⎝ Σ0 ∪

⎞



⎛



Σi ⎠ ∪ ⎝τ (Σ0 ) ∪

i:Si ∩S τ =∅

⎞ τ (Σi )⎠

i:Si ∩S τ =∅

is dense in S. In case g = 0, such Σ0 always exists. Otherwise, Σ0 may or may not exist and so we impose that Σ0 exists as a part of the definition of semi-stable curves with boundary. If Σ0 exists, then Σ0 is always unique. We then put Σ = Σ0 ∪



Σi .

i:Si ∩S τ =∅

See Figure 2.1.1.

Figure 2.1.1 Once we have selected Σ, we can and will choose the marked points zi+ , zi− so that zi+ ∈ Σ. From now on, we will denote the marked semi-stable bordered Riemann surface (S, τ, z, z ± , ori) by (Σ, z, z + ). We note that (Σ, z, z + ) determines (S, τ, z, z ± , ori) in an obvious way. Definition 2.1.18. We denote by Mg,k, the set of all isomorphism classes of marked stable bordered Riemann surface of genus g and with (k, ) marked points.

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We can provide a topology on Mg,k, using the topology on the set Mg,k+2 of stable curves. (See [FuOn99II] Section 10 for details). To give some intuition on the above topology defined on Mg,k, , let us consider the topology of the case g = 0, k = 2,  = 1. In this case, the topology on M0,2,1 is induced from the M0,4 . We would like to describe the limit of a sequence (Σj , zj , {zj+ }) ∈ M0,2,1 such that the sequence does not have a smooth limit and the interior marked points approach to the boundary. When this happens, the two marked points zj+ and zj− collide as j → ∞ by definition of the double. Therefore the corresponding sequence of doubles (Sj , τj , zj , {zj+ , zj− }) as elements in M0,4 converges to a stable map that is the union of two irreducible components 1 1 2 2 2 2 + − (S∞ , τ∞ , z∞ , ∅) ∪ (S∞ , τ∞ , z∞ , {z∞ , z∞ }). 1 2 1 2 Here #(z∞ ) = 3 and #(z∞ ) = 1 and S∞ and S∞ are glued at one element from 1 2 2 z∞ and the unique element in z∞ = {z∞ }. Therefore we conclude that the limit of (Σj , zj , {zj+ }) is the half 1 2 + , ∅) ∪ (Σ2∞ , {z∞ }, {z∞ }) (Σ1∞ , z∞

where Σ1∞ and Σ2∞ are glued similarly. See Figures 2.1.2 and 2.1.3.

Figure 2.1.2

Figure 2.1.3 We also remark that in our compactification sphere bubble at the boundary is regarded as the (trivial) disc bubble plus sphere bubble at the interior of this bubbled disc. (See Figure 2.1.4.)

Figure 2.1.4

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From now on, we will consider only the case of g = 0 with one boundary component, and so denote Mk, in place of M0,k, . In other words, Σ does not have holes or the boundary of the domain Σ is a connected union of a finite number of circles. We also write Mk for Mk,0 . Namely Mk is the moduli space of genus zero bordered Riemann surface with k boundary marked points (and no interior marked point). Some discussion on Mk was given in [FuOh97]. Our discussion here basically follows it, although we need a slight modification. Lemma 2.1.19. Mk, has exactly (k − 1)! connected components. Proof. Let (Σ, z, z + ) = (S, τ, z, z ± , ori) ∈ Mk, . The orientation of S R = S τ determines a cyclic order of {z1 , . . . , zk }. There are (k − 1)! ways to cyclically order a set of order k. It is not difficult to see that this choice of cyclic order determines a connected component of Mk, .  Now, we study orientation of the moduli space Mk, . Let us denote by Mreg k, the ± set of all elements (S, τ, z, z , ori) of Mk, such that S is nonsingular. We remark reg that there exists an obvious fibration Mreg k, → Mk whose fiber is a complex variety and so that it suffices to define an orientation of Mk . We denote the upper half plane by H. Definition 2.1.20. Let k ≥ 3. We consider the pair (D2 , (z1 , . . . , zk )) where (z1 , . . . , zk ), zi ∈ ∂D2 respects the cyclic order of ∂D2 = S 1 with respect to the usual counter clockwise orientation on ∂D2 = S 1 . There is a component of Mk that contains this element. We call this the main component and denote it by Mmain . k main,reg reg main We put Mk = Mk ∩ Mk . When k ≥ 3, the main component Mmain,reg is diffeomorphic to Rk−3 . For k readers’ convenience, we recall the proof of this well-known fact. We define a map Mb,main,reg → Mmain,reg by forgetting the last marked point zk , m k−1 (D2 , (z1 , . . . , zk )) → (D2 , (z1 , . . . , zk−1 )). Identifying D2 with the upper half plane H ∪ ∞, we have the identification (∂D2 , (z1 , . . . , zk )) ∼ = (R ∪ ∞, (x1 , · · · , xk )), where x1 < · · · < xk . Then it is easy to see that the above forgetful map defines consists a fiber bundle whose fiber is diffeomorphic to R. Note that Mmain,reg 3 k−3 ∼ R by induction. (In of a single point. Therefore, we conclude Mmain,reg = k is a smooth manifold (Theorem 7.1.44) Subsection 7.1.3 we will prove that Mmain k, for reader’s convenience.) Using this diffeomorphism, we now define an orientation : A fiber of Mmain,reg → Mmain,reg at (D2 , (z1 , . . . zk−1 )) ∈ Mmain,reg of Mmain,reg k k k−1 k−1 for k ≥ 4 can be identified with an open subset of S 1 ∼ = R∪{∞} which is the interval between xk−1 and x1 . Hence each fiber is naturally oriented, which together with main the orientation of Mmain . Thus by induction, we obtain an k−1 induces that of Mk orientation of the main component. Now, let us look at other components of Mk . Each permutation σ : {1, . . . , k} → {1, . . . , k} induces a homeomorphism σ∗ : Mk → Mk defined by σ∗ : (S, τ, (z1 , . . . , zk )) → (S, τ, (zσ(1) , . . . , zσ(k) )).

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2.1. BORDERED STABLE MAPS AND THE MASLOV INDEX

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Denote by σ0 : {1, . . . , k} → {1, . . . , k} the shift σ0 (i) = i + 1 mod k. This (σ0 )∗ : Mreg → Mreg preserves each component of Mreg and in particular the k k k main component. Lemma 2.1.21. (σ0 )∗ : Mmain,reg → Mmain,reg is orientation preserving if and k k only if k is odd. main,reg be the set of all (z1 , . . . , zk ) ∈ (RP 1 )k such that [CP 1 , Proof. Let M k main,reg main,reg /P SL(2, R) ∼ RP 1 , (z1 , . . . , zk )] ∈ Mmain,reg . Note that M . The = Mk m k main,reg main,reg b,main,reg m map (σ0 )∗ : Mk → Mk lifts to M as a P SL(2, R)-invariant map, which is nothing but the shift map on (RP 1 )k . This shift map σ0 has the parity k − 1 (mod 2). This finishes proof of Lemma 2.1.21.  Proposition 2.1.22. There exists a unique orientation on Mreg such that: k (2.1.23.1) It coincides with the orientation of Mmain,reg described as above. k (2.1.23.2) σ∗ : (S, τ, (z1 , . . . , zk )) → (S, τ, (zσ(1) , . . . , zσ(k) )) is orientation preserving if and only if σ is even. Proof. We note that σ∗ (Mmain,reg ) ∩ Mmain,reg = ∅ if and only if σ = σ0i for k k some i, and that σ0 is even is and only if k is odd. Proposition 2.1.22 then follows to other compofrom Lemma 2.1.21 by transferring the orientation on Mmain,reg k nents through appropriate σ∗ . It is obvious that the corresponding orientation satisfies (2.1.23.1) and (2.1.23.2).  In Chapter 8, we will use Proposition 2.1.22 to provide an orientation of the moduli space of stable maps from genus 0 bordered Riemann surface, which we now define below. We like to mention that M.C. Liu has proved orientability of the moduli space Mg,k, for the higher genus case Theorem 3.9 [Liu02]. Let (M, ω, J) be a symplectic manifold (M, ω) together with a compatible almost complex structure J and let L ⊂ (M, ω) be a Lagrangian submanifold. Definition 2.1.24. A genus 0 stable map from a bordered Riemann surface with (k, ) marked points is a pair ((Σ, z, z + ), w) such that (Σ, z, z + ) is a bordered genus 0 semi-stable curve with (k, ) marked points, and w : (Σ, ∂Σ) → (M, L) is a pseudo-holomorphic map such that the set of all ϕ : Σ → Σ satisfying the following properties is finite: (2.1.25.1) (2.1.25.2) (2.1.25.3)

ϕ is biholomorphic. ϕ(zi ) = zi , ϕ(zi+ ) = zi+ . w ◦ ϕ = ω.

Each genus 0 stable map from open curve ((Σ, z, z + ), w) determines a homology class w∗ ([Σ]) ∈ H2 (M, L; Z) in an obvious way. We say that ((Σ, z, z + ), w) is isomorphic to ((Σ , z  , z + ), w ) if there exists a biholomorphic map ϕ : Σ → Σ such that (2.1.26.1) (2.1.26.2)

w = w ◦ ϕ−1 + ϕ(zi ) = zi , ϕ(zi+ ) = z  i .

Definition 2.1.27. (2.1.28.1) Let β ∈ H2 (M, L; Z) and denote by Mk, (β) the set of all isomorphism classes of genus 0 stable maps from bordered Riemann surface with (k, ) marked

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points, ((Σ, z, z + ), w) such that w∗ ([Σ]) = β. We put reg Msing k, (β) = Mk, (β) \ Mk, (β).

(2.1.28.2)

We denote Mmain z , z + ), w) | (Σ, z, z + ) ∈ Mmain k, (β) = {((Σ,  k, , [w] = β}.

We define a topology of the moduli space Mk, (β) in a way similar to Section 10 [FuOn99II]. (See Subsection 7.1.4.) However there is a case which needs special attention: This is the case when k = 0 and β ∈ H2 (M, L) is mapped to the zero element in H1 (L) under the boundary map H2 (M, L) → H1 (L). In this case the standard stable map moduli space turns out not to be compact and requires a compactification including some map that has continuous family of automorphisms. We refer to Section 3.8 in Chapter 3 and Subsection 7.4.1 in Chapter 7 for the relevant discussions on this phenomenon. Knowing this, we state the following basic result. (See Section A1, for the definition of Kuranishi structure.) Theorem 2.1.29. We assume k = 0. Then, the moduli space Mmain k, (β) of stable maps is compact and Hausdorff, and has a Kuranishi structure. Moreover the family of Kuranishi structures over different choices of (k, ; β) are compatible to one another. Theorem 2.1.29 is Propositions 7.1.1 and 7.1.2 (in case  = 0) and is proved in Subsection 7.1.5 (including the case  = 0). We next state the following Theorem 2.1.30 whose proof will be given in Chapter 8. Let L ⊂ M be a relatively spin Lagrangian submanifold: in other words, we assume that there exists a class st ∈ H 2 (M ; Z2 ) which restricts to the second Stiefel-Whitney class of L. We will prove in Section 8.1 in Chapter 8 that Mk, (β) is orientable under this condition. In fact we will prove the following stronger theorem there. We refer to Section 8.1 for the definition of relative spin structure of the embedding L ⊂ M . Theorem 2.1.30. Let L be relatively spin. Then a choice of relative spin structure of L ⊂ M canonically induces an orientation of Mmain k, (β). We next consider the evaluation maps. k  Definition 2.1.31. We define ev : Mmain k, (β) → L × M by

ev((Σ, z , z + ), w) = (w(z1 ), . . . , w(zk ), w(z1+ ), . . . , w(z+ )). k  We would like to regard Mmain k, (β) as a smooth singular chain on L × M . However in general the space Mmain k, (β) is not a smooth manifold (with boundary or corner), and is not of correct dimension. In this regard, the following is an essential transversality result that will be needed for the construction of our obstruction class. We will make the notion ‘virtual fundamental chain’ in this statement precise in the later chapters. (See Propositions 3.5.2, 7.2.35, Definition A1.28.)

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2.2. THE NOVIKOV COVERING AND THE ACTION FUNCTIONAL

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Theorem 2.1.32. There is a family of multisections s = {s(k, ; β)} of the obstruction bundle of the Kuranishi structure given in Theorem 2.1.29 for which s(k,;β) the perturbed moduli space Mmain of the moduli space Mmain k, (β) k, (β) have a (virtual) fundamental chain (of Q coefficient) of dimension n + μL (β) + k + 2 − 3 where μL (β) is the Maslov index of β ∈ H2 (M, L). Furthermore the perturbations are compatible to one another under the gluing map. s(k,;β) Next we describe the boundary of the chain Mmain . k, (β)

Theorem 2.1.33. Let s be as in Theorem 2.1.32. Then the boundary of the s(k,;β) chain Mmain is the virtual fundamental chain of the space of all equivalent k, (β) s(k,;β) classes of elements ((Σ, z, z+ ), w) ∈ Mmain ) with Σ having exactly one k, (β) singular point. s(k,;β) The chain (Mmain , ev) will be used for identifying the obstruction k, (β) cycles for the definition of Floer cohomology later in this book.

2.2. The Novikov covering and the action functional Let (L0 , L1 ) be a pair of connected compact Lagrangian submanifolds of (M, ω) which are transversal. We like to note that we do not assume our Lagrangian submanifolds are connected at the moment. We consider the space of paths Ω = Ω(L0 , L1 ) = { : [0, 1] → M | (0) ∈ L0 , (1) ∈ L1 }. On this space, we are given the action one-form α defined by " (2.2.1)

1

α()(ξ) =

˙ ω((t), ξ(t)) dt

0

for each ξ ∈ T Ω. Using the Lagrangian property of (L0 , L1 ), a straightforward calculation shows that this form is closed. (See Proposition 2.2.8.) The Floer homology theory is a semi-infinite version of Novikov Morse theory of closed oneforms. Note that Ω(L0 , L1 ) is not connected but has countably many connected components. When we work on a particular fixed connected component of Ω(L0 , L1 ), we specify the particular component by choosing a base path which we denote by 0 . We denote the corresponding component by Ω(L0 , L1 ; 0 ) ⊂ Ω(L0 , L1 ). The base path 0 automatically picks out a connected component from each of L0 and L1 as its initial and final points x0 = 0 (0) ∈ L0 , x1 = 0 (1) ∈ L1 . Then Ω(L0 , L1 ; 0 ) is a subspace of the space of paths between the corresponding connected components of L0 and L1 respectively. Because of this we will always assume that L0 , L1 are connected from now on, unless otherwise said. Next we describe some covering space which we call the Novikov covering of the component Ω(L0 , L1 ; 0 ) of Ω(L0 , L1 ). We first start with describing the universal covering space of Ω(L0 , L1 ; 0 ). Consider the set of all pairs (, w) such that  ∈ Ω(L0 , L1 ) and w : [0, 1]2 → M satisfies the boundary condition  w(0, ·) = 0 , w(1, ·) =  (2.2.2) w(s, 0) ∈ L0 , w(s, 1) ∈ L1 for all s ∈ [0, 1].

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Considering w as a continuous path s → w(s, ·) in Ω(L0 , L1 ; 0 ) from 0 and , the fiber at  of the universal covering space of Ω(L0 , L1 ; 0 ) can be represented by the set of path homotopy classes of w relative to the its end s = 0, 1. 2.2.1. The Γ-equivalence. To carry out the idea of Novikov’s Morse theory [Nov81], it is crucial to have some abelian covering of Ω(L0 , L1 ; 0 ). In this regard, the universal covering space is too large. We will now define a smaller covering space of Ω(L0 , L1 ; 0 ) by modding out the space of paths in Ω(L0 , L1 ; 0 ) by another equivalence relation that is weaker than the homotopy. This is the analog to the Novikov covering space of the contractible loop space introduced by Hofer-Salamon [HoSa95] and the 4-th named author [Ono95]. The deck transformation group of this covering space will be shown to be abelian by construction. Note that when we are given two pairs (, w) and (, w ) from Ω(L0 , L1 ; 0 ), the concatenation w#w : [0, 1] × [0, 1] → M defines a loop c : S 1 → Ω(L0 , L1 ; 0 ). One may regard this loop as a map C : S 1 × [0, 1] → M satisfying the boundary condition C(s, 0) ∈ L0 , C(s, 1) ∈ L1 . Obviously the symplectic area of C, denoted by " ω (2.2.3) Iω (c) = C

depends only on the homotopy class of C satisfying (2.2.2) and so defines a homomorphism on π1 (Ω(L0 , L1 ; 0 )), which we also denote by Iω : π1 (Ω(L0 , L1 ; 0 )) → R. Next we note that for the map C : S 1 × [0, 1] → M satisfying (2.2.2), it associates a symplectic bundle pair (V, λ) defined by VC = C ∗ T M, λC = c∗0 T L0  c∗1 T L1 where ci : S 1 → Li is the map given by ci (s) = C(s, i) for i = 0, 1. This allows us to define another homomorphism Iμ : π1 (Ω(L0 , L1 ), 0 ) → Z;

Iμ (c) = μ(VC , λC )

where μ(VC , λC ) is the Maslov index of the bundle pair (VC , λC ). Using the homomorphisms Iμ and Iω , we define an equivalence relation ∼ on the set of all pairs (, w) satisfying (2.2.2). For given such pair w, w , we denote by w#w the concatenation of w and w along , which defines a loop in Ω(L0 , L1 ; 0 ) based at 0 . We introduce the notion of Γ-equivalence borrowing the terminology Seidel used in [Sei97] in the context of Floer homology of Hamiltonian diffeomorphisms. Definition 2.2.4. We say that (, w) is Γ-equivalent to (, w ) and write (, w) ∼ (, w ) if the following conditions are satisfied Iω (w#w ) = 0 = Iμ (w#w ).  0 , L1 ; 0 ) and call the Novikov We denote the set of equivalence classes [, w] by Ω(L covering space.  0 , L1 ; 0 ): this is just There is a canonical lifting of 0 ∈ Ω(L0 , L1 ; 0 ) to Ω(L  0 , L1 ; 0 ) where 0 is the map 0 : [0, 1]2 → M with 0 (s, t) = 0 (t). [0 , 0 ] ∈ Ω(L  0 , L1 ; 0 ) also has a natural base point which we suppress from the In this way, Ω(L notation.

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We denote by Π(L0 , L1 ; 0 ) the group of deck transformations of the covering  0 , L1 ; 0 ) → Ω(L0 , L1 ; 0 ). It is easy to see that the isomorphism class of space Ω(L Π(L0 , L1 ; 0 ) depends only on the connected component containing 0 . The two homomorphisms Iω and Iμ push down to homomorphisms E : Π(L0 , L1 ; 0 ) → R,

μ : Π(L0 , L1 ; 0 ) → Z

defined by μ(g) = Iμ [C]

E(g) = Iω [C],

for any map C : S 1 × [0, 1] → M representing the class g ∈ Π(L0 , L1 ; 0 ). Corollary 2.2.5. The group Π(L0 , L1 ; 0 ) is an abelian group. Proof. By definition of Π(L0 , L1 ; 0 ), the map E × μ : Π(L0 , L1 ; 0 ) → R × Z is an injective group homomorphism. Therefore we conclude that Π(L0 , L1 ; 0 ) is abelian since R × Z is abelian.  We now define the Novikov ring Λ(L0 , L1 ; 0 ) associated the abelian covering  0 , L1 ; 0 ) → Ω(L0 , L1 ; 0 ) as a completion of the group ring R[Π(L0 , L1 ; 0 )]. Ω(L Here R is a commutative ring with unit. Definition 2.2.6. ΛR k (L0 , L1 ; 0 ) denotes the set of all (infinite) sums  ag [g] g∈Π(L0 ,L1 ;0 ) μ(g)=k

such that ag ∈ R and that the following holds: (2.2.7) order.

For each C, the set {g ∈ Π(L0 , L1 ; 0 ) | E(g) ≤ C, ag = 0} is of finite

We put ΛR (L0 , L1 ; 0 ) =



ΛR k (L0 , L1 ; 0 ).

k

We mainly use R = Q. In case R = Q we write Λ(L0 , L1 ; 0 ) in place of ΛQ (L0 , L1 ; 0 ). The ring structure on ΛR (L0 , L1 ; 0 ) is defined by the convolution product ⎞ ⎛ ⎞ ⎛    ⎝ ag [g]⎠ · ⎝ bg [g]⎠ = ag1 bg2 [g1 g2 ]. g∈Π(L0 ,L1 ;0 )

g∈Π(L0 ,L1 ;0 )

g1 ,g2 ∈Π(L0 ,L1 ;0 )

It is easy to see that the term in the right hand side is indeed an element in ΛR (L0 , L1 ; 0 ), i.e., satisfies the finiteness condition (2.2.7) in its definition. Thus ΛR (L0 , L1 ; 0 ) = ⊕k ΛR k (L0 , L1 ; 0 ) becomes a graded ring under this multiplication. We call this graded ring the Novikov ring associated to the pair (L0 , L1 ) and the connected component containing 0 . 2.2.2. The action functional and the Maslov-Morse index. Now for a given pair (, w), we define the action functional  0 , L1 ; 0 ) → R A : Ω(L

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by the formula

" A(, w) =

w∗ ω.

It follows from the definition of Π(L0 , L1 ; 0 ) that the integral depends only on the Γ-equivalence class [, w] and so pushes down to a well-defined functional on  0 , L1 ; 0 ). A straightforward calculation shows the following the covering space Ω(L whose proof we omit.  0 , L1 ; 0 ) → Ω(L0 , L1 ; 0 ) be the Γ-covering Proposition 2.2.8. Let π : Ω(L space and α be the action one-form on Ω(L0 , L1 ; 0 ). Then we have dA = −π ∗ α. An immediate corollary of this proposition and the definition of α is the following characterization of the critical point set of A. Corollary 2.2.9. The set Cr(L0 , L1 ; 0 ) of critical points of A consists of the pairs of the type [p , w] where p is the constant path with p ∈ L0 ∩ L1 and w is as in (2.2.2). Cr(L0 , L1 ; 0 ) is invariant under the action of Π(L0 , L1 ; 0 ) and so forms a principal bundle over a subset of L0 ∩ L1 with its fiber isomorphic to Π(L0 , L1 ; 0 ). We put Cr(L0 , L1 ) =



Cr(L0 , L1 ; 0,i )

0,i

where 0,i runs over the set of base points of connected components of Ω(L0 .L1 ). Next, we assign an absolute Morse index to each critical point of A. In general, assigning such an absolute index is not a trivial matter because the obvious Morse index of A at any critical point is infinite. For this purpose, we will use the Maslov index of certain bundle pair naturally associated to the critical point [p , w] ∈ Cr(L0 , L1 ; 0 ). We call this Morse index of [p , w] the Maslov-Morse index (relative to the base path 0 ) of the critical point. The definition of μ([p , w]) will somewhat resemble that of A. However to define this, we also need to fix a section λ0 of ∗0 Λ(M ) such that λ0 (0) = T0 (0) L0 , λ0 (1) = T0 (1) L1 . Here Λ(M ) is the bundle of Lagrangian Grassmanians of T M  Λ(M ) = Λ(Tp M ) p∈M

where Λ(Tp M ) is the set of Lagrangian subspaces of the symplectic vector space (Tp M, ωp ). Let  Λori (Tp M ) Λori (M ) = p∈M

be the oriented Lagrangian Grassmanian bundle which is a double cover of Λ(M ). In case when L0 and L1 are oriented, we assume that λ0 is a section of Λori (M ) and respects the orientations at the end points.  0 , L1 ; 0 ) be an element whose projection Let [p , w] ∈ Cr(L0 , L1 ; 0 ) ⊂ Ω(L corresponds to the intersection point p ∈ L0 ∩ L1 .

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Using Proposition 2.1.3, we will associate a symplectic bundle pair (Vw , λw ) over the square [0, 1]2 , which will be defined uniquely upto the homotopy. We first choose Vw = w∗ T M . To define λw , let us choose a path αp : [0, 1] → Λ(Tp M, ωp ) satisfying ⎧ p p ⎪ ⎨ α (0) = Tp L0 , α (1) = Tp L1 ⊂ Tp M, (2.2.10) (αp )(t)  Tp L0 , ⎪ ⎩ αp (t) ∈ U0 (Tp L0 ) for small t, where U0 (Tp L0 ) is as in (2.1.2). Proposition 2.1.3 implies that any two such choices of αp are homotopic to each other. Then we consider a continuous Lagrangian subbundle λw → ∂[0, 1]2 of V|∂[0,1]2 by the following formula: the fiber at each point of ∂[0, 1]2 is given as  λw (s, 0) = Tw(s,0) L0 , λw (1, t) = αp (t), (2.2.11) λw (s, 1) = Tw(s,1) L1 , λw (0, t) = λ0 (0, t). It follows from Proposition 2.1.3 that the homotopy type of the bundle pair constructed as above does not depend on the choice of αp either Definition 2.2.12. We define the Maslov-Morse index of [p , w] (relative to λ0 ) by μ([p , w]; λ0 ) = μ(Vw , λw ). We will just denote μ([p , w]) for μ([p , w]; λ0 ) as we are not going to vary the reference section λ0 . We will make a comment in Remark 2.4.45 on the dependence on the choice of λ0 of the grading of the Floer chain module. 2.3. Review of Floer cohomology I: without anomaly In this section, we will study the set of bounded gradient trajectories u : R → Ω(L0 , L1 ; 0 ) of the action functional A : Ω(L0 , L1 ; 0 ) → R for each given transversal pair (L0 , L1 ). Due to noncompactness of the domain R × [0, 1], one needs to put a certain decay condition of the derivatives of u to study the compactness property and the deformation theory of solutions u. This will be achieved by imposing certain boundedness of the trajectories. The boundedness of the trajectories is measured by the symplectic area of u. 2.3.1. The L2 -gradient equation of A. Recall the identity dA = −π ∗ α.  0 , L1 ; 0 ) with T Ω(L0 , L1 ) via the covering If we identify the tangent space T[,w] Ω(L projection, we can write this identity as " 1 ˙ (2.3.1) dA([, w])(ξ) = ω(ξ(t), (t)) dt 0

 0 , L1 ; 0 ). As in the finite dimensional Morse on the Novikov-covering space Ω(L theory, we will study the gradient flow of A in terms of a given “Riemannian  0 , L1 ; 0 ). metric” on Ω(L

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For each given pair (L0 , L1 ) and a pair J0 , J1 of compatible almost complex structures on (M, ω), we consider a path of compatible almost complex structures {Jt }0≤t≤1 on (M, ω) joining them. We denote Jω = Jω (M ) = the space of compatible almost complex structures. P(Jω ) = C ∞ ([0, 1], Jω ) = the space of parameterized compatible almost complex structures. We denote by {Jt }t an element of P(Jω ). We put (2.3.2)

P(Jω ; J0 , J1 ) = {{Jt }t ∈ P(Jω ) | J0 = J0 , J1 = J1 }.

Then for each given {Jt }t , we define an L2 metric on Ω by the formula " 1 ω(ξ1 (t), Jt ξ2 (t)) dt, ξ1 , ξ2 {Jt }t := 0

using the one parameter family of Riemannian metrics on M gt := ω(·, Jt ·). It follows from (2.3.1) that the L2 -gradient equation of the action functional A becomes ⎧ ⎨ du + J du = 0 t (2.3.3) dτ dt ⎩ u(τ, 0) ∈ L0 , u(τ, 1) ∈ L1 , for a map u : R × [0, 1] → M if one considers u as the path τ → u(τ ) in Ω0 (L0 , L1 ) where u(τ ) is the path defined by u(τ )(t) = u(τ, t). In the point of view of analysis, controlling the geometric energy of u defined by "  2  ∂u 2  1  ∂u    E(u; {Jt }t ) := dt dτ   +  2 ∂τ Jt ∂t Jt is essential for the deformation theory of solutions of (2.3.3). On the other hand, we have the following identity " (2.3.4) E(u; {Jt }t ) = u∗ ω for maps u satisfying ∂u ∂u + Jt = 0, ∂τ ∂t as long as {Jt }t is a family of compatible almost complex structures. In the physics language, they would say that (2.3.4) holds on shell. We note that the symplectic ' area u∗ ω is invariant under the homotopy. Now let L0 , L1 be Lagrangian submanifolds of (M, ω) which are transverse. For each given J ∈ Jω , we study the Cauchy-Riemann equation (2.3.3) and define the space of bounded solutions thereof by    "   reg ∗ ( (2.3.5) M (L1 , L0 ; {Jt }t ) = u  u satisfies (2.3.3) and u ω 0, Floer established one-one correspondence between the solution space M(p, q) of (4.3) and that of gradient trajectories satisfying χ˙ − gradg f (χ) = 0 issued at p ending at q. Then using the Morse homology argument from [Mil63, Wit82, Con78], he proves that HF ∗ (Graph( tdf ), oN ; {Jt }t ) is isomorphic to H ∗ (N ; Z2 ). Floer then concludes that this consideration of the cotangent bundle, via the Darboux-Weinstein theorem, implies that HF ∗ (ψ1H (L), L; {Jt }t ) is isomorphic to H ∗ (L; Z2 ) for the case π2 (M, L) = {e}. However we would like to point out that this last step of proof for the general (M, L) satisfying π2 (M, L) = {e} is not completely trivial in that it requires to prove that as ψ = ψ1H converges to identity all the elements of M(ψ(L), L; {Jt }t ) become thin so that the images of them are contained in a Darboux-Weinstein neighborhood. For the Floer case π2 (M, L) = {e}, this is indeed not difficult to show by a little bit of convergence argument of the Floer trajectory space when combined with the uniform C 1 -bound which will follow from the fact that the assumption π2 (M, L) = {e} prevents the bubbling phenomenon. We refer to [Oh96I] for a detailed explanation on this point. We will come back to these points in the next section when we introduce the Bott-Morse version of Floer cohomology and mention its relation to HF ∗ (ψ(L), L; {Jt }t ) for ψ = id. 2.3.3. Bott-Morse Floer cohomology. When one considers the Hamiltonian isotopic pair (L0 , L1 ) with L0 = L,

L1 = ψ1H (L)

for a Hamiltonian H : [0, 1] × M → R, one would hope to be able to use the invariance property of HF (ψ1H (L), L; {Jt }t ) and compute the Floer cohomology HF (ψ1H (L), L; {Jt }t ) by choosing a special type of the pair ({Jt }t , H) as Floer did for the zero section in the cotangent bundle. However when there exist non-constant bubbles around in the picture, one needs to analyze how the bubbles interact with the Floer boundary map. When ψ H is far from the identity or more precisely when ψ1H (L) is far from L, the bubblings are mixed up with the Floer trajectories and hard to de-couple from the latter. On the other hand, when ψ H is C 1 -close to the identity, the Floer boundary map undergoes the thick-thin decomposition: in this decomposition thin trajectories, or more precisely the trajectories whose image are contained in a given Darboux-Weinstein neighborhood of L, correspond to the Morse gradient trajectories when one writes ψ1H = Graph( df ),

H = f ◦ π

in the Darboux neighborhood. All other trajectories, called thick trajectories, are combinations of bubbles and the Morse gradient trajectories. In fact, many nontrivial results concerning symplectic topology of compact Lagrangian embeddings were previously obtained by studying this thick-thin decomposition in the Floer cohomology. See [Chek98, Oh96I] for such applications, and [Fuk97III, Oh96I,

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96II] for the description of the limiting picture as → 0. A precise study of this adiabatic degeneration turns out to be highly non-trivial and technical. One might even hope to study the degenerate case (2.3.14)

H = 0 (or ψ H = id),

Jt ≡ J0 time independent

directly instead of taking the above adiabatic limit. Indeed the definition of Floer cohomology can be generalized to the pair (L0 , L1 ) with clean intersections. In that case, our functional A is an infinite dimensional analogue of Bott-Morse functional [Bot59]. For example, one can directly consider the pair L0 = L1 = L. We refer to Subsection 3.7.5 for the details of this construction. Bott-Morse theory in Floer homology was first carried out by the first named author in the gauge theory setting [Fuk96II] in 1992. (Mrowka’s work [Mro89] which appeared before [Fuk96II] is also closely related to it.) In the Floer theory setting for the Hamiltonian diffeomorphisms, the construction was proposed by Piunikhin [Piu94] and also by Ruan-Tian [RuTi95], and a somewhat different approach (using a similar moduli space to one which had been introduced in [Fuk97III]) was outlined by Piunikhin-Salamon-Schwarz [PSS96]. (According to line 11 page 171 in [PSS96] the analytic detail is not given in [PSS96].) Chapter 7 Sections 7.1 and 7.2 of this book provide materials to work out the analytic details needed to establish this Bott-Morse theory indicated in [Piu94] and [RuTi95]. Recently, Zhu and the second named author [OhZhu07] developed the scale-dependent gluing, which provides the analytic details for the approach adopted in [PSS96], although one could avoid rescaling target manifolds and could write down a proof following the more standard approach of Floer’s gluing but in Bott-Morse setting. A related study in the Lagrangian intersections had been carried out before by Pozniak [Poz99] when there is no bubbling phenomenon around. (See Subsection 7.2.2 for more discussion.) One advantage of directly working with the Bott-Morse setting (2.3.14) is that the bubbling phenomenon or the quantum contributions can be completely decoupled from the classical part of the cohomology. Here we use the singular cohomology instead of the Morse cohomology, while the latter will emerge if one takes the approach of adiabatic degeneration as → 0 for non-zero Morse function f . This enables one to reduce computation of Floer cohomology to a study of holomorphic discs in some favorable cases. This idea has been exploited by Cho and second named author for the computation of Floer cohomology of the Lagrangian torus fibers of Fano toric manifolds [Cho04I, ChOh03], which mathematically proves a mirror-symmetry prediction made by Hori [Hor01] in relation to the A-model on the toric manifolds and its Landau-Ginzburg B-model. Our construction in Chapter 3 of the A∞ algebra associated to a Lagrangian submanifold can be regarded as the Bott-Morse version of the Floer theory (with product structure), where all the objects of Lagrangian submanifolds simultaneously collapse to a single Lagrangian submanifold. In particular its m1 -cohomology, when defined, will be isomorphic to the Floer cohomology HF ∗ (ψ1H (L), L). We refer to Chapter 5 Section 5.3 for the proof of this latter statement. 2.4. Review of Floer cohomology II: anomaly appearance Let Li , i = 0, 1 be two Lagrangian submanifolds of (M, ω), which are transverse to each other. In the beginning we will not impose any conditions on the Lagrangian

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submanifolds Li , but introduce them as we go along when needed. In this section R denotes a commutative ring with unit. 2.4.1. The Floer cochain module. We recall that the critical point set of the action functional A is  0 , L1 ; 0 ) | p ∈ L0 ∩ L1 }. Cr(L0 , L1 ; 0 ) = {[p , w] ∈ Ω(L  0 , L1 ; 0 ) preserves the critical The action of the group Π(L0 , L1 ; 0 ) on Ω(L point set Cr(L0 , L1 ; 0 ): for each g ∈ Π(L0 , L1 ; 0 ) and [p , w] ∈ Cr(L0 , L1 ; 0 ), we have (2.4.1)

g · [p , w] = [p , g#w].

This action induces a natural R[Π(L0 , L1 ; 0 )]-module structure on the free Rmodule generated by Cr(L0 , L1 ; 0 ). This free module is not finitely generated, but it is free and finitely generated as an R[Π(L0 , L1 ; 0 )]-module and its rank is #(L0 ∩ L1 ). On the other hand, the Maslov-Morse index μ([p , w]) provides a natural grading and the action functional A provides a filtration on Cr(L0 , L1 ; 0 ). This makes the R-module becomes a graded filtered module. In particular, the module has a natural non-Archimedean topology. As noted by Novikov [Nov81] and Floer [Flo89I], it is crucial to complete this module with respect to this topology. We now explain this completed module, which we call the (Novikov) Floer cochain module: Definition 2.4.2. For each k ∈ Z, we consider the formal (infinite) sum  (2.4.3) x= a[p ,w] [p , w], a[p ,w] ∈ R. μ([p ,w])=k

(2.4.4.1) We call those [p , w] with a[p ,w] = 0 generators of the sum x and write [p , w] ∈ x. (2.4.4.2) We define the support of x by supp(x) := {[p , w] ∈ Cr(L0 , L1 ; 0 ) | a[p ,w] = 0 in the sum (2.4.3) }. (2.4.4.3) We call the formal sum x above a (Novikov) Floer cochain of degree k, if it satisfies the finiteness condition # (supp(x) ∩ {[p , w] | A([p , w] ≤ λ}) < ∞ for any λ ∈ R. We define CFRk (L1 , L0 ; 0 ) to be the set of Floer cochains of degree k. (2.4.4.4) We then define the Z-graded free R module CFR∗ (L1 , L0 ; 0 ) by CFRk (L1 , L0 ; 0 ). CFR∗ (L1 , L0 ; 0 ) = k

We now describe the graded module structure of CFR∗ (L1 , L0 ; 0 ) over the graded ring, the Novikov ring ΛR (L0 , L1 ; 0 ) = ΛR k (L0 , L1 ; 0 ). k

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Let y =



REVIEW: FLOER COHOMOLOGY

 bg [g] ∈ ΛR (L0 , L1 ; 0 ) and x = a[p ,w] [p , w]. We define  y·x= bg a[p ,w] (g · [p , w])

where g · [p , w] is defined as in (2.4.1). It is not difficult to see that the term in the right hand side defines a Novikov Floer cochain i.e., satisfies the finiteness condition (2.4.4.3) in Definition 2.4.2 and so lies in CFR∗ (L1 , L0 ; 0 ). This action makes CFR∗ (L1 , L0 ; 0 ) a graded ΛR (L1 , L0 ; 0 )-module. Then we would like to define the Floer coboundary map δ0 : CFR∗ (L1 , L0 ; 0 ) → CFR∗+1 (L1 , L0 ; 0 ) satisfying δ0 ◦ δ0 = 0 as proposed in Definition 2.3.12. This map will be defined by assigning the matrix element δ0 ([p , w]), [q , w ] for each given pair [p , w], [q , w ]. As in the case of Floer homology of periodic orbits, one would like this matrix element to be defined using the Floer moduli space of connecting orbits from [p , w] to [q , w ]. While this will be partially the case, the story becomes much more complex (and interesting) for the Lagrangian intersection Floer theory, which is partially responsible for the volume of the present book. 2.4.2. The Floer moduli space. A solution u : R × [0, 1] → M of Equation (2.3.3) naturally defines a path in Ω(L0 , L1 ; 0 ) which can be lifted to an L2  0 , L1 ; 0 ). Each choice [p , w] ∈ gradient trajectory of A to the covering space Ω(L  0 , L1 ; 0 ) of a lifting of the asymptotic path u(−∞, ·) = p lifts the moduli Ω(L (reg (p, q; B; {Jt }t ) to the moduli space of gradient trajectories of A which space M we denote by (reg ([p , w], [q , w ]; {Jt }t ) M where (2.4.5)

[q , w ] = [q , w#B].

Here ‘reg’ in the notation stands for ‘regular’. We would like to point out that for given [p , w], [q , w ] there could be many different choices of B ∈ π2 (p, q) satisfying (2.4.5). The ambiguity can be precisely described in the following: Lemma & Definition 2.4.6. Let B, B  ∈ π2 (p, q) and [p , w] be given. Then  0 , L1 ; 0 ) if and only if [B#B  ] ∈ π1 (Ω(L0 , L1 ; 0 )) [q , w#B] = [q , w#B  ] in Ω(L satisfies (2.4.7)

[B#B  ] ∈ Ker Iω ∩ Ker Iμ .

We say that B and B  are Γ-equivalent in π2 (p, q) and denote B ∼ B  if (2.4.7) holds. We also denote by [B] the corresponding equivalence class of B in π2 (p, q), by G(p, q) = G(L0 , L1 ; p, q) the set of such equivalence classes of ∼. When we are given two critical points [p , w], [q , w ] we define the moduli space M ([p , w], [q , w ]; {Jt }t ) by the union  (reg (p, q; B; {Jt }t ) (reg ([p , w], [q , w ]; {Jt }t ) := M (2.4.8) M (reg

B∈[w#w ]

where [w#w ] ∈ G(p, q) is the equivalence class corresponding to the glued map w#w ∈ M ap(p, q).

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It is clear from Lemma 2.4.6 that for a given pair [p , w], [q , w ] there are exactly #(Ker Iω ∩ Ker Iμ ) different homotopy classes of B ∈ π2 (p, q) satisfying B ∈ [w#w ]. In general #(Ker Iω ∩ Ker Iμ ) could be infinite and so it is not obvious whether the union (2.4.8) is a finite union. However (2.4.8) is actually a finite union by the following proposition and the identity ω(B) = A([q , w ]) − A([p , w]) < ∞. Proposition 2.4.9. Let {Jt }t ∈ P(Jω ) be given and C > 0. Then there are (reg (p, q; B; {Jt }t ) = ∅ and ω(B) ≤ C. only a finite number of B for which M Proof. Suppose to the contrary that there is an infinite sequence Bj whose elements are all different in π2 (p, q), which satisfies ω(Bj ) ≤ C, and for which M(p, q; Bj ; {Jt }t ) = ∅. Let uj ∈ M(p, q; Bj ; {Jt }t ) be any element respectively for each j. Then we have the energy E(uj ; {Jt }t ) = ω(Bj ) ≤ C for all j and (2.4.10)

lim u(τ, ·) = p ,

τ →−∞

lim u(τ, ·) = q .

τ →∞

Using the energy bound and the asymptotic condition, we apply Gromov-Floer compactness theorem to extract a subsequence, again denoted by uj , such that uj converges to a stable broken Floer trajectory from p to q. It is known that this convergence preserves the homotopy class in π2 (p, q) and in particular there must exist some N ∈ N such that Bj = [uj ] = [uN ] = BN for all j ≥ N . This contradicts to the assumption that Bj are mutually different in π2 (p, q). This finishes the proof.  (reg ([p , w], [q , w ]; {Jt }t ) as Therefore we can also define the moduli space M the set of maps u : R × [0, 1] → M satisfying (2.4.11.1) (2.4.11.2)

u(R × {0}) ⊂ L0 , u(R × {1}) ⊂ L1 , u satisfies ⎧ ⎪ ⎨ ∂u + Jt ∂u = 0 ∂τ ∂t ⎪ ⎩ lim u(τ, t) = p, lim u(τ, t) = q, τ →−∞

τ →+∞

(2.4.11.3) w#u ∼ w , where w#u is the obvious concatenation of w and u along the constant path p . For each τ0 ∈ R, u(τ, t) → u(τ + τ0 , t) defines an R action on the moduli space (reg ([p , w], [q , w ]; {Jt }t ). We put M (2.4.12)

Mreg ([p , w], [q , w ]; {Jt }t ) =

(reg ([p , w], [q , w ]; {Jt }t ) M . R

Unlike the case where π2 (M, L) = {e}, the space Mreg ([p , w], [q , w ]; {Jt }t ) will not satisfy the properties stated in Definition 2.3.12 in general because of the presence of bubbling phenomena. There are two different aspects on what the bubbling phenomenon affects on the Floer cohomology theory:

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(2.4.13.1) One comes from the phenomenon of bubbling off multiple covers of negative holomorphic spheres or discs. This is the phenomenon which already exists in the Floer cohomology of periodic orbits or of the closed strings. (2.4.13.2) The other is the unique phenomenon of anomaly appearance δ0 ◦ δ0 = 0 for the original Floer coboundary map δ0 in the case of open strings. This is closely related to the fact that bubbling off a disc is a phenomenon of codimension one, while bubbling off a sphere is that of codimension two in general. As well-known by now, the problem (2.4.13.1) of bubblings of negative Chern number or of negative Maslov index can be treated by considering the Kuranishi structure and the perturbation theory of multi-valued sections. In Floer theory, we need to work in the chain level of this virtual moduli cycle machinery, unlike the case of Gromov-Witten invariants for which only the homology level theory is needed. In a broad sense, a large part of difficulties present in our geometric construction of various moduli objects in this book are due to (2.4.13). On the other hand, even when we consider the case where the phenomenon (2.4.13.1) does not occur as in the semi-positive case, the anomaly appearance (2.4.13.2) cannot be ruled out. This is largely responsible for our introduction of A∞ algebras associated to Lagrangian submanifolds and their deformation theory. The following simple example already illustrates the “instanton effects” that cause an anomaly on the coboundary property of Floer coboundary map. A similar example was already looked at in [Oh93]. Example 2.4.14. Consider (M, ω) = (C, ω0 ) and √ L0 = R + −1 · 0, L1 = S 1 = ∂D2 (1). Both L0 and M are not compact. However one can conformally compactify C to P1 ∼ = S 2 so that both R and S 1 become equators. L0 and L1 intersect at two points which we denote p = (−1, 0), q = (1, 0).

Figure 2.4.1 We now look at the moduli spaces M(p, q), M(q, p) and M(p, p) respectively. It is easy to see that π2 (p, q) = π2 (L0 , L1 ; p, q) is a principal homogeneous space of π2 (C, S 1 ) ∼ = π1 (S 1 ). Denote by B1 ∈ π2 (p, q) the homotopy class represented by the obvious upper semi-disc. Similarly we denote by B2 ∈ π2 (q, p) the class represented by the lower semi-disc. Then we denote by B ∈ π2 (p, p) the homotopy class given by B = B1 #B2 . By a simple Maslov index calculation, we derive μ(p, q; B1 ) = μ(q, p; B2 ) = 1 and so μ(p, p; B) = 2. By a simple application of the Riemann mapping theorem with boundary, we prove that M(p, q; B1 ) has the unique element which is represented by u1 : R × [0, 1] → C whose image is the obvious upper semi-disc. Similarly M(q, p; B2 ) has

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the unique element whose image is the lower semi-disc. And by a simple linear analysis of Riemann-Hilbert problem, one can also prove that these maps are also regular in that its linearization map is surjective. By a dimensional consideration, we derive the formula δ0 ([p , w]) = [q, w#B1 ],

δ0 ([q , w ]) = [p, w #B2 ]

for all bounding discs w of p and w of q . Now we analyze the moduli space M(p, p; B). Considering each element of ( p; B), M(p, p; B) as the unparameterized curve corresponding to an element of M(p, we can prove by the Riemann mapping theorem that it consists of the holomorphic maps u : R × [0, 1] → C for −1 <  < 1 that satisfy u(R × {0}) ⊂ R × {0}, u(R × {1}) ⊂ ∂D2 , [u ] = B u (−∞, ·) = u (∞, ·) = p , u (0, 0) = (, 0).

Figure 2.4.2 By the symmetry consideration, we derive that u must also satisfy u (0, 1) = (1, 0). One can show that the above family indeed comprise all the elements of M(p, p; B) and all of these maps are Fredholm regular. From this description, a natural compactification of M(p, p; B) is the one obtained by adding the broken trajectory u1 #u2 which corresponds to the limit as  → 1, and the other end in the limit as  → −1. We note that the map u satisfies (2.4.15.1) u (R × {0}) ⊂ {(t, 0) | −1 < t ≤ } and in particular as  → −1 the u |R×{0} converges to the constant map (−1, 0). (2.4.15.2) The image of u |R×{1} wraps around the boundary ∂D2 exactly once. One can also check that on any compact subset K ⊂ R×[0, 1]\{(0, 1)}, we have du ∞,K → 0 as  → −1, and |du (0, 1)|  ∞. From this analysis, we conclude that the real scenario behind the above picture as  → −1 is appearance of the

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stable trajectory corresponding to the following singular curve (u∞ , R × [0, 1], {(0, 1)}) ∪ (w, D2 , {pt}) where (2.4.16.1) (u∞ , R × [0, 1], (0, 1)) is the principal component u∞ : R × [0, 1] → C is the constant map p = (−1, 0) with its domain given by (R × [0, 1], {(0, 1)}) ∼ = (D2 \ {±1}, {i}), (2.4.16.2) w : (D2 , ∂D2 ) → (C, L1 ) is the obvious inclusion map with its domain being the disc D2 with one marked point on the boundary ∂D2 . We note that this configuration is an admissible stable trajectory: u∞ is stable because its domain has 3 special points on the boundary, and w is stable because it is a non-constant map and so both maps have trivial automorphism groups. This analysis of M(p, p; B) together with (2.4.15) computes the matrix element (δ0 ◦ δ0 )([p , w]), [p , w#B] = 1, and

(δ0 ◦ δ0 )([p , w]), [p , w#B  ] = 0 for any other B  ∈ π2 (p, p) with B  = B. This clearly shows δ0 ◦ δ0 = 0. This example illustrates that the boundary of M(p, p; B) with μ(B) = 2 can have a boundary component which is not of the type of broken trajectories. And we note that the Maslov index of the disc w above is two. We will see later that this latter fact is not coincidental. As indicated above, a compactification of the space Mreg ([p , w], [q , w ]; {Jt }t ) requires to add an object consisting of the union of Floer trajectories and holomorphic spheres or discs. We will describe this compactification in Subsection 3.7.4. We write M([p , w], [q , w ]; {Jt }t ) the compactified moduli space. We define a topology on it in a similar way as done in [Fl088II, FuOn99II] which makes our moduli space M([p , w], [q , w ]; {Jt }t ) a compact Hausdorff space whose induced topology on Mreg ([p , w], [q , w ]; {Jt }t ) is the strong C ∞ topology. We refer to Subsection 7.1.4 or [FuOn99II] for a detailed description of the topology. At this stage, we would like to emphasize that this compactification is defined as a topological space for any choice of {Jt }t for a transversal pair L0 , L1 of Lagrangian submanifolds. The topological space M([p , w], [q , w ]; {Jt }t ) will not be a smooth manifold even for a generic choice of {Jt }t in case the pair (L0 , L1 ), in general. However there is a particular class of Lagrangian submanifolds, that of monotone Lagrangian submanifolds, for which the moduli space M([p , w], [q , w ]; {Jt }t ) becomes a smooth manifold with corners of the expected dimension for a dense set of {Jt }t ∈ P(Jω )reg (L0 , L1 ) ⊂ P(Jω ) at least for strata of dimension ≤ 2. We will discuss this case in Subsection 2.4.5. 2.4.3. The Novikov ring ΛR (L). As Example 2.4.14 illustrates, a proper description of a compactification of the Floer moduli space Mreg ([p , w], [q , w ]) in general requires a study of pseudo-holomorphic spheres and discs with boundary lying either on L0 or L1 . As usual by now, we will use a stable map type compactification of Mreg ([p , w], [q , w ]). This requires a careful study of the moduli

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space of pseudo-holomorphic discs with Lagrangian boundary condition provided in Subsection 2.1.2. To encode the effects of the pseudo-holomorphic discs on the coboundary map of Floer cochain complex, we also need to study the Novikov ring associated to Lagrangian submanifolds. The definition of this Novikov ring is now in order. Let L ⊂ M be a compact Lagrangian submanifold. We recall that there are natural group homomorphisms Iω : π2 (M, L) → R,

Iμ : π2 (M, L) → Z :

For each given A ∈ π2 (M, L), the values of these are defined by the symplectic area ω(w) and the Maslov index μL (w) of a map w : (D2 , ∂D2 ) → (M, L) representing A. These homomorphisms factor through the Hurewitz map π2 (M, L) → H2 (M, L) by definition. Definition 2.4.17. We define the equivalence relation β ∼ β  in π2 (M, L) by setting ω(β) = ω(β  ) and μL (β) = μL (β  ), and denote the set of equivalence classes by Π(L) = Π(M ; L). We will abuse the notation β also for the elements of Π(L). As in the case of Π(L0 , L1 ; 0 ), it follows that Π(L) is a free abelian group of its rank less than the rank of H2 (M, L). Therefore we can define a completed ring of the group ring R[Π(L)] in the same way as ΛR (L0 , L1 ; 0 ) for the pair (L0 , L1 ). We denote this by ΛR (L) and call the Novikov ring of L. We remark that in later chapters, we use universal Novikov ring Λ0,nov rather than ΛR (L) or ΛR (L0 , L1 ; 0 ). The relation between them will be explained in Section 5.1. We note that there exist natural group homomorphisms Π(Li ) → Π(L0 , L1 ; 0 ), i = 0, 1.

(2.4.18)

We will just describe this for i = 0 since the one for i = 1 is entirely similar. To  0 , L1 ; 0 ) define this homomorphism, it suffices to find an action of Π(L0 ) on Ω(L  0 , L1 ; 0 ) → Ω(L0 , L1 ; 0 ). Let β ∈ Π(L0 ) that is compatible with the covering Ω(L and wβ : (D2 , ∂D2 ) → (M, L0 ) be its representative. We define the map  0 , L1 ; 0 ) → Ω(L  0 , L1 ; 0 ) gβ : Ω(L by the formula gβ ([p , w]) = β · [p , w] := [p , w#wβ ] where w#wβ is the gluing of wβ to the w|[0,1]×{0} : More precisely, we take any path joining 0 (0) and wβ (1) on L0 and use this path to define a boundary connected sum w#wβ . It is easy to see that the Γ-equivalence class of (, u#wβ ) is independent of the choice of the path and the representatives wβ and u. We note that the following diagrams commute g→g·[0 ,w]  0 ,L1 ; 0 ) Π(L0 ,L1) −−−−−−−→ Ω(L ⏐ ⏐ ⏐ ⏐ A−A([0 ,w]) E

R

=

R

g→g·[0 ,w]  0 ,L1 ; 0 ) Π(L0 ,L1) −−−−−−−→ Ω(L ⏐ ⏐ ⏐μ ⏐μ−μ([ ,w]) 0  

Z

Diagram 2.4.1

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Z

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respectively. This proves that this map is well-defined and respects the covering  0 , L1 ; 0 ) → Ω(L0 , L1 ; 0 ). Obviously gβ is a bijective map and satisfies map Ω(L gβ1 β2 = gβ1 · gβ2 for all β1 , β2 ∈ Π(L0 ). This finishes the construction of the homomorphism (2.4.18). Example 2.4.19. We first consider the Clifford torus T n ⊂ Pn T n = {[z0 : z1 : · · · : zn ] ∈ Pn | |zi | = 1}. This is an example of monotone Lagrangian submanifolds considered in [Oh93]. It is easy to check that π2 (Pn , T n ) splits π2 (Pn , T n ) ∼ = π1 (T n ) ⊕ π2 (Pn ). If we denote by βi the class of the disc √ −1θi0

Di = {[z0 : z1 : · · · : zn ] ∈ Pn | zj ≡ e

fixed except for j = i},

then we have the relation β0 + · · · + βn = α

in π2 (Pn , T n )

where α is the (positive) generator of π2 (Pn ). Furthermore it follows that (2.4.20.1) (2.4.20.2)

c1 (α) , n+1 ω(α) = 2π. ω(βi ) = n+1 μ(βi ) = 2 =

Therefore all βi define the same element, which we denote by β, in Π(Pn ; T n ) and α = (n + 1)β. Hence we have Π(Pn ; T n ) ∼ = Z. Next we consider other standard tori given by T(c0 ,··· ,cn ) = {[z0 : · · · : zn ] ∈ Pn | |zi | = ci > 0} and denote by βi and α the classes defined similarly as in the above case where c0 = · · · = cn . Now (2.4.20.1) still holds but (2.4.20.2) is no longer the case. Depending on the rational dependence of the numbers {ω(β0 ), · · · , ω(βn ), ω(α) = 2π} the ranks of Π(Pn ; T(c0 ,··· ,cn ) ) are changing between 1 and n + 1. Let J be a compatible (time-independent) almost complex structure on M . For each given β ∈ Π(M ; L), we define the moduli space of pseudo-holomorphic discs attached to L. Definition 2.4.21. We define ( β; J) = {w : D2 → M | w is J holomorphic, w(∂D2 ) ⊂ L, [w] = β}. M(L; ( ( β; J) in case no confusion can occur. However we We write M(β) for M(L; would like to emphasize that the almost complex structure J may vary for a different choice of the Lagrangian submanifold L.

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( β; J) by We remark that the group P SL(2; R) = Aut(D, jD ) acts on M(L; −1 ϕ · w = w ◦ ϕ . We put ( β; J)/P SL(2; R). M(L; β; J) = M(L; This moduli space will not be a smooth manifold even for a generic choice of J for a Lagrangian submanifold again due the phenomenon of multiple cover of spheres or discs of negative indices. In Subsection 2.1.2 (and more in Subsection 7.1.4), we have provided a description of the moduli space of stable maps (with marked points) which will be used to overcome the latter phenomenon similarly as in the moduli space of stable maps from a closed Riemann surface. 2.4.4. Monotone Lagrangian submanifolds. In this subsection, we will restrict to a special class of Lagrangian submanifolds studied by the second named author in [Oh93,96I] in his study of Floer homology and its application to the symplectic topology of such Lagrangian submanifolds. We begin with the definition of monotone Lagrangian submanifolds used in [Oh93], which is in turn the analogue to monotone symplectic manifolds that Floer [Flo89I] used for the periodic orbit problem. Let Iω , Iμ : π2 (M, L) → R, Z respectively which are the homomorphisms given in Subsection 2.4.3. Definition 2.4.22. A compact Lagrangian submanifold L ⊂ (M, ω) is called monotone if Iω = λIμ for some λ ≥ 0. We define the minimal Maslov number Σ(L) to be the positive generator of the image of Iμ . Note that this case includes the so called weakly exact case i.e., the case of Iω = 0 in a trivial way. This case can be studied in the exactly same way as in the Floer’s case π2 (M, L) = {0}, if one ignores the grading problem. Therefore we assume that L ⊂ M is monotone with λ > 0 and Iμ = 0 from now on. The RP n ⊂ CP n or the Clifford torus considered in Example 2.4.19 are examples of monotone Lagrangian submanifolds. As pointed out in [Oh93], it is easy to see that if M allows any monotone Lagrangian submanifold, (M, ω) itself must be a monotone symplectic manifold in the sense of Floer [Flo89I]: a symplectic manifold (M, ω) is called monotone if there exists λ > 0 such that " c1 (α) = λ ω α

for any homology class α ∈ H2 (M, Z) in the image of the Hurewitz homomorphism π2 (M ) → H2 (M, Z). The following proposition illustrates some special symplectic topology of this class of Lagrangian submanifolds against the general ones. Proposition 2.4.23. Let L ⊂ (M, ω) be a monotone Lagrangian submanifold. Then Π(L) is a free abelian group of rank 1 and its associated Novikov ring ΛR (L) becomes a field. The following theorem can be proven by a dimension counting argument based on the Sard-Smale theorem. We refer readers [Oh93] for the details of its proof.

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For J, J  ∈ Jω we put P(Jω ; J, J  ) = {{Jt }t ∈ P(Jω ) | J0 = J, J1 = J  }. Theorem 2.4.24. (Oh) We assume that L0 , L1 are monotone with non-zero minimal Maslov number. Then there exists a subset Jω0 (L0 , L1 ) ⊂ Jω and a subset P 0 (Jω ; J0 , J1 ) ⊂ P(Jω ; J0 , J1 ) for J0 , J1 ∈ Jω0 (L0 , L1 ), such that for any {Jt }t ∈ P 0 (Jω ; J0 , J1 ) the followings hold: (2.4.25.1) For all [p , w], [q , w ] ∈ Cr(L0 , L1 ) with μ([q , w ]) = μ([p , w]), the moduli space M([p , w], [q , w ]; {Jt }t ) is empty unless [p , w] = [q , w ]. (2.4.25.2) For all [p , w], [q , w ] ∈ Cr(L0 , L1 ) with μ([q , w ]) = μ([p , w]) + 1, the moduli space M([p , w], [q , w ]; {Jt }t ) is a compact manifold and so is a finite set. (2.4.25.3) If μ([q , w ]) < μ([p , w]), then M([p , w], [q , w ]; {Jt }t ) = ∅. This theorem implies that for a generic choice of {Jt }t , the hypotheses (2.3.12.1) and (2.3.12.2) of Definition 2.3.12 hold. This allows us to be able to at least define the Floer coboundary map δ0 for any monotone pair for such a generic choice of {Jt }t . Let us make this statement more precise incorporating the effects of the deck  0 , L1 ; 0 ) and on its transformation Π(L0 , L1 ; 0 ) on the Novikov covering space Ω(L Morse theory. Another important issue is that of coherent orientation on our moduli space M([p , w], [q , w ]; {Jt }t ). We will prove in Subsection 8.1.3 Theorem 8.1.14 of Chapter 8 that if the pair (L0 , L1 ) is relatively spin, then there exists a canonical way to put a coherent orientation on the collections of the moduli spaces M([p , w], [q , w ]; {Jt }t ). Therefore if the pair (L0 , L1 ) is also relatively spin then integer n([p , w], [q , w ]) = #(M([p , w], [q , w ]; {Jt }t )) for μ[q , w ] = μ([p , w]) + 1 is well-defined. We use it as a matrix element to define the standard Floer coboundary map δ0 : CFRk (L1 , L0 ; 0 ) → CFRk+1 (L1 , L0 ; 0 ) as follows.  (2.4.26) δ0 ([p , w]) = n([p , w], [q , w ]) · [q , w ]. μ(q ,w )=μ(p ,w)+1

Here R = Z in case (L0 , L1 ) is a relatively spin pair and R = Z2 in general. Proposition 2.4.9 implies that the right hand side is in CFR∗ (L1 , L0 ; 0 ). Refining Floer’s proof in [Flo88IV], the second named author [Oh93] proved that δ0 ◦ δ0 = 0 in the case of monotone Lagrangian submanifolds with the minimal Maslov number > 2 with some topological restriction on the pair (L0 , L1 ). This latter topological restriction will not be needed once we use the Novikov Floer cochain modules CFR∗ (L1 , L0 ) as defined in this book. (See Theorem 2.4.42.) However the restriction on Maslov index turns out to be something that cannot be entirely removed even with the usage of Novikov rings as Example 2.4.14 illustrates. We will amplify this example by relating the matrix element (δ0 ◦ δ0 )([p , w]), [q , w ]

with certain singular chains on L0 or on L1 or on both in the next subsection. This non-zero contribution arises from the presence of extra boundary components of the compactification M([p , w], [q , w ]; {Jt }t ) when μ([q , w ]) − μ([p , w]) = 2 other than the broken trajectories as illustrated by Example 2.4.14.

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2.4.5. Appearance of the primary obstruction. We go back to the general pair (L0 , L1 ) for the moment. We first describe the complement of the smooth moduli space Mreg ([p , w], [q , w ]) in its compactification M([p , w], [q , w ]). We consider Msing ([p , w], [q , w ]) := M([p , w], [q , w ]) \ Mreg ([p , w], [q , w ]). For the study of boundary of M([p , w], [q , w ]), we need to describe the stratum “codimension one”, that is the top dimensional stratum of the singular locus Msing ([p , w], [q , w ]). We write it as ∂M([p , w], [q , w ]). In a transversal case, this stratum will be given by either a broken trajectory with two components or a trajectory with a disc attached to its boundary. We denote ◦

◦

M1,0 ([p , w], [q , w0 ]) ∼ = M0,1 ([p , w], [q , w0 ]) (reg ([p , w], [q , w ]). =M

(2.4.27)

0

◦

This is a special case of Definition 3.7.24, where Mk1 ,k0 ([p , w], [q , w0 ]) is defined. ◦

◦

We define a map ev : M1,0 ([p , w], [q , w0 ]) → L1 , or ev : M0,1 ([p , w], [q , w0 ]) → L0 , by ev(u) = u(0, 1),

(2.4.28)

or

ev(u) = u(0, 0),

respectively. (We remark that when we reagard u as an element of the moduli ◦

space M1,0 ([p , w], [q , w0 ]), the point (0, 1) ∈ R × [0, 1] is regarded as a marked point. On the other hand when we reagard u as an element of the moduli space ◦

M0,1 ([p , w], [q , w0 ]), the point (0, 0) ∈ R × [0, 1] is regarded as a marked point.) Definition 2.4.29. Let [p , w], [q , w ] ∈ Cr(L0 , L1 ) and β(0) ∈ Π(L0 ) and β(1) ∈ Π(L1 ). For each i = 0, 1, we put Int N ([p , w], [q , w ] : L1 , β(1) ) =

 β(1) ,w1 ;w1 #β(1) =w

◦

M1,0 ([p , w], [q , w1 ])ev ×ev0 M1 (L1 ; β(1) ; J1 ) , R

Int N ([p , w], [q , w ] : L0 , β(0) ) =

 β(0) ,w0 ;w0 #β(0) =w

◦

M0,1 ([p , w], [q , w0 ])ev ×ev0 M1 (L0 ; β(0) ; J0 ) . R

We define the moduli space N ([p , w], [q , w ] : L1 , β(1) ) as the closure of the moduli space Int N ([p , w], [q , w ] : L1 , β(1) ) in M([p , w], [q , w ]). The definition of N ([p , w], [q , w ] : L0 , β(0) ) is similar. (See Figure 2.4.3.)

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Figure 2.4.3: N ([p , w], [q , w ] : L1 , β(1) ) As we pointed out before, we would like to say that N ([p , w], [q , w ] : Li , β(i) ) is of codimension one. This statement, in a naive sense, will not make sense even for a generic choice of {Jt }t in general. (It does make sense in general if we regard them as spaces with Kuranishi structures.) However this is really the case if one can choose such {Jt }t and if (L0 , L1 ) is monotone. We restrict ourselves to monotone case here. We recall that a pseudo-holomorphic map u : Σ → M (from a Riemann surface Σ which may or may not be bordered) is said to be somewhere injective, if there exists p ∈ Σ and a neighborhood U of p such that u(p) ∩ u(Σ \ U ) = ∅, u is an immersion at p.

(2.4.30.1) (2.4.30.2)

We recall that the following facts hold for generic J ∈ Jω [McD87]. (2.4.31.1) If ∂Σ = ∅ then any pseudo-holomorphic map u : Σ → M decomposes u = u ◦ π where π : Σ → Σ is a branched covering and u : Σ → M is somewhere injective. (2.4.31.2) If u : Σ → M is somewhere injective then the moduli space of pseudoholomorphic maps is smooth and of correct dimension in a neighborhood of u. (We do not use somewhere-injectivity except this section. So the reader can skip the rest of this section and proceed to the next chapter if he is not familiar with somewhere-injectivity.) Theorem 2.4.32. Suppose that L0 , L1 are monotone, J0 , J1 ∈ Jω0 (L0 , L1 ) and {Jt }t ∈ P 0 (Jω ; J0 , J1 ) are as in Theorem 2.4.24. We assume that μ([q , w ]) − μ([p , w]) ≤ 2.

(2.4.33)

Then N ([p , w], [q , w ] : Li , β(i) ) for both i = 0, 1 are smooth manifolds of dimension μ([q , w ]) − μ([p , w]) − 2. Proof. This will follow from a simple dimension counting of the fiber product in Definition 2.4.29. We consider only the case i = 1. By definition, N ([p , w], [q , w ] : L1 , β(1) ) is the fiber product ◦

(2.4.34)

M1,0 ([p , w], [q , w1 ])ev ×ev0 M1 (L1 ; β(1) ; J1 ) (ev × ev0 )−1 (Δ) = . R R

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◦

Here Δ ⊂ L1 × L1 is the diagonal and M1,0 ([p , w], [q , w1 ]) is as in (2.4.27). We recall: ◦ vir dim M1,0 ([p , w], [q , w ]) = μ([q , w ]) − μ([p , w]), vir dim M1 (L1 , J1 ; β(1) ) = n + μ(β(1) ) − 2 and

μ(β(1) ) = μ([q , w ]) − μ([q , w ]). By monotonicity, we find that μ(β(1) ) > 0 if M(L1 ; β(1) ; J1 ) = ∅. (Note β(1) = 0.) Hence, (2.4.33) implies μ([q , w ]) − μ([p , w]) ≤ 1. We first consider the case when μ([q , w ]) − μ([p , w]) = 1. Then, by Theorem 2.4.24, the moduli space ◦

M1,0 ([p , w], [q , w1 ]) ⊆ M([p , w], [q , w ]) × R is smooth and of dimension μ([q , w ]) − μ([p , w]) = 1. It follows that every element of M(L1 ; β(1) ; J1 ) is somewhere injective. In fact if v ∈ M(L1 ; β(1) ; J1 ) is not somewhere injective, then  [KwOh00, Lazz00] imply that there exists vi (i = 1, · · · , a, a ≥ 2) such that [vi ] = [v] and M(L1 ; [vi ]; J1 ) is nonempty. This is impossible since μL1 (vi ) > 0. Hence M1 (L1 ; β(1) ; J1 ) is smooth and of dimension n − 1. Then by an argument which is by now standard, we can use somewhere-injectivity to show that the evaluation map ◦

ev × ev0 : M1,0 ([p , w], [q , w1 ]) × M1 (L1 ; β(1) ; J1 ) → L1 × L1 is transverse to Δ for generic J0 , J1 , {Jt }t . Therefore, the moduli space (2.4.34) becomes a smooth manifold of dimension μ([q , w1 ]) − μ([p , w]) + n + μ(β(1) ) − 2 − n (2.4.35)

= μ([q , w1 ]) − μ([p , w]) + μ(β(1) ) − 2 = μ([q , w1 ]) − μ([p , w]) − 2,

as required. We next consider the case μ([q , w1 ]) − μ([p , w]) ≤ 0. Let us assume that (2.4.34) is nonempty. Then, Theorem 2.4.24 implies that [q , w1 ] = [p , w] and that ◦

M1,0 ([p , w], [q , w ]) must consist of a stationary element u ≡ p. Moreover if (u, v) ◦

is an element of (2.4.34) (where u ∈ M1,0 ([p , w], [q , w ]) and v ∈ M1 (L1 ; β(1) ; J1 )) then v must pass through p. Furthermore we have μ(β(1) ) ≤ 2. If v is not somewhere injective, then the image of v is contained in a union of vi : (D2 , ∂D2 ) → (M, L1 ) such that μ([vi ]) < μ([v]) ≤ 2 ([KwOh00, Lazz00]). Hence M1 (L1 ; [vi ]; J1 ) is a compact and smooth manifold of dimension dim M1 (L1 ; [vi ]; J1 ) ≤ n − 1. Therefore, by re-choosing J1 if necessary, the image of the evaluation map ev : M1 (L1 ; [vi ]; J1 ) → L1 can avoid the zero dimensional set L0 ∩ L1 . Thus we may assume that v is always somewhere injective if (u, v) is an element of (2.4.34). It follows that M1 (L1 ; β(1) ; J1 ) is a smooth manifold and the fiber product (2.4.34) is transversal. The proof of Theorem 2.4.32 is complete. 

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Therefore we derive Theorem 2.4.36. Let L0 , L1 and {Jt }t be as in Theorem 2.4.32. We assume (2.4.33). Then we have the identity ∂M([p , w], [q , w ])  =

M([p , w], [r , w ]) × M([r , w ], [q , w ])

μ([r ,w ])=μ([q ,w ])−1

+



N ([p , w], [q , w ] : L0 , β(0) ) +

β(0)



N ([p , w], [q , w ] : L1 , β(1) ).

β(1)

Proof. Let ui be a divergent sequence of elements of M([p , w], [q , w ]). By Gromov’s compactness theorem, one of the followings must occur in the limit (after taking a subsequence if necessary): (2.4.37.1) ui splits into a sum of various connecting orbits. (2.4.37.2) Bubbling-off spheres occur at some points of R × [0, 1]. (2.4.37.3) Bubbling-off discs occur at some boundary points of R × [0, 1], i.e., at some points in R × {0, 1}. (2.4.37.1) corresponds to the first term of Theorem 2.4.36 and (2.4.37.3) corresponds to the second and third terms there. (2.4.37.2) is a phenomenon of codimension 2 and so does not occur in the boundary of codimension 1. The last dimension counting argument is allowed by Theorem 2.4.32. Hence the proof.  By the standard cobordism argument, Theorem 2.4.36 implies the identity  0 = (δ0 ◦ δ0 )([p , w]), [q , w ] + #N ([p , w], [q , w ] : L0 , β(0) ) β(0)

(2.4.38) +



#N ([p , w], [q , w ] : L1 , β(1) )

β(1) 

where (δ0 ◦ δ0 )([p , w]), [q , w ] denotes the coefficient of [q , w ] in (δ0 ◦ δ0 )([p , w]). Here # is the order with sign. We do not explain sign (that is the orientation of the moduli space) here, since it will be discussed in detail in later chapters, especially in Chapter 8. Now we would like to study the following two cases separately as in [Oh93, 96I]: (2.4.39.1) (2.4.39.2) 2.

the case where L0 , L1 have the minimal Maslov number > 2, the case where one or both of L0 , L1 have the minimal Maslov number

We start with the case (2.4.39.1). Based on the observation that if β(i) ∈ Π(M ; Li ) satisfies μ(β(i) ) > 2, then N ([p , w], [q , w ] : Li , β(i) ) = ∅ for a generic choice of {Jt }t when μ([q , w ]) − μ([p , w]) = 2, the second named author proved [Oh93] that the extra terms in (2.4.38) become zero by a dimension counting argument. We recall this argument below in the more current language used in this book. Let us consider only the case i = 1. In [Oh93] the following moduli space ( 1 ; β(1) ) × S 1 )/P SL(2; R) M∂ (L1 ; β(1) ) = (M(L

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2.4. REVIEW OF FLOER COHOMOLOGY II: ANOMALY APPEARANCE

75

and the evaluation map ev ∂ : M∂ (L1 ; β(1) ) → L1 ;

ev ∂ ([w, t]) = w(t)

are considered. Note that the dimension of M∂ (L1 ; β(1) ) is μ(β(1) ) + n − 2. We also remark that M∂ (L1 ; β(1) ) in fact coincides with M1 (L1 ; β(1) ) and ev ∂ coincides with ev0 : M1 (L1 ; β(1) ) → L1 . Therefore if we assume the minimal Maslov numbers of L0 , L1 are greater than 2 and so μ(β(1) ) > 2, we have dim M1 (L1 ; β(1) ) > dim L1 when M1 (L1 ; β(1) ) is transverse and has the correct dimension. In this case, the chain ev0 : M1 (L1 ; β(1) ) → L1 is degenerate and so becomes zero as a chain of dimension dim M1 (L1 ; β(1) ) on L1 , which turns out to be the reason behind the argument below. We recall that (δ0 ◦ δ0 )([p , w]), [q , w ] is nonzero only when μ([q , w ]) − μ([p , w]) = 2. Therefore from the dimension formula (2.4.35) μ([q , w1 ]) − μ([p , w]) + μ(β(1) ) − 2 of the fiber product ◦

M1,0 ([p , w], [q , w1 ])ev ×ev0 M1 (L1 ; β(1) ) (2.4.40) R and from Theorem 2.4.24, we derive its dimension to be μ([q , w1 ]) − μ([p , w]) + μ(β(1) ) − 2 = 0. Therefore we have (2.4.41)

μ([q , w1 ]) − μ([p , w]) = 2 − μ(β(1) ) < 0

due to the hypothesis (2.4.39.1). (2.4.41), (2.4.25.3) implies: ◦

M1,0 ([p , w], [q , w1 ]) = ∅ and so N ([p , w], [q , w ] : L1 , β(1) ) must be empty. This proves the following theorem which was proved by the second named author in [Oh93]. (The statement below is slightly improved. For example R = Z in case L1 , L0 is a relatively spin pair.) Theorem 2.4.42. (Oh) Let L0 , L1 be monotone Lagrangian submanifolds. We assume that their minimal Maslov number are larger than 2. Then the coboundary operator δ0 : CFR∗ (L1 , L0 ; 0 ) → CFR∗+1 (L1 , L0 ; 0 ) defined by (2.4.26) satisfies δ0 ◦ δ0 = 0. Here we put R = Z if L0 , L1 is a relatively spin pair and R = Z2 in general. Hence, in case (2.4.39.1), we define Floer cohomology HF (L1 , L0 ; 0 ) = Ker δ0 / Im δ0 . We can prove that it is independent of Hamiltonian isotopy classes of L0 , L1 . We do not give its detail here since we prove more general results later in Chapter 5. Now we study the case (2.4.39.2) where the minimal Maslov number of L1 or of L0 is 2. In this case (2.4.41) is replaced by μ([q , w1 ]) − μ([p , w]) = 2 − μ(β(1) ) ≤ 0

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76

2.

REVIEW: FLOER COHOMOLOGY

and so we must have μ([q , w1 ])−μ([p , w]) = 0 and μ(β(1) ) = 2. By Theorem 2.4.24, μ([q , w1 ]) − μ([p , w]) = 0 implies that p = q and v must pass through p, where (u, v) is an element of (2.4.40). Furthermore the chain ev0 : M1 (L1 ; β(1) ) → L1 defines an n dimensional compact cycle and so a multiple of the fundamental class [L1 ]. This cycle can indeed be non-zero as Example 2.4.14 illustrates and so the corresponding fiber product can really exist. The outcome is that δ0 ◦ δ0 = 0 can really occur even for the monotone Lagrangian submanifolds for which most of the needed transversality properties can be obtained just by perturbing the almost complex structures. In the case when L1 = ψ(L0 ) for some Hamiltonian diffeomorphism ψ, the equality δ0 ◦ δ0 = 0 holds under the milder assumption that the minimal Maslov number is not smaller than 2. See [Oh93] or Subsection 3.6.3 of this book. One important idea introduced in the present book to define Floer cohomology of Lagrangian submanifold beyond the case of Theorem 2.4.42 is that we impose a homological condition on Li that the chain ev0 : M1 (Li ; β(i) ) → Li , which becomes a cycle when β(i) is of the minimal possible symplectic area, is homologous to zero, and then use a chain bounding this cycle to modify the standard Floer coboundary map δ0 to δ in order to achieve δ ◦ δ = 0. It turns out that a coherent explanation on the this dependence on the parameters together with the transversality issues requires a bulk of algebraic machinery of new homological algebra and deformation theory of filtered A∞ algebra. Remark 2.4.43. Using the terminology of filtered A∞ bimodule introduced in Section 3.7, Theorem 2.4.36 can be written as: (2.4.44)

(n0,0 ◦ n0,0 )([p , w]), [q , w ] = ± n1,0 (m0 (1), [p , w]), [q , w ]

± n0,1 ([p , w], m0 (1)), [q , w ] .

Namely the second and the third terms of (2.4.38) is the right hand side of (2.4.44). On the other hand, the first term of (2.4.38) is the left hand side of (2.4.44). We remark that (2.4.44) follows from the equality (n0,0 ◦ n0,0 )(x) ± n1,0 (m0 (1), x) ± n0,1 (x, m0 (1)) = 0, which is a special case of the defining equation of the filtered A∞ bimodule. Actually it is the Hom(D, D) = Hom(B0 C1 [1] ⊗ D ⊗ B0 C0 [1], D) component of dˆ2 = 0. See Definition 3.7.5 (1). Remark 2.4.45. We remark the role of the based path 0 in our construction. The point of view that our Lagrangian intersection theory should be about the based space of paths, rather than the space of paths will be important in defining the chain homomorphism under the Hamiltonian isotopy of Lagrangian submanifolds (See Section 5.3). The choice of section λ0 of ∗0 Λ(M ) (see Subsection 2.2.2) is also needed to provide an absolute grading of the Floer complex and to define a degree preserving chain homomorphism between the Floer cohomology over the Novikov rings under the Hamiltonian isotopy. We refer to Section 5.1 for more explanations on this point.

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https://doi.org/10.1090/amsip/046.1/03

CHAPTER 3

The A∞ algebra associated to a Lagrangian submanifold 3.1. Outline of Chapter 3 As we explained in the previous chapter, Floer cohomology for Lagrangian intersection can not be defined, in general. This trouble comes from the existence of pseudo-holomorphic discs bubbling off at the boundary of a pseudo-holomorphic strip. When one considers the stable map compactification of moduli spaces of certain pseudo-holomorphic discs or strips to define the “coboundary operator” in the usual way, the bubbling-off discs appear in the real codimension one boundary components. On the other hand, it is known that Floer cohomology of the periodic Hamiltonian system can be defined for arbitrary closed symplectic manifolds ([FuOn99II, LuTi98, Rua99]). In this case, we use the moduli spaces of stable maps of genus zero without boundary to define the coboundary operator, where the bubbling-off spheres appear in the strata of real codimension two (i.e., complex codimension one). Therefore, after we settle transversality argument by using the Kuranishi structure introduced in [FuOn99II], we can prove that these bubbling off spheres do not contribute to the coboundary operator. Thus the theory can be formulated in the (co)homology level. In the Lagrangian intersection case, however, it is necessary to develop our theory in the (co)chain level, because of this existence of the real codimension one strata corresponding to bubbling off discs. For this purpose, we introduce and construct a certain homotopical algebra called a filtered A∞ algebra. Roughly speaking, an A∞ algebra is a graded module ⊕C • with infinitely many operations m = {mk }k=0,1,2,··· ; mk : C ∗ ⊗ · · · ⊗ C ∗ −→ C ∗ , 



(k = 0, 1, 2, · · · )

k times

which satisfy certain relations called the A∞ formula, or the A∞ relation. We call m an A∞ structure. See Definition 3.2.3 (unfiltered A∞ algebra) and Definition 3.2.20 (filtered A∞ algebra) for the precise definitions. We will provide the basic material of A∞ algebras and filtered A∞ algebras in Section 3.2. Historically, J. Stasheff introduced the notion of (unfiltered) A∞ algebras in [Sta63] inspired by M. Sugawara’s work [Sug57], to study the homotopy theory of loop spaces. (See [Ada78, BoVo73] for related works.) One geometric aspect of the A∞ structure is that it describes the codimension one degeneration of trees (contractible 1-dimensional CW complex). Since all the configurations of bubbling off holomorphic discs (genus zero) can be described by trees, the A∞ structure appears naturally in our story. An A∞ algebra can be regarded as generalizations of D.G.A. (differential graded algebra) and has played an important role in the deformation theory (see [GoMi88,90, GugSt86] for example). We will investigate a deformation theory of Lagrangian 77

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

submanifolds and their Floer cohomology in terms of our A∞ algebra. This is the main theme of Chapters 4 and 5. Before we state the main results of this chapter, we like to mention the issue of orientations. The moduli space of pseudo-holomorphic discs with the Lagrangian boundary condition is not always orientable, while it is well known that the moduli space of pseudo-holomorphic maps from a closed Riemann surface is always orientable and has a canonical orientation. Moreover, since we use the stable map compactification of pseudo-holomorphic maps and the Kuranishi structure, the fundamental chain of the moduli spaces are defined over Q, not over Z in general. In particular, we can not work over Z/2Z-coefficients in general and hence can not avoid the orientation problem. These points will be studied in Chapter 8 in detail. Here we introduce the notion of relative spinness which gives a condition for orientability of the moduli space of pseudo-holomorphic discs with Lagrangian boundary condition. The proof will be given in Section 8.1. Definition 3.1.1. (1) Let L be an oriented Lagrangian submanifold in a symplectic manifold M . We call L a relatively spin Lagrangian submanifold, if there exists st ∈ H 2 (M ; Z/2Z) such that st|L = w2 (T L) ∈ H 2 (L; Z/2Z). (2) A pair of oriented Lagrangian submanifolds (L(0) , L(1) ) in M is called a relatively spin pair, if there exists st ∈ H 2 (M ; Z/2Z) such that st|L(0) = w2 (T L(0) ) and st|L(1) = w2 (T L(1) ) simultaneously. For example, if L is spin, it is obviously relatively spin. In particular, when dimR M ≤ 6, every orientable Lagrangian submanifolds is automatically relatively spin. To specify an orientation on the moduli space of pseudo-holomorphic discs with a Lagrangian boundary condition, we need some more data, which we call relative spin structure. See Definition 8.1.2 in Chapter 8. Throughout this chapter, we assume that L is a relatively spin Lagrangian submanifold and (L(0) , L(1) ) is a relatively spin pair, unless otherwise mentioned. The first main result (Theorem 3.5.11) of this chapter is the construction of a filtered A∞ algebra associated to a relatively spin Lagrangian submanifold L which encodes contributions of all pseudo-holomorphic discs attached to L. The obstruction to defining Floer cohomology can be seen in terms of this filtered A∞ algebra. We use universal Novikov ring Λ0,nov (R). See (Conv.4) in Chapter 1. (In the algebraic part of the story we mostly work with Λ0,nov (R) for a commutative ring R with unit such as R = Z or R = Z/2. Those algebraic materials are used in [FOOO09I].) The relationship between this ring and the standard Novikov ring in the literature (and in Chapter 2) will be explained in Section 5.1, Chapter 5. (k) (k) We denote by Λnov and Λ0,nov the grading k parts of them, respectively. When we need to specify the ring R, we write Λnov (R), Λ0,nov (R) etc. (We also write them R as ΛR nov , Λ0,nov sometimes.) λ The rings Λnov , Λ0,nov , Λ+ 0,nov have filtrations F Λnov etc. See (Conv.6), (1.37). Thus Λnov and Λ0,nov become filtered graded rings which are complete. We note that Λnov is a localization of Λ0,nov . We will construct a filtered A∞ algebra over Λ0,nov (Q) associated to a relatively spin Lagrangian submanifold. We will also construct a deformed Floer cohomology with Λ0,nov (Q)-coefficients, when it is possible. In this case the parameter λi geometrically stands for the symplectic area (or the energy) of a pseudo-holomorphic

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79

disc and 2ni for its Maslov index. The ring Λnov (Q) will be used when we study the invariance of Floer cohomology under the Hamiltonian isotopy. We start with the geometric realization of an (unfiltered) A∞ algebra in the classical picture. In this case, we can construct an A∞ algebra over Z since we can use a system of single-valued sections but not of multi-values ones in the construction of associated fundamental chains. (See Section 3.4 for the details.) Let S k (L; Z) be the free Z module generated

by all codimension k smooth singular simplices of L. We regard S(L; Z) = k S k (L; Z) as a cochain complex. Choose a countable set XL of smooth singular simplices and denote by ZXL the free Z submodule of S(L; Z) generated by XL . Then we prove the following theorem in Section 3.4 (Theorem 3.4.8). We take an appropriate smooth simplicial decomposition of L and represent the fundamental class as a smooth singular chain P D[L]. (Here we write P D[L] in place of [L] since we regard a codimension k singular chain as a k cochain. This identification may be regarded as a Poincar´e duality.) Theorem 3.1.2. For any oriented manifold L, there exists XL such that we can construct an A∞ structure denoted by mL over Z on ZXL . It has a homotopy unit P D[L]. Moreover, the cohomology group of (ZXL , mL 1 ) is isomorphic to the cohomology group (over Z-coefficient) of L. We will give the definition of a homotopy unit of an A∞ algebra in Section 3.3 (Definition 3.3.2). Here we have to choose a suitable countable set XL because we need and use the transversality argument and the Baire category theorem. Remark 3.1.3. In Section 4.6, we will prove that the weak homotopy equivalence class of the A∞ algebra (ZXL , mL ) constructed above is uniquely determined by L. Moreover, when we put RXL = ZXL ⊗ R and extend mL to RXL , then (RXL , mL ) is homotopy equivalent to the de Rham D.G.A. of L over R. See Section 7.5 as for more discussions on the relationship between our results and the rational homotopy theory due to Quillen and Sullivan [Qui69, Sull78]. Next, we will construct a filtered A∞ algebra over Λ0,nov (Q) associated to a Lagrangian submanifold L. Denote by X1 (L) a countable set of smooth singular simplices such that X1 (L) ⊃ XL where XL is chosen as in Theorem 3.1.2 above. We sometimes write an element in X1 (L) as P = (|P |, f ). Here |P | is an n − k dimensional simplex and f : |P | → L is a smooth map. We denote by C k (L; Q) a submodule of S k (L; Q) generated by X1 (L). We define C • (L; Λ0,nov ) to be the completion of C • (L; Q) ⊗ Λ0,nov ; C • (L; Λ0,nov ) := (C • (L; Q) ⊗Q Λ0,nov ) with respect to the uniform structure induced by a filtration similar to (1.38). To an element xT λ en ∈ C(L; Λ0,nov ) with x ∈ C(L; Q), we assign a degree defined by deg(xT λ en ) := deg x+2n, where deg x is the degree of x as cochain. We put C[1]k = C k+1 and denote by deg the shifted degree, that is, deg (xT λ en ) = deg(xT λ en )−1. Let Mmain k+1 (β) be the main component of moduli space of bordered stable pseudo-holomorphic maps to (M, L) of genus zero with k + 1 marked points on the boundary, whose homotopy class is β ∈ Π(M ; L) = π2 (M, L)/ ∼. (See Definition 2.4.17.) For Pi ∈ X1 (L), we consider a fiber product (3.1.4)

Mmain k+1 (β) (ev1 ,··· ,evk ) ×f1 ×···×fk (P1 × · · · × Pk ),

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

where evi (i = 0, 1, · · · , k) is the evaluation map at the i-th marked point. Here and hereafter we write Pi in place of |Pi | in case no confusion can occur. We can show that (3.1.4) is a space with Kuranishi structure. We denote the set (3.1.4) by   Mmain k+1 (β; P ), where P = (P1 , · · · , Pk ). The following is our first main theorem in this chapter. Theorem 3.1.5. Let L be a relatively spin Lagrangian submanifold. Then for any given XL satisfying Theorem 3.1.2 there exist choices of X1 (L) ⊃ XL and of a  system of multisections sβ,P of Mmain k+1 (β; P ) with which we can construct a filtered • A∞ structure m = {mk } over Λ0,nov on C (L; Λ0,nov ). Moreover, it has a homotopy unit and satisfies the G(L)-gapped condition. We will give the definition of the G-gapped condition in Section 3.2 (Definition 3.2.26), which is related to Gromov’s compactness property of pseudo-holomorphic curves. Let G ⊂ R≥0 × 2Z be a submonoid of R≥0 × 2Z. (Namely, (0, 0) ∈ G, β, β  ∈ G ⇒ β + β  ∈ G). We assume the following: Condition 3.1.6. (3.1.7.1) Let π : R≥0 × 2Z → R≥0 be the projection. Then π(G) ⊂ R≥0 is discrete. (3.1.7.2) G ∩ ({0} × 2Z) = {(0, 0)}. (3.1.7.3) G ∩ ({λ} × 2Z) is a finite set for any λ. In Theorem 3.1.5, we take G(L) as follows: We define G(L)0 by G(L)0 = {(ω[β], μL (β)) | β ∈ Π(L), M(L; β; J) = ∅}, where ω[β] is the symplectic area and μL (β) is the Maslov index. See Definition 2.4.17 as for the notation Π(L). Now we define (3.1.8)

G(L) = the submonoid of R≥0 × 2Z generated by G(L)0 .

Then G(L) satisfies Condition 3.1.6 by Gromov’s compactness theorem. (Recall that the Maslov index for an oriented Lagrangian submanifold is always even.) We remark that G(L) may depend on the compatible almost complex structure J, in general. There are nontrivial issues of the transversality and of the sign that we have to solve in relation to the choice of X1 (L) and to the construction of the system of multisections sβ,P in the theorem above. Postponing the study of the transversality till Chapter 7 and the sign problems till Chapter 8, we will prove Theorem 3.1.5 in Section 3.5 (Theorem 3.5.11). Since we use the notions of Kuranishi structures and of multisections, various fundamental chains are defined over Q, not Z in general. This is the reason why we take R = Q in this construction. By reducing the coefficient ring Λ0,nov to Q, we discard the contribution of pseudo-holomorphic discs as we mentioned before. Then we obtain the following result (3.5.12.3) Theorem 3.5.11. Its proof will be completed in Subsection 7.2.8 using the results of Subsection 7.2.9 and Section 8.9.) Theorem 3.1.9. Let (C(L; Q), m) be the A∞ algebra over Q with m0 = 0 as the reduction of (C(L; Λ0,nov ), m) in Theorem 3.1.5. Then the restriction of m to QXL ⊂ C(L; Q) coincides with mL which is constructed in Theorem 3.1.2. Thus our filtered A∞ algebra (C(L; Λ0,nov ), m) can be interpreted as a quantum deformation of the rational (more precisely real) homotopy type of L. The word of

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‘quantum’ here means to include all nonlinear contributions coming from pseudoholomorphic discs (called disc instantons in physics literature). Due to the presence of non-zero m0 in a filtered A∞ algebra (C(L; Λ0,nov ), m), the operator m1 in Theorem 3.1.10 does not satisfy m1 ◦ m1 = 0 in general. This is the obstruction to defining Floer cohomology of L. On the other hand, using any b ∈ C[1]0 = C 1 , we can deform m to mb by mbk (x1 , · · · , xk )  = mk+ i (b, · · · , b, x1 , b, · · · , b, · · · , b, · · · , b, xk , b, · · · , b)         0 ,··· ,k

0

1

k−1

k

= m(e x1 e x2 · · · xk−1 e xk e ) b

b

b

b

for k = 0, 1, 2 · · · , so that (C, mb ) is also a filtered A∞ algebra. (See Definition 3.6.9). Here we use a notation eb := 1 + b + b ⊗ b + b ⊗ b ⊗ b + · · · . In addition, if b satisfies an A∞ version of Maurer-Cartan equation: (3.1.10)

m(eb ) := m0 (1) + m1 (b) + m2 (b, b) + m3 (b, b, b) + · · · = 0,

we achieve mb0 = 0 (Proposition 3.6.10). Then mb1 ◦ mb1 = 0 and we can define the cohomology of (C(L; Λ0,nov ), mb1 ). We call a solution b of the Maurer-Cartan equation (3.1.10) a bounding cochain. (We also assume that b ≡ 0 mod Λ+ 0,nov for the sake of convergence of the left hand side in (3.1.10).)     (C(L; Λ0,nov ), m) = M(L) and say that We denote the set of solutions by M  L is unobstructed if M(L) = ∅. If L is unobstructed, we define Floer cohomology  deformed by b ∈ M(L); HF (L, b; Λ0,nov ) := H(C(L; Λ0,nov ), mb1 ). The next problem is to obtain some condition for L to be unobstructed. For this purpose, we will construct a sequence of elements {[ok (L)]}k=1,2,··· of H(L; Q), which we call the (Floer) obstruction classes. Since G(L) satisfies Condition 3.1.6, we may put G(L) = {(λi , μi ) = (ω[βi ], μL (βi )) | i = 0, 1, 2, · · · } such that i < j ⇒ λi ≤ λj . (Note that λ0 = μ0 = 0 and 0 < λ1 .) If ω[β] = ω[β  ] for some β, β  , there is more than one way of enumerating G(L). However the results we obtain are independent of the enumeration. Theorem 3.1.11. Let (C(L; Λ0,nov ), m) be the filtered A∞ algebra constructed in Theorem 3.1.5. Then we have the sequences of cochains ok (L) and bk (L) (k = 1, 2, · · · ) with the following properties: (3.1.12.1) [ok (L)] ∈ H 2−μk (L, Q) and bk (L) ∈ C 1−μk (L; Q). Here μk = μL (βk ) is the Maslov index of βk which is even. (3.1.12.2) The cocycle ok (L) is defined if bj (L) and oj (L) for j with λj  λk are defined. [ok (L)] depends on bj (L) and oj (L) for j with λj  λk . (3.1.12.3) bk (L) is defined to be a cochain satisfying −m1,0 (bk ) = ok if [ok (L)] = 0.

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(3.1.12.4) L is unobstructed if and only if there exists an inductive choice of bk (L) such that [ok (L)] are all zero. The obstruction classes can be constructed purely algebraically in any abstract filtered A∞ algebra satisfying the G-gapped condition. In fact, we carry out this construction in Subsection 3.6.2 (Theorem 3.6.18) in this general setting. Moreover, we can define a refinement of the obstruction classes denoted by [ok (L; weak)] where the preceding obstruction classes are assumed to vanish only for non-positive Maslov index. See Theorem 3.6.43. The refined obstruction classes [ok (L; weak)] are related to a weaker condition to define the Floer cohomology as follows. Recall that our A∞ algebra (C(L; Λ0,nov ), m) has a homotopy unit. For an A∞ algebra with homotopy unit, we introduce a weaker condition for defining Floer cohomology in Subsection 3.6.3. Let (C, m) be an A∞ algebra over Λ0,nov with unit e. (See Definition 3.2.20 (3).) For b ∈ C[1]0 , we consider the equation m(eb ) = cee

(3.1.13) +(0)

for some c ∈ Λ0,nov weaker than the Maurer-Cartan equation. (Here deg e = 0 and deg e = 2. Thus deg(cee) − 1 = +1.) We set weak (C) = {b ∈ C[1]0 | m(eb ) = cee for some c ∈ Λ+(0) }. M 0,nov We say that an A∞ algebra over Λ0,nov with unit e is weakly unobstructed if weak (C) = ∅. For an A∞ algebra (C, m) with homotopy unit, we have an asM sociated A∞ algebra (C + , m+ ) with unit e+ (Definition 3.3.2). We say that an A∞ weak (C + ) = ∅. For algebra (C, m) with homotopy unit is weakly unobstructed if M + weak (C ), we define δb,b by b∈M  m+ · · · , b, x, b, · · · , b), δb,b (x) = k1 +k2 +1 (b,     k1 ,k2 ≥0

k1

k2

+  which is nothing but m+b 1 (x). Then it follows that δb,b ◦ δb,b = 0 for b ∈ Mweak (C ) (Lemma 3.6.33). Therefore if (C(L; Λ0,nov ), m) is weakly unobstructed, we can still weak (L) := M weak (C(L; Λ0,nov )+ ) as define a Floer cohomology deformed by b ∈ M

HF (L, b; Λ0,nov ) := H(C(L; Λ0,nov )+ , m+b 1 ). Theorem 3.6.43 shows that if all the refined obstruction classes [ok (L; weak)] vanish then L is weakly unobstructed. Now considering pseudo-holomorphic discs with one interior marked point in addition to the boundary marked points, we will show the following in Section 3.8. (See Theorem 3.8.11.) Theorem 3.1.14. Let L ⊂ M be a relatively spin Lagrangian submanifold of dimension n. Then (1) If [GW0,1 (M )(L)] = 0, then M(L) = ∅. (2) If [GW0,1 (M )(L)] = 0, then the obstruction [ok (L)] lies in the kernel of the Gysin homomorphism i! : H 2−μk (L; Q) → H n+2−μk (M ; Q). Here [GW0,1 (M )(L)] ∈ H(M ; Q) ⊗ Λ0,nov

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is the Gromov-Witten invariant of genus zero with two marked points, one of that is attached to L. See Subsection 3.8.1 for the precise definition. See Corollaries 3.8.16 - 3.8.19 in Subsection 3.8.2 for some applications. Theorem 3.1.15. There exists a sequence of obstruction classes [ok (L; def)] ∈

H 2−μk (L; Q) Im i∗ : H 2−μk (M ; Q) → H 2−μk (L; Q)

and bk ∈ C21−μk (L; Q), bk ∈ C 2−μk (M ; Q), (δM bk = 0) with the following properties. (3.1.16.1) The cocycle ok (L; def) is defined if bj and oj for j with λj  λk are defined. [ok (L; def)] depends on bj and oj for j with λj  λk . (3.1.16.2) bk , bk are defined if [ok (L; def)] = 0. (3.1.16.3) L is unobstructed after bulk deformations if all of the obstruction classes [ok (L; def)] are defined and are zero. Corollary 3.1.17. If i∗ : H 2k (M ; Q) → H 2k (L; Q) is surjective for all k, then L is unobstructed after bulk deformations. If L is unobstructed after bulk deformation, we can also define Floer cohomoldef (L), (Lemma 3.8.60). See Definition 3.8.40 for the ogy deformed by (b, b) ∈ M  set Mdef (L). weak,def (L) Involving the unit or the homotopy unit, we can similarly define M and a sequence of obstruction cocycles ok (L; weak,def) which gives a generalization of Theorem 3.1.15. See Subsection 3.8.5. So far, we have considered the case of single Lagrangian submanifolds. Next, we consider the case of pairs (L(0) , L(1) ) of relatively spin Lagrangian submanifolds. We first assume that L(0) is transversal to L(1) , in Subsection 3.7.4. For such a pair (L(0) , L(1) ) we define a graded filtered Λ0,nov module, C(L(1) ,L(0) ;Λ0,nov ) (see Definition 3.7.20). On the other hand, we have the filtered A∞ algebra (C(L(i) , Λ0,nov ), m(i) ) for each L(i) (i = 0, 1). Then we will construct a filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ) (Theorem 3.7.21). Theorem 3.1.18. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian submanifolds which are transverse to each other. Then there exists a left (C(L(1) ; Λ0,nov ), m(1) ) and right (C(L(0) ; Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ). It is G(L(1) , L(0) )-gapped. Furthermore the pair of the homotopy units {e1 , e0 } of (C(L(1) ; Λ0,nov ), m(1) ) and (C(L(0) ; Λ0,nov ), m(0) ) acts as a homotopy unit. The precise definitions of various notions appearing in the theorem will be given in Subsection 3.7.1. See also (3.7.43) for the definition of G(L(1) , L(0) ). Moreover we can show the following (Theorem 3.7.45). Theorem 3.1.19. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian sub (0) ), manifolds which are transversal to each other. Then for each b0 ∈ M(L  (1) ), there exists a cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) which we b1 ∈ M(L call Floer cohomology of (L(0) , L(1) ) deformed by (b0 , b1 ).

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Let Ω(L(0) , L(1) ) be the space of all paths  : [0, 1] → M joining L(0) and L(1) . We remark π0 (Ω(L(0) , L(1) )) = i∗ (π1 (L(0) ))\π1 (M )/i∗ (π1 (L(1) )), if L(0) , L(1) are connected. Each point p ∈ L(0) ∩ L(1) determines a constant path t → p and hence an element of π0 (Ω(L(0) , L(1) )). It induces the following decomposition of Floer cohomology group (See Remark 3.7.46.) (3.1.20)

HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) ∼ HF ((L(1) , b1 ), (L(0) , b0 ); 0 ; Λ0,nov ). = [0 ]∈π0 (Ω(L(0) ,L(1) ))

We note that when L(0) and L(1) are weakly unobstructed, we can not define weak (L(i) ) in general, while a Floer cohomology of (L(0) , L(1) ) deformed by bi ∈ M (i) we can define the Floer cohomology HF (L , bi ; Λ0,nov ) for each i = 0, 1. This fact is an important point for various applications. To describe a condition for the deformed Floer cohomology of the pair (L(0) , L(1) ) to be defined, we introduce a function weak (L) → Λ+(0) PO : M 0,nov called a potential function by PO(b) = c in the equation (3.1.13). Then we can show the following (Proposition 3.7.17). Theorem 3.1.21. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian subweak (L(0) ), manifolds which are transversal to each other. Then, for each b0 ∈ M (1)  b1 ∈ Mweak (L ) with PO(b0 ) = PO(b1 ), we can define a Floer cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) deformed by b0 , b1 . A similar decomposition as (3.1.20) also exists. weak,def (L) → Λ+(0) in a similar way. We can define PO : M 0,nov In Subsection 3.7.5, we will study the case that L(0) intersects L(1) cleanly. (Namely Bott-Morse version of Floer cohomology.) The connected components of the intersection L(1) ∩ L(0) are smooth manifolds but not necessary orientable. By a careful consideration of orientation sheaves, we can obtain the same conclusion as in the transversal case. Theorem 3.1.22. Let (L(0) , L(1) ) be a relatively spin pair. Assume that L(0) intersects L(1) cleanly. Then the conclusions in Theorem 3.1.18, 3.1.19 and 3.1.21 still hold. We note that the filtered A∞ algebra (C(L; Λ0,nov ), m) itself can be regarded as a left (C(L; Λ0,nov ), m) and right (C(L; Λ0,nov ), m) filtered A∞ bimodule (see Example 3.7.6). On the other hand, when L(0) = L(1) (=: L) in Theorem 3.1.22, we have a left (C(L; Λ0,nov ), m) and right (C(L; Λ0,nov ), m) filtered A∞ bimodule as well. Then we can show that the two A∞ bimodules can be identified if {Jt }t , a family of almost complex structures which is used to construct the A∞ bimodule structure, is independent of t (Proposition 3.7.75). Thus when L is unobstructed or weakly unobstructed, we have HF (L, b; Λ0,nov ) ∼ = HF ((L, b), (L, b); Λ0,nov ).

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Furthermore, if we consider the case that L(1) = ψ(L(0) ) where ψ is a Hamiltonian diffeomorphism in Theorem 3.1.22, we have a deformed Floer cohomology HF ((ψ(L), ψ∗ (b)), (L, b); Λ0,nov ). (See Section 4.1 Theorem 4.1.3 for ψ∗ (b).) As we will see in examples given in Subsection 3.7.6, this is not independent of ψ in general. However, we will prove in Chapter 5 that HF ((ψ(L), ψ∗ (b)), (L, b)); Λnov ) := HF ((ψ(L), ψ∗ (b)), (L, b); Λ0,nov ) ⊗Λ0,nov Λnov is independent of the Hamiltonian diffeomorphism ψ. By combing these, therefore, we have HF (L, b; Λnov ) ∼ = HF ((ψ(L), ψ∗ (b)), (L, b)); Λnov ). The organization of this chapter is in order. We introduce the notions of filtered A∞ algebras and A∞ homomorphisms in Section 3.2, and the homotopy unit, the unital or homotopy-unital A∞ homomorphism in Section 3.3. These are presented in the purely algebraic context and Lagrangian submanifolds do not appear there. After that, we give a geometric realization of the classical A∞ algebra (ZXL , mL ) with the homotopy unit on each oriented manifold L. In the following Section 3.5, we will construct the filtered (quantum) A∞ algebra (C(L, Λ0,nov ), m) with homotopy unit associated to a relatively spin Lagrangian submanifold L. We note that throughout in Sections 3.4 and 3.5, we do not assume any unobstructedness. In Section 3.6, we will introduce notions of the bounding cochains and the Maurer-Cartan equation in the context of filtered A∞ algebras and construct the obstruction classes. We define notions of unobstructedness and weak unobstructedness. We also introduce the potential function PO therein and study the relation between the weak unobstructedness condition and existence of Floer cohomology of L. In Section 3.7, we consider a relatively spin pair (L(0) , L(1) ) of Lagrangian submanifolds. We deform the standard Floer coboundary operator for (L(0) , L(1) ) using the constructions provided in the previous sections. For this purpose, we introduce the algebraic notions of a filtered A∞ bimodule structure on a graded filtered Λ0,nov module and of A∞ bimodule homomorphisms in Subsections 3.7.1 and 3.7.2 respectively. We discuss the weak unobstructedness in Subsection 3.7.3. Our main task in Section 3.7 is to construct a left (C(L(1) , Λ0,nov ), m(1) ) and right (C(L(0) , Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ). This is done for the case that L(0) is transversal to L(1) in Subsection 3.7.4 and for the case that L(0) and L(1) intersect cleanly in Subsection 3.7.5. When we define the A∞ bimodule structure, it is not necessary to assume that the A∞ algebras (C(L(0) , Λ0,nov ), m(0) ) and (C(L(1) , Λ0,nov ), m(1) ) are unobstructed or weakly unobstructed. (Namely filtered A∞ bimodule is constructed for arbitrary relatively spin pair.) If they are unobstructed, then we can obtain the deformed Floer cochain complex (C(L(1) , L(0) ; Λ0,nov ), δb1 ,b0 ) by using the A∞ bimodule structure and bounding cochains b0 , b1 chosen respectively for L(0) , L(1) . We will give some simple examples of calculation of Floer cohomology in Subsection 3.7.6. In Section 3.8, we consider the moduli space of pseudo-holomorphic discs with interior marked points in addition. By considering the moduli space of pseudo-holomorphic map from bordered Riemann surface with one interior marked point and no boundary marked point, we will construct a sequence of operators pk and study the properties. Using the operators p, we study properties of our obstruction classes constructed in Subsection 3.6.2 and Subsection 3.6.3 and give some useful conditions for (weak)

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unobstructedness in subsections Subsections 3.8.1-3.8.3. Furthermore, by considering the case of several interior marked points in Subsection 3.8.4, we will construct a sequence of operators q. These operators give a bulk deformation of our filtered A∞ algebra by elements of H 2 (M ; Λ+ 0,nov ). We can also construct a relative version of the obstruction classes in Subsection 3.8.5. The outline of the construction of q is given in Subsection 3.8.6. This idea is also used in the story of our filtered A∞ bimodules in Subsection 3.8.7 and we get a sequence of operators r, whose construction is explained in Subsection 3.8.8. The detail of the construction of these operators will be given in Section 7.4. In the last subsection we generalize the construction of operators pk by combining it with the construction of q and obtain a sequence of operators p,k . We will discuss some properties of it there. These operators will be used in the spectral sequence argument in Chapter 6. We prove several formulas which provides the basic properties of operators p, q, r in Section 3.8. Those formulas are reinterpreted in Section 7.4 using the language of L∞ structures and the detail of the construction of them is also given there. In order to construct the A∞ algebras geometrically, we need to resolve certain transversality problems. This point will be discussed in Chapter 7 in more details. We need a careful consideration of the sign (or of the orientation). The precise consideration of the signs will be carried out in Chapter 8. So in this chapter, we just state our results in a rigorous manner and prove them modulo the transversality and the sign problems. We note that our construction in this chapter will be done by choosing and fixing various structures, for example, compatible almost complex structures, Kuranishi structures, multisections, triangulation, countable sets XL , X (L), etc. We have to study the dependence or independence by these choices. This will be done in the following Chapters 4 and 5. 3.2. Algebraic framework on filtered A∞ algebras In this section, we introduce the notions of the filtered A∞ algebra and of the A∞ homomorphism in a purely algebraic context. So we do not mention Lagrangian submanifolds in this section. We note that our definition of A∞ algebras is slightly different from the standard one in the literature in ([Sta63]) that our coefficient ring is a filtered commutative ring Λ0,nov . Because of this, in Subsection 3.2.1, we first recall the notion of an A∞ algebra (without filtrations) and of an A∞ homomorphism. After that we will introduce our filtered A∞ algebra and filtered A∞ homomorphism in the next subsection. To distinguish the ordinary (unfiltered) A∞ algebra from our filtered A∞ algebra, we will put ‘bar’s over the notations for the former. 3.2.1. A∞ algebras and homomorphisms. Let R be a commutative ring

m with unit. Let C = m∈Z C be a free graded R module. We always assume that C is generated by countably many elements. We put (C[1])m = C (C[1])m1 ⊗ · · · ⊗ (C[1])mk . Bk (C[1]) =

m+1

and

m1 ,··· ,mk

In some literature one denotes by Bk (C) the right hand side above instead of Bk (C[1]). However throughout this book, we use the notation Bk (C[1]). (Sometimes we write Bk C[1] in place of Bk (C[1]).) Suppose that we have a sequence of

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3.2. ALGEBRAIC FRAMEWORK ON FILTERED A∞ ALGEBRAS

87

maps m = {mk }k≥1 of degree +1 mk : Bk (C[1]) → C[1], The direct sum B(C[1]) := coproduct Δ is defined by:

k

Δ(x1 ⊗ · · · ⊗ xn ) =

for k = 1, · · · .

Bk (C[1]) has a structure of graded coalgebra. The n 

(x1 ⊗ · · · ⊗ xk ) ⊗ (xk+1 ⊗ · · · ⊗ xn ) .

k=0

We can extend mk uniquely to a coderivation ˆk : m



Bn (C[1]) →

n



Bn−k+1 (C[1]),

n

by the formula

(3.2.1)

ˆ k (x1 ⊗ · · · ⊗ xn ) = m

n−k+1 

(−1)deg x1 +···+deg x−1 +−1 x1 ⊗ · · · ⊗

=1

mk (x , · · · , x+k−1 ) ⊗ · · · ⊗ xn ˆ k = 0 for k > n. Here and hereafter deg x means the degree of x for k ≤ n and m before we shift it. m+1 Hereafter for x ∈ (C[1])m = C we put deg x = m = deg x − 1.

(3.2.2)

Using this notation the sign in (3.2.1) is (−1)deg  ˆ . d= m We put 



x1 +···+deg x−1

.

k

Definition 3.2.3. m = {mk }k≥1 defines a structure of A∞ algebra over R on C, if  d◦ d = 0. Remark 3.2.4. (1) The operator mk is defined for k > 0. (m0 = 0.) For a filtered A∞ algebra (which we define in the next subsection) we consider mk for k ≥ 0. Since m0 = 0, we find that m1 m1 = 0. So we have a cochain complex (C[1], m1 ). (See Remark 3.2.21 (1).) (2) When we regard a differential graded algebra (D.G.A.) over R as an A∞ algebra, we just put mk = 0 for k ≥ 3. But we note that the sign of m2 is different from that of product in the D.G.A. (3.2.5)

m1 (x) = (−1)deg x dx, m2 (x, y) = (−1)deg x(deg y+1) x · y.

Here x · y is the product in the D.G.A. (See also Remark 3.2.21 (2) below.) Next, we define the notion of an A∞ homomorphism. Let (C i , mi ), i = 1, 2, be A∞ algebras over a commutative ring R with unit. For k = 1, 2, · · · , let us

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consider the family of maps fk : Bk (C 1 [1]) → C 2 [1] of degree 0. These maps induce f : B(C [1]) → B(C [1]), which on B (C [1]) is give by 1 2 k 1  f(x ⊗ · · · ⊗ x ) = fk1 (x1 , · · · , xk1 ) ⊗ · · · ⊗ 1 k 0 n. When k = 0, we put m0 (1) in the right hand side. for k ≤ n and m ˆ 0 by Namely we define m ˆ 0 (x1 ⊗ · · · ⊗ xn ) = m

n+1 

(−1)deg x1 +···+deg x−1 +−1 x1 ⊗ · · · ⊗ x−1 ⊗

=1

m0 (1) ⊗ x ⊗ · · · ⊗ xn .  We next consider a completion B(C[1]) of B(C[1]). We define a filtration F Bk (C[1]) on Bk (C[1]) by  + λ1 m1 , F C ⊗ · · · ⊗ F λk C mk . F λ Bk (C[1]) = λ

λ1 +···+λk ≥λ

k (C[1]) be the completion of Bk (C[1]) with respect to this filtration. We Let B call

this filtration energy filtration. There is another filtration of BC[1] that it k≥k0 Bk C[1], k0 = 1, 2, · · · . We call it the number filtration.   Definition 3.2.16. B(C[1]) is the set of all formal sum k xk where xk ∈ k (C[1]) such that B k (C[1]) x k ∈ F λk B with limk→∞ λk → ∞. ∞   ˆ k. We define dˆ : B(C[1]) → B(C[1]) by dˆ = k=0 m Lemma 3.2.17. If (3.2.12) is satisfied, then d is well-defined as a map from   B(C[1]) to B(C[1]).  k (C[1])-component of Proof. Let xk be as in Definition 3.2.16. The B    ˆ  (xk+−1 ). By (3.2.12) and Definition 3.2.16, we have d xk is  m k (C[1]). ˆ  (xk+−1 ) ∈ F λk+−1 B m k (C[1]) implies that d xk converges. The proof Therefore the completeness of B of Lemma 3.2.17 is complete.  Now we introduce the following condition for an element e of C 0 = C[1]−1 . (Compare also [Fuk97II] Section 11.) Condition 3.2.18. (3.2.19.1) mk+1 (x1 , · · · , e, · · · , xk ) = 0, for k ≥ 2 or k = 0. (3.2.19.2) m2 (e, x) = (−1)deg x m2 (x, e) = x.

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Definition 3.2.20. (1) The structure of a filtered A∞ algebra on C is a collection m = {mk }k≥0 that satisfy (3.2.12.1) - (3.2.12.7), (3.2.13.1) - (3.2.13.3) and  d ◦ d = 0. We call B(C[1]) the (completed) bar complex associated to the A∞ algebra (C, m). If (C, m) is a filtered A∞ algebra then (C, mk ) as in (3.2.13) is an A∞ algebra. We call (C, mk ) the R-reduction of (C, mk ). In this book we frequently denote the R reduction of (C, mk ) etc. by (C, mk ) without mentioning. (2) We say a filtered A∞ algebra (C, m) is strict if m0 = 0. (3) If a (filtered) A∞ algebra has an element e of degree 0 (before shifted) satisfying Condition 3.2.18, we call the A∞ algebra an (filtered) A∞ algebra with unit and e a unit. Sometimes we simply call it a unital filtered A∞ algebra. For a unital filtered A∞ algebra (C, m) we put Cred = C/Λ0,nov [e] and we always assume that there exists a splitting C = Cred ⊕ Λ0,nov [e] in this book. For the unfiltered A∞ algebra (C, m) over R, we define the unit and assume the splitting above in the same way. (See Subsections 4.4.4, 4.4.7 as for some discussion on Cred .) Remark 3.2.21. (1) The equation d◦ d = 0 produces infinitely many relations among mk ’s. For example, the first few relations are given by m1 (m0 (1)) = 0, m2 (m0 (1), x) + (−1)deg x+1 m2 (x, m0 (1)) + m1 (m1 (x)) = 0, m3 (m0 (1), x, y) + (−1)deg x+1 m3 (x, m0 (1), y) + (−1)deg x+deg y+2 m3 (x, y, m0 (1)) + m2 (m1 (x), y) + (−1)deg x+1 m2 (x, m1 (y)) + m1 (m2 (x, y)) = 0, ········· In general, it is easy to show that d ◦ d = 0 is equivalent to that for each k   (−1)deg x1 +···+deg xi−1 +i−1 (3.2.22) k1 +k2 =k+1 i mk1 (x1 , · · · , mk2 (xi , · · · , xi+k2 −1 ), · · · , xk ) = 0. We call this identity the A∞ formula or the A∞ relation. If m0 = 0, then m1 m1 = 0. So in this case m1 plays the role of a boundary operator. (2) Suppose m0 = 0 and m3 = · · · = 0. Then d ◦ d = 0 implies (3.2.23)

m2 (m2 (x, y), z) + (−1)deg x+1 m2 (x, m2 (y, z)) = 0.

(3.2.23) is consistent with (3.2.5) and the associativity of the product in D.G.A. (3) We put (3.2.24)

lk (x1 , · · · , xk ) =

1  (−1) (σ) mk (xσ(1) , · · · , xσ(k) ), k! σ∈Sk

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

where (3.2.25)

(σ) =



(deg xi + 1)(deg xj + 1).

iσ(j)

This can be regarded as a symmetrization of the operators mk . The A∞ formula for mk leads to certain relations among the operators lk , which define an L∞ structure. See Section A3 (and for example [LaMa95]) on the symmetrization of an A∞ structure. The sign convention is slightly different from ours in some of the literatures. (4) Later in Section 3.5 we will construct a filtered A∞ algebra associated to a Lagrangian submanifold. This filtered A∞ algebra does not have a unit in the sense of Definition 3.2.20 (3). But in Section 7.3, we will show that it has a homotopy unit. We will define the notion of homotopy unit in the A∞ algebra in Section 3.3.  (5) Note the completion BC[1] is not a coalgebra in the standard sense but there exists an operation     BC[1] Δ : BC[1] → BC[1] ⊗  ◦ Δ = (1⊗Δ)   denotes a completion such that (Δ⊗1) ◦ Δ. Here and hereafter ⊗ of the tensor product over Λ0,nov . We call such an algebraic structure formal coalgebra. (Such a notion appears in the theory of formal group [Die73].) Our d is a coderivation of the formal coalgebra. We next define the gapped condition. Let G ⊂ R≥0 × 2Z be a submonoid. We put G = {β = (λ(β), μ(β)) ∈ R≥0 × 2Z | β ∈ G}. Definition 3.2.26. Suppose that G satisfies Condition 3.1.6. We say that a filtered A∞ algebra (C, m) is G-gapped, if mk has the decomposition  T λ(β) eμ(β)/2 mk,β mk = β∈G

for some family of R module homomorphisms mk,β : Bk C[1] → C[1] for k = 0, 1, 2, · · · and β = (λ(β), μ(β)) ∈ G. Here we identify C = C ⊗R Λ0,nov . (C, m) is simply said to be gapped if it is G-gapped for some G. Note (3.1.7.2) implies (3.2.13.3). The gapped condition will be used when we construct a spectral sequence as well as various arguments involving inductive constructions over the filtration. (See Chapters 4, 5 and 6.) Next, we define the notion of the filtered A∞ homomorphisms. Let (Ci , mi ), i = 1, 2, be filtered A∞ algebras over the ring Λ0,nov . For k = 0, 1, 2, · · · , we consider the family of Λ0,nov module homomorphisms fk : Bk (C1 [1]) → C2 [1] of degree 0 such that (3.2.27.1)

fk (F λ Bk (C1 [1])) ⊆ F λ C2 [1],

and (3.2.27.2)



f0 (1) ∈ F λ C2 [1]

for some λ > 0.

Note that f0 : Λ0,nov → C2 [1] and we do not assume f0 (1) = 0 in this case. These maps induce f : B(C  1 [1]) → B(C  2 [1]),

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3.2. ALGEBRAIC FRAMEWORK ON FILTERED A∞ ALGEBRAS

93

which on Bk (C1 [1]) is given by the formula  f(x1 ⊗ · · · ⊗ xk ) = fk1 (x1 , · · · , xk1 ) ⊗ · · · 0≤k1 ≤···≤kn ≤k

(3.2.28)

· · · ⊗ fki+1 −ki (xki +1 , · · · , xki+1 ) ⊗ · · · · · · ⊗ fk−kn (xkn +1 , · · · , xk ),

and (3.2.27) implies that f converges. We note that when f0 appears in the right hand side of (3.2.28), we put f0 (1) there. Thus, in particular, f(1) is given by f(1) = 1 + f0 (1) + f0 (1) ⊗ f0 (1) + · · · = ef0 (1) in our notation of exponential function. (See also (3.6.1) below.) We remark that the right hand side converges because of (3.2.27.2). Then it is easy to see that f : B(C  1 [1]) → B(C  2 [1]) is a (formal) coalgebra homomorphism. (See Lemma 3.6.2). Definition 3.2.29. Let fk : Bk (C1 [1]) → C2 [1] satisfy (3.2.27) as above. We assume that the R[e, e−1 ] module homomorphism fk

−1 mod Λ+ ] → C[1] ⊗R R[e, e−1 ] 0,nov : Bk C[1] ⊗R R[e, e

is induced from an R module homomorphism fk : Bk C[1] → C[1]. (1) We call f = {fk }k≥0 a filtered A∞ homomorphism from C1 to C2 if f ◦ d1 = d ◦ f. We simply write f : C1 → C2 . (2) A filtered A∞ homomorphism f is called strict if f0 = 0. (3) Let G ⊂ R≥0 × 2Z be a discrete submonoid satisfying Condition 3.1.6. Let Ci (i = 1, 2) be gapped filtered A∞ algebras. We say that f is G-gapped if there exist fk,β : Bk C 1 [1] → C 2 [1] for k = 0, 1, 2, · · · , β ∈ G such that  fk,β T λ(β) eμ(β)/2 . (3.2.30) fk = 2

We remark that if G is another discrete submonoid satisfying Condition 3.1.6 and containing G, then every G-gapped filtered A∞ homomorphism is G -gapped. When the submonoid G is not explicitly specified, we just say f is gapped. Let fik : Bk (Ci [1]) → Ci+1 [1] (i = 1, 2) define a filtered A∞ homomorphism. Definition 3.2.31. The composition f2 ◦f1 = {(f2 ◦f1 )k } of f1 and f2 is defined by the formula (f2 ◦ f1 )k (x1 , · · · , xk )   = f2m (f1k1 (x1 , · · · , xk1 ), · · · , f1km (xk−km +1 , · · · , xk )). m k1 +···+km =k

Lemma 3.2.32. f2 ◦ f1 defines a filtered A∞ homomorphism from C1 to C3 . Proof. By noticing that we have f(1) = ef0 (1) , the proof is similar to that of Lemma 3.2.9.  Let (Ci , mi ) (i = 1, 2) be filtered A∞ algebras over Λ0,nov and f : (C1 , m1 ) → (C2 , m2 ) a filtered A∞ homomorphism. Then f naturally induces an A∞ homomorphism f : (C 1 , m1 ) → (C 2 , m2 ), where (C i , mi ) is the R-reduction of (Ci , mi ).

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Definition 3.2.33. Let (Ci , mi ) (i = 1, 2) be filtered A∞ algebras over Λ0,nov such that mi0 = 0. For these filtered A∞ algebras, we say that a filtered A∞ homomorphism f : (C1 , m1 ) → (C2 , m2 ) is a weak homotopy equivalence, if the induced A∞ homomorphism f : (C 1 , m1 ) → (C 2 , m2 ) is a weak homotopy equivalence in the sense of Definition 3.2.10. Furthermore, when (Ci , mi ) and f are G-gapped, we call it a G-gapped weak homotopy equivalence. Definition 3.2.34. For a filtered A∞ algebra (C, m), we say that a filtered A∞ algebra (C  , m ) is an A∞ -deformation of (C, m), if there exists a weak homotopy  equivalence f : (C, m) → (C , m ) between their R-reductions. (We also say that   a filtered A∞ algebra (C , m ) is an A∞ -deformation of an unfiltered A∞ algebra  (C, m), if there exists a weak homotopy equivalence f : (C, m) → (C , m ). 3.3. Algebraic framework on the homotopy unit In Section 3.5 we will construct a filtered A∞ algebra associated to a Lagrangian submanifold. We do not know in general whether this filtered A∞ algebra has a unit in the sense of Definition 3.2.20 (3). In Section 7.3, however, we will see that it has a homotopy unit. In this section, we will give an algebraic definition of homotopy unit. A geometric realization of the homotopy unit of our A∞ algebra associated to a Lagrangian submanifold will be given in Section 7.3. In Subsection 3.3.2, we define the notion of a unital (resp. homotopy-unital) A∞ homomorphism, which preserves the unit (resp. the homotopy unit). 3.3.1. Definition of the homotopy unit. Let (C, m) be a filtered A∞ algebra. (We mainly consider Λ0,nov as the coefficient ring, but obviously the following framework is also available for an A∞ algebra over R without filtration.) We do not assume that it has a unit. Firstly, we prepare some notations. We denote by Bk (B(C[1])) the tensor product of k copies of B(C[1])’s and put B(B(C[1])) = Bk (B(C[1])) = B(C[1]) ⊕ (B(C[1]) ⊗ B(C[1])) ⊕ · · · . k≥1

 B(C[1]))  Let B( be a completion of B(B(C[1])) with respect to the energy filtration. To avoid confusion between the tensor product in B(C[1]) with the one in B(B(C[1])), we use ⊗ for the latter, and we use the bold face letter x etc. to denote elements of B(C[1]). Namely, for example, x1 ⊗ x2 is an element of B2 (C[1]) and x1 ⊗ x2 is an element of B2 (B(C[1])). If xi = xi,1 ⊗ · · · ⊗ xi,i ∈ Bi (C[1]) then x1 ⊗ · · · ⊗ xk ∈ Bk (B(C[1])) and x1 ⊗ · · · ⊗ xk ∈ B1 +···+k (C[1]). (x1 ⊗ · · · ⊗ xk = x1,1 ⊗ x1,2 ⊗ · · · ⊗ x1,1 ⊗ x2,1 ⊗ · · · ⊗ xk,k .) Now we put (3.3.1)

C + = C ⊕ Λ0,nov e+ ⊕ Λ0,nov f,

where deg e+ = 0, deg f = −1. (Here deg is a degree before shifted.) Definition 3.3.2. Let (C, m) be an A∞ algebra. We say an element e ∈ C 0 = C[1]−1 is a homotopy unit of (C, m) if there exist maps hk : Bk (B(C[1])) → C[1]

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of degree 3 − 2k for k = 1, 2, · · · which preserve the energy filtration, such that (3.3.3)

h1 = m

and the following (C + , m+ ) becomes an A∞ algebra with unit e+ ∈ C + [1]−1 in the sense of Definition 3.2.20 (3). Here C + = C ⊕ Λ0,nov e+ ⊕ Λ0,nov f as in (3.3.1) and m+ is defined by following: + deg x + m+ m2 (x, e+ ) = x, 2 (e , x) = (−1)

(3.3.4)

+ m+ k (· · · , e , · · · ) = 0,

k = 2

for x ∈ C + , and m+ (x1 ⊗ f ⊗ x2 ⊗ f ⊗ · · · ⊗ f ⊗ xk ) = hk (x1 ⊗ · · · ⊗ xk )

(3.3.5.1)

except for k = 2 and x1 , x2 ∈ Λ0,nov (= Λ0,nov · 1). For the latter case, we put + m+ 1 (f) = e − e + h2 (1 ⊗ 1)

(3.3.5.2) with

+ h2 (1 ⊗ 1) ≡ 0 mod Λ+ 0,nov C .

(3.3.5.3)

We also assume that m+ k satisfies (3.2.13). In other words, we assume that hk

−1 mod Λ+ ] → C[1] ⊗R R[e, e−1 ] 0,nov : Bk (B(C[1])) ⊗R R[e, e

is induced by an R module homomorphism (3.3.6)

hk : Bk (B(C[1])) → C[1].

We call (C, m) satisfying (3.3.3)-(3.3.6), a filtered A∞ algebra with homotopy unit, or simply, homotopy-unital filtered A∞ algebra. Remark 3.3.7. (1) If we count the degree of ⊗ as deg ⊗ = −2 (= shifted degree of f), the operators hk will always have degree +1. (2) We now explain the geometric origin of our definition of the homotopy unit. Let (X, x0 ) be a pointed space. Assume that there exists a product structure · : X × X → X such that its restrictions to {x0 } × X and X × {x0 } are homotopic to identity. (In other words X is an H space.) Let H1 : {x0 } × X × [0, 1] → X, H2 : X × {x0 } × [0, 1] → X be such homotopies respectively, i.e., satisfy H1 (x0 , x, 1) = H2 (x, x0 , 1) = x, H2 (x, x0 , 0) = x · x0 . H1 (x0 , x, 0) = x0 · x, Then x0 is a homotopy unit. We consider the disjoint union X ∪ [0, 1] and identify x0 and 0. Namely we put X+ = X ∪[0, 1]/x0 ∼ 0. We extend · to · : X+ ×X+ → X+ as follows: If s, s ∈ [0, 1] and x, x ∈ X, then s · x = H1 (x0 , x , s),

x · s = H2 (x, x0 , s ).

It follows that 1 · x = x · 1 = x. Namely 1 is a unit. The construction of (C + , m+ ) in Definition 3.3.2 is an algebraic version of this construction. More precisely, the observation above provides us with the geometric meaning of the homotopy unit, at least, at the classical level. In this case, the formula involving the term h2 (1 ⊗ 1) in (3.3.5.1), which is nothing but (3.3.5.2), does not appear. In Section 7.3 we will

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explain the geometric meaning of the term h2 (1 ⊗ 1) in our filtered A∞ algebra in more detail. (3) There is an ambiguity of the choice of the maps h in Definition 3.3.2. So the A∞ algebra (C + , m+ ) is not uniquely determined from (C, m). However, we will show that the homotopy equivalence class as a filtered A∞ algebra (see Section 4.2) of (C + , m+ ) is uniquely determined by (C, m). In fact, we will show that (C + , m+ ) is homotopy equivalent to (C, m) as a filtered A∞ algebra. See Lemma 4.2.55 and Proposition 4.2.52. In this sense, C can be regarded as something similar to a deformation retract of C + . The A∞ formulae for (C + , m+ ) imply certain relations between the maps hk ’s. We now write down these relations explicitly. We define ˆk : B(B(C[1])) → B(B(C[1])), h

ˆ : B(B(C[1])) → B(B(C[1])) h

as follows: Let Δ : B(C[1]) → B(C[1]) ⊗ B(C[1]) be the coproduct. For k ≥ 2, we put ˆk (x1 ⊗ · · · ⊗ xn ) h      = (−1)deg x1 +··· deg xi−1 +deg xi,a x1 ⊗ · · · ⊗ xi−1 a,b,i

  ⊗ xi,a ⊗ hk (xi,a ⊗ xi+1 ⊗ · · · ⊗ xi+k−2 ⊗ xi+k−1,b ) ⊗ xi+k−1,b

⊗ xi+k ⊗ · · · ⊗ xn ,  ˆk maps Bn (B(C[1])) where Δxi = a xi,a ⊗ xi,a . (Note that the homomorphism h to Bn−k+1 (B(C[1])), since ⊗ in the second line is not ⊗ .) Here deg is the degree after shifted. Namely, if x = x1 ⊗ · · · ⊗ xm , then deg x = deg x1 + · · · + deg xm − m. See (3.2.2). For k = 1, we put    ˆ1 (x1 ⊗ · · · ⊗ xn ) = h (−1)deg x1 +··· deg xi−1 x1 ⊗ · · · ⊗ xi−1 a,i

 i ⊗ xi+1 ⊗ · · · ⊗ xn . ⊗ dx Here we recall

 i= dx



(−1)deg



x3;1 i,a

3;2 3;3 x3;1 i,a ⊗ m(xi,a ) ⊗ xi,a ,

a

where we use (Conv.3) (1.34). ˆ = h ˆk and extend it to B(  B(C[1])).  We also introduce a family of We put h maps bk,i : Bk (B(C[1])) → Bk−1 (B(C[1])), (i = 1, · · · , k − 1) by bk,i (x1 ⊗ · · · ⊗ xk ) = (−1)deg



x1 +···+deg xi

x1 ⊗ · · · ⊗ (xi ⊗ e ⊗ xi+1 ) ⊗ · · · ⊗ xk

for a given e ∈ C 0 = C[1]−1 . (If we count deg ⊗ = −2, then bk,i is of degree +1.) Proposition 3.3.8. Let (C, m) be an A∞ algebra. Then (3.3.9) is equivalent to (3.3.10). (3.3.9)

The (C + , m+ ) defined in Definition 3.3.2 is an A∞ algebra with unit e+ .

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(3.3.10) The maps hk : Bk (B(C[1])) → C[1] satisfy (3.3.5.2), (3.3.5.3), (3.3.6) and have the following properties: (3.3.10.0)

h1 = m. k 

(3.3.10.1)

=1

ˆk−+1 − h ◦ h

k−1 

hk−1 ◦ bk,i + Pk = 0.

i=1

Here Pk = 0 if k = 2. Moreover, P2 = 0 on the complement of     B0 (C[1]) ⊗ B1 (C[1]) ⊕ B1 (C[1]) ⊗ B0 (C[1]) ⊂ B2 (B(C[1])). (3.3.10.2) −P2 (x⊗1) = P2 (1⊗x) = x,     on B0 (C[1]) ⊗ B1 (C[1]) ⊕ B1 (C[1]) ⊗ B0 (C[1]) ⊂ B2 (B(C[1])). The statement (3.3.10) above enables us to directly formulate the definition of a homotopy unit of (C, m) in terms of the original A∞ algebra (C, m) itself without introducing (C + , m+ ). The proof of Proposition 3.3.8 is a straightforward calculation, which we omit. 3.3.2. Unital (resp. homotopy-unital) A∞ homomorphisms. In this subsection, we will give the definition of an A∞ homomorphism preserving unit or homotopy unit. Let (C1 , m1 ), (C2 , m2 ) be filtered A∞ algebras and f : C1 → C2 be a filtered A∞ homomorphism. We first assume that Ci has a unit ei (i = 1, 2). Definition 3.3.11. We say f preserves unit, or sometimes call it a unital A∞ homomorphism, if (3.3.12.1) (3.3.12.2)

fk (x1 , · · · , xi−1 , e1 , xi+1 , · · · , xk ) = 0 for k ≥ 2, f1 (e1 ) = e2 .

Unfiltered version can be defined in the same way. We next consider the case of A∞ algebras with homotopy unit. We assume that filtered A∞ algebras (C1 , m1 ) and (C2 , m2 ) have homotopy unit e1 and e2 , respectively. We then define + Ci+ = Ci ⊕ Λ0,nov e+ i ⊕ Λ0,nov fi

as in (3.3.1). Definition 3.3.13. We say a filtered A∞ homomorphism f : C1 → C2 preserves homotopy unit, or call f : C1 → C2 a homotopy-unital A∞ homomorphism, if it extends to a unital filtered A∞ homomorphism f+ : C1+ → C2+ . 3.4. A∞ deformation of the cup product In this section, we give a geometric construction of the classical A∞ algebra associated to each oriented, not necessarily relatively spin, Lagrangian submanifold by using the space of metric ribbon trees and the moduli space of constant pseudoholomorphic discs. (Actually the construction works for any oriented manifold L.) The meaning of “classical” is the A∞ algebra without filtration. It is defined over

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Z. We should note that in our geometric construction of the classical A∞ algebra, the problem of the transversality at the diagonal plays a crucial role. The proofs related to this transversality will be carried out in Section 7.2. In this section we state the main propositions needed in a precise manner and explain the main geometric ingredients needed for the construction of the deformation. First, we recall the notion of metric ribbon trees. We refer to [FuOh97] for more details, where the idea of using the metric ribbon tree in Morse theory and in topological field theory appeared in a rigorous manner for the first time. A ribbon tree is a pair (T, i) where T is a tree and i : T → D2 is an embedding such that (3.4.1.1) (3.4.1.2) (3.4.1.3)

no vertex of T has 2 edges, if v ∈ T is a vertex with one edge then i(v) ∈ ∂D 2 , i(T ) ∩ ∂D2 consists of vertices of one edge.

Let Gk be the set of (T, i, v1 ) where (T, i) is an isotopy class of ribbon tree such that #i(T ) ∩ ∂D2 = k and that v1 ∈ i(T ) ∩ ∂D2 . We call a vertex exterior if it lies on ∂D2 and interior otherwise. We order the set of exterior vertices so that v1 is the first one and that the order is compatible with the counter clockwise orientation on ∂D2 . We call an edge exterior if it contains an exterior vertex, and interior otherwise. 0 For each t = (T, i, v1 ) ∈ Gk , we denote by Cext (t) the set of all exterior vertices 0 1 1 and by Cint (t) the set of all interior vertices. Similarly we denote by Cext (t), Cint (t) the set of exterior and interior edges respectively. 1 For each t = (T, i, v1 ) ∈ Gk , let Gr(t) be the set of all maps  : Cint (t) → R+ . We put Grk = ∪t∈Gk Gr(t) and call its element a metric ribbon tree. We regard (e) as the length of the edge and set the length of exterior edge to be infinite. We refer to [FuOh97] for the description of the topology put on Grk . Denote the set 1 (t) → R+ ∪ {∞} by Gr(t) and their unions by Grk . The space of all maps  : Cint Grk is a compactification of Grk . , following Section 10 in [FuOh97]. We next define a map Θ : Grk → Mmain,reg k Let (t, ) ∈ Grk . We take an Euclidean rectangle Le = [0, (e)] × [0, 1] for each 1 1 e ∈ Cint (t) and Le = (−∞, 0] × [0, 1] for e ∈ Cext (t). We remove {0, (e)} × {1/2} ⊂ 0 [0, (e)] × [0, 1], {0} × {1/2} ⊂ (−∞, 0] × [0, 1]. For given v ∈ Cint (t), we consider   edges e, e containing v where e is the edge next to e according to the cyclic order we put on the set of edges containing v. Now if both e and e are outgoing edges, we glue {0} × (1/2, 1] ⊂ ∂Le and {0} × [0, 1/2) ⊂ ∂Le . We carry out a similar gluing process for other cases of the orientations put on e and e . See Figure 3.4.1.

Figure 3.4.1 We have thus obtained a space X0 (t, ) together with an (incomplete) flat metric. We can conformally fill the holes of X0 (t, ) and obtain a space X(t, )

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equipped with a complex structure and a singular Riemannian metric. We denote Θ(t, ) := [X(t, )] which is the isomorphism class of X(t, ). In Theorem 10.4 [FuOh97], the first and the second named authors proved that the map Θ : Grk → Mmain,reg is a homeomorphism. (Recall that Mmain,reg is diffeomork k k−3 , see Section 2.1.) We used this homeomorphism to define a smooth phic to R structure on Grk . We remark that Θ can be extended to the compactification Grk of Grk and leads to a homeomorphism Θ : Grk → Mmain . We have thus chosen a singular metric k main,reg . We modify this family of metrics in a neighborhood on each element of Mk of singular points and obtain a smooth family of smooth Riemannian metrics. We will use these metrics to define appropriate Sobolev norms in Section 7.1. Now we are going to construct the classical A∞ algebra associated to each oriented Lagrangian submanifold L in our symplectic manifold M . Let Δk+1 ⊂ Lk+1 = L × · · · × L (k + 1 times direct products of L) be the main diagonal where Δk+1  L and let NΔk+1 Lk+1 be the normal bundle of Δk+1 in Lk+1 . We identify NΔk+1 Lk+1 with a tubular neighborhood TubeΔk+1 Lk+1 of Δk+1 in Lk+1 . For β ∈ Π(M, L), we denote by Mmain k+1 (β) the main component of the moduli space of genus 0 stable pseudo-holomorphic maps from (Σ, ∂Σ) to (M, L) in class β with k + 1 marked points on the boundary (see Definition 2.1.27). For the case β0 = 0, k + 1 > 2, main Mk+1 (β0 ) is the moduli space of constant maps. This moduli space is transversal, namely it has a correct dimension k − 2 + n. However the evaluation map k+1 ev : Mmain is not a submersion. So we take the following Kuranishi k+1 (β0 ) → L structure on it. (See Section A1 for the definition of Kuranishi structure.) We take Uk+1 = TubeΔk+1 Lk+1 × Grk+1 . We pull back the normal bundle NΔk+1 Lk+1 to Uk+1 and take it as an obstruction bundle Ek+1 . We can find a section sk+1 : Uk+1 → Ek+1 which vanishes only at Δk+1 × Grk+1 by identifying TubeΔk+1 Lk+1 with NΔk+1 Lk+1 . Then (Uk+1 , Ek+1 , sk+1 ) defines a Kuranishi structure, (with trivial automorphism groups) and ev = (ev1 , · · · , evk , ev0 ) : Uk+1 → Lk+1 (which is actually the composition of an open embedding and the projection) is a submersion. It means that ev is weakly submersive (Definition A1.13) with respect to this Kuranishi structure. Next we have to choose a suitable Kuranishi structure on Mmain k+1 (β) to construct the filtered A∞ algebra. This is because of the transversality problem which we mentioned above. We will discuss this point in Chapter 7 Sections 7.1 and 7.2 in details. In this section we just state the following proposition without elaborating the discussion of the Kuranishi structure at this stage. Proposition 3.4.2. There exists an oriented Kuranishi structure on the moduli space Mmain k+1 (β) such that it is compatible with the compactification and the Kuranishi structure on Mmain k+1 (β0 ) coincides with the above Kuranishi structure. We will prove Proposition 3.4.2 in Section 7.1, where the precise meaning of the compatibility mentioned in the proposition is given (Proposition 7.1.2). Roughly speaking it means the following. The boundary component that appears in the compactification of Mmain k+1 (β) can be identified with a fiber product of various main  other Mk +1 (β )’s. The fiber product of the space with Kuranishi structure has the induced Kuranishi structure (see Section A1.2). The compatibility means that

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the Kuranishi structure on the boundary induced by the fiber product coincides with the Kuranishi structure induced from that of Mmain k+1 (β) as the boundary. We next state the main result on the existence of appropriate perturbations of the original s, the Kuranishi map. We consider a countable set XL of smooth singular simplices on L such that if P ∈ XL then each of its face ∂i P is contained in XL . We take such a countable set because we use the method of “smooth correspondence” and need the transversality argument for which we have to use the Baire category theorem. This point will be discussed in Chapter 7 in details. We write an element in XL as P = (|P |, f ). Here |P | is a simplex and f is a smooth map |P | → L. (We write P in place of |P | sometimes by abuse of notation.) Now, we consider a free Z module generated by XL . We write it as ZXL . We are going to define the structures of A∞ algebra on ZXL . Let (Pi , fi ) ∈ XL . We put main (3.4.3) Mmain k+1 (β0 ; P1 , · · · , Pk ) = Mk+1 (β0 )(ev1 ,··· ,evk ) ×f1 ×···×fk (P1 × · · · × Pk ).

Here the right hand side of (3.4.3) is the fiber product in the sense of the space of Kuranishi structure. We define it in general in Section A1.2 Lemma A1.39. Here we explain what it is in our case. We defined the Kuranishi structure (Uk+1 , Ek+1 , sk+1 ) of Mmain k+1 (β0 ). The evaluation map ev = (ev1 , · · · , evk , ev0 ) : Uk+1 → Lk+1 is a submersion hence the fiber product (3.4.4)

Uk+1 (P1 , · · · , Pk ) = Uk+1 (ev1 ,··· ,evk ) ×f1 ×···×fk (P1 × · · · × Pk )

is transversal and is a manifold (with possibly corners) equipped with a smooth triangulation. The bundle Ek+1 → Uk+1 can be pulled back to give a bundle, which we denote by Ek+1 (P1 , · · · , Pk ). The section sk+1 of Ek+1 (the Kuranishi map) induces a section of Ek+1 (P1 , · · · , Pk ), which we write sP1 ,··· ,Pk . Then, (Uk+1 (P1 , · · · , Pk ), Ek+1 (P1 , · · · , Pk ), sP1 ,··· ,Pk ) defines a Kuranishi structure on the fiber product of the right hand side of (3.4.3), regarded as a topological space. From now on we regard that Mmain k+1 (β0 ; P1 , · · · , Pk ) is a space with Kuran(β ; P ishi structure and ev0 : Mmain 0 1 , · · · , Pk ) → L is defined and is a strongly k+1 continuous map. (See Definition A1.13 for its definition). Now we will prove the following theorem in Section 7.2 (Proposition 7.2.35). We put P = (P1 , · · · , Pk ), Pi ∈ XL . Proposition 3.4.5. There exist a choice of XL and a system of perturbations  (single valued sections of obstruction bundles) sP of Mmain k+1 (β0 ; P ), such that the following holds. (3.4.6.1) sP is transversal to 0. (3.4.6.2) sP is compatible at the compactification. −1 (3.4.6.3) There is a smooth triangulation of s−1  (0) so that (sP  (0), ev0 ) can be P regarded as a Z linear combination of elements of XL . (3.4.6.4) The inclusion ZXL ⊂ C(L; Q) induces an isomorphism of H(ZXL , δ) to the cohomology H(L; Q) of L. Some explanation on (3.4.6.3) is in order. By (3.4.6.1), s−1  (0) is a smooth P manifold. We take a smooth triangulation together with an order of vertices. Each

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simplex of this triangulation can be canonically identified to the standard simplex by using order of the vertices. We restrict ev0 : s−1  (0) → L to one of the simplices. P We then obtain a smooth singular simplex of L. (3.4.6.3) asserts that it is an element of XL . −1 Let Δda , a ∈ A be the set of all simplices of s−1  (0) of dimension d = dim sP  (0) P Hereafter we write  −1 s ±(Δda , ev0 ), Mmain k+1 (β0 ; P1 , · · · , Pk ) = (sP  (0), ev0 ) = a∈A

which is regarded as a smooth singular chain of L. Here the sign ± in the above  formula is defined as follows. Our moduli space Mmain k+1 (β0 ; P ) has an orientation d (Chapter 8). Each of the simplex Δa is identified with the standard simplex in Rd with standard orientation. We take + sign if and only if these two orientations coincide. (See Section 7.2 Remark 7.2.15 (3).) ∞ Now we define the operations mL = {mL k }k=0 which define the A∞ algebra structure on ZXL as follows: (3.4.7.1) (3.4.7.2) (3.4.7.3)

mL 0 = 0, n mL 1 (P ) = (−1) ∂P , L s mk (P1 , · · · , Pk ) = Mmain k+1 (β0 ; P1 , · · · , Pk ) ,

where n = dim L. The notation ∂ in (3.4.7.2) is the usual (classical) boundary operator. Here we regarded P as a chain. See Remark 3.5.8 (1). We note that Proposition 3.4.2 and the choice of XL yield that we can define the fiber product in the right hand side in (3.4.7.3) in the sense of Kuranishi structure. The orientation of the moduli space in (3.4.7.3) is defined in Definition 8.4.1, Chapter 8. Here we equip Δk × Grk+1 with the orientation transferred from the one on the moduli main ∼ space Mmain k+1 (β0 ) of constant maps via the identification Θ : Gr k+1 → Mk+1 = main Mmain k+1 (β0 ). The orientation on Mk+1 (β0 ) will be given in Chapter 8, Section 8.6. On the other hand, we have defined another orientation on Mmain,reg in Subsection k+1 main 2.1.2. By analyzing orientation on Mmain and using the map Θ : Gr k+1 → Mk+1 , k+1 we can find an orientation on Δk × Grk+1 . This orientation might be different from the one defined above as the zero set s−1 k+1 (0). One can see the difference exactly but we omit it, because we do not need it. Then we can obtain the following theorem. Theorem 3.4.8. Let L be a smooth oriented manifold. Then there exists a countable set XL of smooth singular simplices on L which satisfies the following properties: (3.4.9.1) (ZXL , mL ) is an A∞ algebra over Z. (3.4.9.2) There is a smooth simplicial decomposition of L by which we regard P D[L] as an element of ZXL . (Here P D denotes the Poincar´e duality.) Then P D[L] is a homotopy unit of (ZXL , mL ). (3.4.9.3) The cohomology group of (ZXL , mL 1 ) is isomorphic to the cohomology group (over Z-coefficient) of L. Here (3.4.9.1) follows from Proposition 3.4.5 (especially (3.4.6.2)). The detailed proof is given in Subsections 7.2.4-8. We now explain (3.4.9.2). We take a smooth simplicial decomposition of L and order its vertices. Then each simplex of it can be identified with the standard

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simplex and hence determines a singular simplex. The sum  L ±Δdim (3.4.10) a a

of all such singular simplices of dimension dim L represents the fundamental homology class [L]. (Here ± is determined whether the orientation of the standard simplex coincides with the orientation of L or not.) (See Subsection 7.3.2.) In this book, we regard singular chain complex as a cochain complex. Hence the sum (3.4.10) above is regarded as an element of S 0 (L; Z). We write it as P D[L]. The property (3.4.9.2) will be proved in Section 7.3. (3.4.9.3) follows from (3.4.6.4). We call (ZXL , mL ) the classical A∞ algebra associated to L. We would like to mention that there are works [McC04, Wil05] closely related to Theorem 3.4.8. Remark 3.4.11. We remark that in Section 7.2 we construct A∞ structure not directly by using the multisections sP . Instead, we construct an AK structure for each K directly using sP (see Definition 4.4.6) and then obtain an A∞ structure which is AK homotopy equivalent thereto (Definition 4.4.8). The reason why we can not directly construct an A∞ structure at once will be explained in Subsection 7.2.3. This point is quite technical. The same remark applies to other constructions of this chapter. However we do not make this remark again. In practice it does not matter whether we construct an A∞ structure at once or construct an AK structure for each K and then obtain an A∞ structure in somewhat indirect way. In Chapter 4 (as a part of Theorem 4.1.1 proven there) we will find that the weak homotopy equivalence class of the A∞ algebra (ZXL , mL ) is uniquely determined by L. Namely, the A∞ algebra (ZXL , mL ) is independent of the choices of the perturbed section sk constructed in Theorem 3.4.8 and the countable set XL , up to weak homotopy equivalence. Moreover, in Section 7.5, we prove that when we put RXL = ZXL ⊗ R and extend mL to RXL , then (RXL , mL ) is weak homotopy equivalent to the de Rham complex of L over R (Theorem X). 3.5. The filtered A∞ algebra associated to a Lagrangian submanifold In this section, we show that the construction of Chapter 2 and its generalization lead to a geometric realization of the filtered A∞ algebra on each relatively spin Lagrangian submanifold L of M . Our main theorem in this section is Theorem 3.5.11. We again emphasize that we do not yet impose the condition [ok (L)] = 0 on the obstruction classes ok (L). In this section we put R = Q. Let S k (L; Q) be the free Q module generated by codimension k smooth singular simplices of L. Let X1 (L) be a countable set of smooth singular simplices with X1 (L) ⊃ XL , X1 (L) = XL where XL is as in Theorem 3.4.8. We denote by C k (L; Q) the submodule of S k (L; Q) generated by X1 (L). (As in the previous section, we use the method of “smooth correspondence”. For this purpose, we need to use the transversality argument and the Baire category theorem which is why the countably generated complex C k (L; Q) is used. See Section 7.2 for the construction of C k (L; Q).) We sometimes write an element of X1 (L) as P = (|P |, f ) as in Section 3.4. We define C • (L; Λ0,nov ) to be the completion of C • (L; Q) ⊗ Λ0,nov . For the

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103

convenience of notations, we put C • = C • (L; Λ0,nov ) := (C • (L; Q) ⊗ Λ0,nov ) . Recall that the element T λ eμ in Λ0,nov has degree 2μ. Thus the degree in C (L; Λ0,nov ) is the sum of the degree in C • (L; Q) and the degree of the coefficients in Λ0,nov . Using the filtration on Λ0,nov , we can uniquely define the filtration on C k (L; Λ0,nov ) which satisfies the following conditions: •

C k (L; Q) ⊂ F 0 C k (L; Λ0,nov ) and C k (L; Q) ∩ F λ C k (L; Λ0,nov ) = {0} We now define the homomorphisms

for λ > 0.

mk : Bk (C(L; Λ0,nov )[1]) → C(L; Λ0,nov )[1] for k ≥ 0. To do this, we recall that Mk+1 (β) is the set of pairs ((Σ, z), w) where (Σ, z) ∈ Mk+1 and w : (Σ, ∂Σ) → (M, L) is a pseudo-holomorphic map which represents the class β. Let Mmain k+1 (β) be the subset of Mk+1 (β) consisting of elements ((Σ, z), w) where (Σ, z) is in the main component. (See Definitions 2.1.20 and 2.1.27.) Let Pi ∈ X1 (L), we consider fiber product (3.5.1)

Mmain k+1 (β) (ev1 ,··· ,evk ) ×(f1 ×···×fk ) (P1 × · · · × Pk )

as in the last section. Let P = (P1 , · · · , Pk ), Pi ∈ X1 (L). The fiber product (3.5.1)  is a space with Kuranishi structure, we denote (3.5.1) by Mmain k+1 (β; P ). Proposition 3.5.2. Suppose that L is a relatively spin Lagrangian submanifold. Then there exist a choice of X1 (L) and a system of multisections sβ,P of Mmain (β; P ) with the following properties: k+1

(3.5.3.1)

 The (virtual) dimension of Mmain k+1 (β; P ) is given by  n− (gi − 1) + μL (β) − 2,

where n = dim L, (Pi , fi ) ∈ C gi (L; Q) and μL (β) is the Maslov index. (3.5.3.2) sβ,P is transversal to 0. (3.5.3.3) sβ,P is compatible at the boundary. (3.5.3.4) (s−1  (0), ev0 ) is a Q linear combination of elements of X1 (L). β,P More precisely (3.5.3.4) is stated as follows. By Lemma A1.26 the set s−1  (0) β,P has a smooth triangulation. Take one of such triangulations. We also fix an order of the vertices of this triangulation. Each simplex Δda of dimension d = dim s−1  (0) β,P of it comes with multiplicity mulΔda . (See Definition A1.27.) The restriction of ev0 to Δda defines a smooth singular simplex (Δda , ev0 ). Then (3.5.3.4) means that each of (Δda , ev0 ) is an element of X1 (L). We remark that the virtual fundamental chain (s−1  (0), ev0 ) is defined to be β,P (s−1  (0), ev0 ) = β,P



±mulΔda (Δda , ev0 )

a∈A

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

where {Δda | a ∈ A} is the set of all simplices of dimension d = dim s−1  (0) and ± β,P is determined by considering the orientation. (See Definition A1.28.) Remark 3.5.4. We should emphasize that the choice of the multisection sβ,P depends on β and P . We have to choose the multisection sβ,P satisfying certain compatibility conditions (3.5.3.3). (See Compatibility Conditions 7.2.38 and 7.2.44.) We also need to choose the triangulation of s−1  (0) to be compatible for β,P various choices of β, P . We will discuss these point in Section 7.2 in detail. Proof. We will explain our construction of the Kuranishi structure and state Propositions 7.1.1, 7.1.2 and 7.2.35 in Chapter 7 which are about how we choose C(L; Q) and the multisection s above. As for the orientations on Mmain k+1 (β) and for the fiber product of Kuranishi structures, see Chapter 8 Sections 8.4 and 8.5. Here we prove only the statement on the dimension in (3.5.3.1). We compute the dimension   μL (β) + k + 1 − 3 + n − gi = μL (β) + n − (gi − 1) − 2, 

as required.

Definition 3.5.5. Let the multisection s and the countably generated sub s module C(L; Q) be as in Proposition 3.5.2. We define Mmain k+1 (β; P ) by −1  s Mmain k+1 (β; P ) := (sβ,P  (0), ev0 ).

(As for the orientation of the above moduli space, see Definition 8.4.1 in Chapter 8). Now we define the maps mk . We recall that we have the element β0 = 0 ∈ G(L), (see (3.1.8) in Section 3.1 for the definition of G(L)). It satisfies μL (β0 ) = 0 and ω(β0 ) = 0. Definition 3.5.6. (1) For (P, f ) ∈ C k (L, Q), we define  M1 (β)s for β = β0 m0,β (1) = 0 for β = β0 ,  Mmain (β; P )s for β = β0 2 m1,β (P, f ) = (−1)n ∂P for β = β0 , and

e = P D([L]) ∈ C 0 (L) = C[1]−1 (L), where [L] is the fundamental cycle of L and P D denotes the Poincar´e duality. The notation ∂ in the definition of m1,β0 is the usual (classical) boundary operator. (2) For each k ≥ 2 and Pi ∈ C gi (L, Q), we define mk,β by  s mk,β (P1 , . . . , Pk ) = mk,β (P1 ⊗ · · · ⊗ Pk ) = Mmain k+1 (β; P ) . (3) Then we define mk (k ≥ 0) by  mk,β ⊗ T ω(β) eμL (β)/2 . (3.5.7) mk = β∈G(L)

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105

Remark 3.5.8. (1) In the definition of m1,β0 above, we see P as a chain and ∂ is the usual boundary operator as we noted. When we see P as a cochain, we have to change the sign into the following; m1,β0 (P ) = (−1)n+deg P +1 dP, where deg P is the degree of P before shifted. Here we use the convention of the sign on the Poincar´e duality as follows: For a chain S in L, the Poincar´e dual P D(S), regarded as a current, satisfies " " α|S = P D(S) ∧ α S

L

(L). We note that this convention is consistent with the confor any α ∈ Ω vention on the orientation of the normal bundle given in Section 8.6. Under this convention, we can see that P D(∂S) = (−1)deg S+1 d(P D(S)). (2) We note that mk depends on the almost complex structure J and perturbation s of Kuranishi map (that is a multisection of the obstruction bundle. See Section A1.) We will study this dependence in detail later in Section 4.6. (3) We note that the condition (3.2.12.7) is satisfied since m0,β0 (1) = 0. (4) As in (3.2.13), we can reduce the coefficient Λ0,nov of (C(L, Λ0,nov ), m) to Q to get an A∞ algebra (C, m), since Condition (3.2.13.3) is a consequence of (3.1.7.2). Because m0,β0 = 0, we have m0 = 0 in this case. Furthermore, we should note that (C, m) is not a D.G.A., but an A∞ algebra. Let us consider the case β = β0 (= 0). If P1 = P2 , Mmain (β0 ; P1 , P2 ) is (as a set) nothing but 2 P1 ∩ P2 . But an essential point is that we can not define self intersection at the chain level. Similarly we can not expect that mk = 0 for k ≥ 3, because we do not have the transversality for the diagonal case. We will discuss this problem in more detail at Section 7.2 in Chapter 7. In the previous section Section 3.4, for the Lagrangian submanifold L, we constructed the classical A∞ algebra (ZXL , mL ) over Z, admitting the transversality argument. Then we can show that (C(L, Λ0,nov ), m) is an A∞ deformation of (QXL , mL ) in the sense of Definition 3.2.34. (See Theorem 3.5.11 below.) We also remark one more point about the sign. According to (3.2.5), we should check dim S

(3.5.9)

Mmain (β0 ; P1 , P2 ) = (−1)deg P1 (deg P2 +1) P1 ∩ P2 , 3

for β = β0 (= 0). This will be proved in Corollary 8.6.4 in Chapter 8. Lemma 3.5.10. The formal sum (3.5.7) converges and defines a continuous operation mk : Bk (C(L; Λ0,nov )[1]) → C(L; Λ0,nov )[1]. Proof. G(L) satisfies Condition 3.1.6 by Gromov’s compactness theorem.  Recall we put deg β = μL (β) and shift the degree of (P, f ) by 1. Then the dimension formula (3.5.3.1) implies that the degree of mk on B(C[1]) is +1. The following is the main theorem in this section: Theorem 3.5.11. Suppose that L is a relatively spin Lagrangian submanifold. Then we have the following: (3.5.12.1)

(C(L; Λ0,nov ), m) is a filtered A∞ algebra with homotopy unit e.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

(3.5.12.2) (C(L; Λ0,nov ), m) is G(L)-gapped. (3.5.12.3) (C(L; Λ0,nov ), m) is an A∞ deformation of the classical A∞ algebra (QXL , mL ) in Theorem 3.4.8, in the sense of Definition 3.2.34. We note (3.5.12.3) means the following: when we reduce the coefficient ring Λ0,nov to Q as in (3.2.13), we have an A∞ algebra (C(L; Q), m) over Q with m0 = 0: Then the restriction of m to QXL ⊂ C(L; Q) coincides with mL constructed in Section 3.4. Remark 3.5.13. As mentioned in Remark 3.4.11, we will not directly construct the filtered A∞ structure out of the multisection s. Instead, for any energy E ≥ 0 and k fixed, we will construct an An,K structure using only the moduli spaces of pseudo-holomorphic discs with energy bounded by E. See Section 7.2.6 for the precise definition of the An,K structure and details of the constructions are in Sections 7.2.4-8. Proof. The properties (3.2.12.1)-(3.2.12.5) follow directly from our construction of (C(L; Λ0,nov ), m). We now prove that {mk }k≥0 satisfy the A∞ formulae, assuming the sign rules in (3.5.17) below. The sign will be precisely checked in Section 8.5. Let us prove that d ◦ d = 0. This is a consequence of analysis of the boundary of Mmain k+1 (β; P1 , · · · , Pk ). (See (3.5.1).) We find that its boundary is the sum of  Mmain k+1 (β; P1 , · · · , ∂Pi , · · · , Pk ) i

and the terms described by Figure 3.5.1 below.

Figure 3.5.1 On the other hand, in order to prove d ◦ d = 0, we note that it is enough to show that    (−1)deg P1 +···+deg Pi−1 +i−1 (3.5.14) β1 +β2 =β k1 +k2 =k+1 i mk1 ,β1 (P1 , · · · , mk2 ,β2 (Pi , · · · , Pi+k2 −1 ), · · · , Pk ) = 0. (See Remark 3.2.21 (1).) Here we write Pi in place of (Pi , fi ) for simplicity. We divide the left hand side of (3.5.14) into 3 terms, corresponding to the cases β1 = 0

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3.6. BOUNDING COCHAINS AND THE A∞ MAURER-CARTAN EQUATION

107

and k1 = 1, β2 = 0 and k2 = 1, and the others. Then we can rewrite the left hand side of (3.5.14) as follows: m1,0 mk,β (P1 , · · · , Pk )  + (−1)deg P1 +···+deg Pi−1 +i−1 mk,β (P1 , · · · , m1,0 (Pi ), · · · , Pk ) i

(3.5.15)





β1 +β2 =β, k1 +k2 =k+1; β1 =0 or k1 =1, β2 =0 or k2 =1

i

+

(−1)deg P1 +···+deg Pi−1 +i−1

mk1 ,β1 (P1 , · · · , mk2 ,β2 (Pi , · · · , Pi+k2 −1 ), · · · , Pk ). By Definition 3.5.6, we have m1,0 = (−1)n ∂, where ∂ is the classical boundary map. Hence the first term in (3.5.15) is nothing but (−1)n ∂Mmain k+1 (β : P1 , · · · , Pk ),

(3.5.16.0)

and the second term in (3.5.15) is the sum of i−1

(−1)n+

(3.5.16.1)

j=1 (deg Pj +1)

Mmain k+1 (β : P1 , · · · , ∂Pi , · · · , Pk ).

The third term in (3.5.15) geometrically corresponds to the moduli spaces described by Figure 3.5.1. This is the sum of (3.5.16.2)

(−1)

i−1

j=1 (deg Pj +1)

Mmain k−k2 +2 (β1 ; P1 , · · · , Mmain k2 +1 (β2 ; Pi , · · · , Pi+k2 −1 ), · · · , Pk ).

Moreover, as for the orientations of these spaces, we can show the following: (3.5.17.1)

(−1)n Mmain k+1 (β : P1 , · · · , ∂Pi , · · · , Pk ) ⊆ (−1)n+1+

i−1

j=1 (deg Pj +1)

∂Mmain k+1 (β; P1 , · · · , Pk )

and (3.5.17.2)

main Mmain k−k2 +2 (β1 ; P1 , · · · , Mk2 +1 (β2 ; Pi , · · · , Pi+k2 −1 ), · · · , Pk )

⊆ (−1)n+1+

i−1

j=1 (deg Pj +1)

∂Mmain k+1 (β; P1 , · · · , Pk ).

The proofs of (3.5.17.1) and (3.5.17.2) will be given in Proposition 8.5.1 in Chapter 8. Therefore we find that (3.5.16.0) and the sum of (3.5.16.1) and (3.5.16.2) cancel each other and so (3.5.15) becomes zero. Thus we have proved that d ◦ d = 0. (3.5.12.2) is immediate from construction. For the construction of a homotopy unit, we also have to study transversality. The proofs will be postponed until Section 7.3 (and Section 8.10 for the signs). (3.5.12.3) is immediate from its construction which will be given in Subsections 7.2.4 - 7.2.7, except the sign in (3.5.9) which will be checked in Corollary 8.6.4.  3.6. Bounding cochains and the A∞ Maurer-Cartan equation So far we have constructed a filtered A∞ algebra for a relatively spin Lagrangian submanifold without assuming any unobstructedness conditions. Because of the presence of m0 , the operator m1 can not define the cohomology in general. In this section, we study the obstruction to deforming m1 so that the deformed m1 defines the cohomology. For this purpose we first introduce an algebraic notion

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of bounding cochains (solutions of Maurer-Cartan equation) in a filtered A∞ algebra in Subsection 3.6.1, and we construct a sequence of the classical cohomology classes in Subsection 3.6.2 which describes the obstruction to the existence of the bounding cochains. Further in Subsection 3.6.3 we introduce a weaker unobstructedness condition by using the unit or the homotopy unit in the A∞ algebra. These constructions can be done in a purely algebraic manner. 3.6.1. Bounding cochains and deformations. Let (C, m) be a filtered A∞ algebra over Λ0,nov and b ∈ (C[1])0 = C 1 with b ≡ 0 (mod Λ+ 0,nov ). We put (3.6.1)

 eb := 1 + b + b ⊗ b + · · · + b ⊗ · · · ⊗ b + · · · ∈ B(C[1]).   k times

We do not put the factorials here unlike the usual definition of the exponential. This is because we used only the main component among the k! components of Mk+1 when we realized the map mk,β geometrically in Section 3.5. The condition that b ≡ 0 (mod Λ+ 0,nov ) implies that the right hand side in (3.6.1) converges with respect to the topology defined by the energy filtration, and so it defines an element  of B(C[1]). We remark the following:  Lemma 3.6.2. For a non-zero element x of B(C[1]), x = eb for some b ∈  B1 (C[1]) = C[1] if and only if Δx = x ⊗ x, where Δ is the coproduct on B(C[1]) defined by (3.2.14). Proof. It is easy to check the ∞statement for “only if” and so we will only prove the one for “if”. Write x = k=0 xk , xk ∈ Bk (C[1]) and assume (3.6.3)

Δx = x ⊗ x.

Note Δ : BC[1] → BC[1] ⊗ BC[1] where we write ⊗ for the tensor product of the right hand side of the above formula. We say the BCk1 [1] ⊗ Bk2 C[ 1] k1 +k2 =k

component of BC[1] ⊗ BC[1], the k-th order term and denote the k-th order term of y by (y)k . We also write BCk [1] component of y ∈ BC[1] by (y)k . Using the definition (3.2.14) of the coproduct Δ and comparing the 0-th order term of the two sides of (3.6.3) we get the equation (x0 )2 = x0 ,

x0 ∈ Λ0,nov

and hence x0 = 0 or 1. In case x0 = 1, comparison of the 1st-order terms of (3.6.3) gives rise to the equation 1 ⊗ x1 + x1 ⊗ 1 = 1 ⊗ x1 + x1 ⊗ 1. In other words, the equation gives no restriction on the 1st-order term. We set b := x1 . Now for the second order terms in general, it is easy to see that the coproduct satisfies (Δx2 )2 = 1 ⊗ x2 + π(x2 ) + x2 ⊗ 1. Here π : B2 C[1] → B1 C[1] ⊗ B1 C[1] is defined by π(x ⊗ y) = x ⊗ y.

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Therefore the second order term of (3.6.3) becomes 1 ⊗ x2 + π(x2 ) + x2 ⊗ 1 = 1 ⊗ x2 + x1 ⊗ x1 + x2 ⊗ 1 and hence x2 = x1 ⊗ x1 = b ⊗ b. Repeating this straight-forward order-by-order comparisons of the two sides of (3.6.3), we prove x = eb . In case x0 = 0, a similar reasoning proves x = 0. This finishes the proof.  Definition 3.6.4. We say that b ∈ (C[1])0 with b ≡ 0 (or mod Λ+ 0,nov ) is b  a bounding cochain or a solution of Maurer-Cartan equation if de = 0. (Recall  ˆ k and m ˆ k is defined in (3.2.15).) A filtered A∞ algebra is said to that d = m be unobstructed if there exists a bounding cochain and obstructed otherwise. We  denote by M(C) the set of all bounding cochains b.  b = 0 is an inhomogeneous A∞ -version of the Remark 3.6.5. Our equation de Maurer-Cartan or the Batalin-Vilkovisky master equation [BaVi93]. In fact we can write it as  δbβ + mk,βk+1 (bβ1 , · · · , bβk ) = 0 k=0,1,2,··· βk+1 ,β1 ,··· ,βk ∈G, βk+1 +β1 +···+βk =β (k,βk+1 )=(1,(0,0))

 where b = T λ(β) eμ(β)/2 bβ . The relation of the BV master equation to the deformation theory is discussed in [ASKZ97, GugSt86, HiSc97, Kon03, Sche98]. (See also [Fuk02III].) Let (C, m) be a filtered A∞ algebra with bounding cochains b1 , b0 . We define δb1 ,b0 : C[1] −→ C[1] as follows. Definition 3.6.6. δb1 ,b0 (x) =

 k1 ,k0 ≥0

mk1 +k0 +1 (b1 , · · · , b1 , x, b0 , · · · , b0 ).     k1

k0

Symbolically we may simply write this definition as δb1 ,b0 (x) = m(eb1 xeb0 ) in an obvious way. It then follows that (3.6.7)

 b1 xeb0 ) = eb1 δb ,b (x)eb0 + d(e  b1 )xeb0 + (−1)deg x+1 eb1 xd(e  b0 ). d(e 1 0

 b1 ) = d(e  b0 ) = 0. The second and the third term vanishes if d(e  b1 ) = d(e  b0 ) = 0, then δb ,b ◦ δb ,b = 0. Lemma 3.6.8. If d(e 1 0 1 0 Proof. By (3.6.7) we have  d(e  b1 xeb0 )) = d(e  b1 δb ,b (x)eb0 ) = eb1 ((δb ,b ◦ δb ,b )(x))eb0 , 0 = d( 1 0 1 0 1 0 from which the lemma immediately follows.



We remark that we can also deform the structure of the filtered A∞ algebra for any cochain b ∈ C[1]0 with b ≡ 0 mod Λ+ 0,nov , which is not necessarily a bounding cochain as follows. Later in Subsection 5.2.2, we will discuss deformations of the filtered A∞ homomorphisms.

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Definition 3.6.9. For any cochain b ∈ C[1]0 with b ≡ 0 mod Λ+ 0,nov , we put mbk (x1 , · · · , xk )  = mk+ i (b, · · · , b, x1 , b, · · · , b, · · · , b, · · · , b, xk , b, · · · , b)         0 ,··· ,k

0

1

k−1

k

= m(e x1 e x2 · · · xk−1 e xk e ) b

b

b

b

for k = 0, 1, 2, · · · . We note that mb0 (1) = m(eb ). Then we can easily see the following. Proposition 3.6.10. (C, mb ) is also a filtered A∞ algebra. And mb0 = 0 if and  b ) = 0. In this case mb is nothing but δb,b in Definition 3.6.6. only if b satisfies d(e 1  which is given by Proof. We extend mbk to db as a coderivation on B(C[1]) db (x1 , · · · , xk ) =

k+1  k+1  =1

(−1)deg x1 +···+deg x−1 +−1 x1 ⊗ · · · ⊗ x−1

 =

⊗ m(eb x eb · · · eb x −1 eb ) ⊗ x ⊗ · · · ⊗ xk . When  = , we put m(eb ) in place of m(eb x eb · · · eb x −1 eb ) in the right hand side.  b x1 eb · · · eb xk eb ) = 0 and obtain the result We compare this formula with d ◦ d(e that db ◦ db = 0. Moreover by noticing the following identities  b ) = eb m(eb )eb = eb mb (1)eb , d(e 0 we have proven the second assertion.



 we put Definition 3.6.11. For b1 , b0 ∈ M(C), HF (C, b1 , b0 ; Λ0,nov ) = H(C, δb1 ,b0 ). When b1 = b0 , we simply write HF (C, b; Λ0,nov ) = H(C, mb1 ). We recall that when we consider a homotopy unit in an A∞ algebra C, we extended C to C + = C ⊕ Λ0,nov e+ ⊕ Λ0,nov f as in (3.3.1). We next study the relationship between bounding cochains of C and of C + . Proposition 3.6.12. If b ∈ C[1]0 = C 1 is a bounding cochain of (C, m), then it is also a bounding cochain of (C + , m+ ). The natural inclusion (C, mb ) → (C + , m+b ) induces an isomorphism H(C, mb1 ) → H(C + , m+b 1 ). Remark 3.6.13. From Theorem 4.2.45, we will find that the second assertion implies that the natural inclusion (C, mb ) → (C + , m+b ) gives a homotopy equivalence between the two filtered A∞ algebras. See Section 4.2 for the definition of the notion of the homotopy equivalence between A∞ algebras. Proof. The first half of the assertions is obvious. We prove the second half of the assertions. Clearly the natural inclusion is a cochain map and is injective. The quotient cochain complex is generated by [e+ ] and [f]. Then (3.3.5.2) implies d[f] = [e+ ]. Hence the quotient complex is acyclic. This finishes the proof.  Now we apply the above discussion to our geometric situation. We have the filtered A∞ algebra C(L; Λ0,nov ) associated to a relatively spin Lagrangian submanifold L.

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111

Definition 3.6.14. We say that b ∈ C(L; Λ0,nov )[1]0 ∈ C 1 (L; Λ0,nov ) is a  b = 0. We denote by M(L;  J, s) the set of all bounding bounding cochain if de  cochains b. We say L is unobstructed, if M(L; J, s) is nonempty.  J, s) may depend on the given almost complex structure The solution set M(L; J and a system s of multisections. However, unobstructedness of L is independent of J and s, which will be proved later in Section 4.6. When no confusion can occur,  J, s) as M(L)  we write M(L; for simplicity. 3.6.2. Obstruction for the existence of bounding cochain. In the last subsection, we introduced the notion of bounding cochains b and defined the cohomology group H(C, mb1 ). For geometric applications, it is important to know when   M(L) is nonempty. This subsection states conditions for M(C) to be nonempty in terms of the cohomology of (C, m1 ), (which is the usual cohomology of L over Q when C = C(L; Λ0,nov )). For this purpose we algebraically define a sequence of cohomology classes, which we call obstruction classes in H(C, m1 ). Let G ⊂ R≥0 × 2Z be a submonoid satisfying Condition 3.1.6. We write G = {(λi , μi )} such that i < j ⇒ λi ≤ λj . (Note that λ0 = μ0 = 0 and 0 < λ1 .) We consider a G-gapped filtered A∞ algebra C. (See Definition 3.2.26.) Remark 3.6.15. We remark that if G is another submonoid satisfying Condition 3.1.6 and containing G, then every G-gapped filtered A∞ algebra is also G -gapped. Definition 3.6.16. Let G ⊇ G. A G-gapped filtered A∞ algebra is called to be unobstructed over G if there exists a bounding cochain b written as b=

∞ 



T λi e

μ i 2

bi

i=1 

for some bi ∈ C[1]−μi = C

1−μi

, where we put G = {(λi , μi ) | i = 1, 2, · · · }.

Lemma 3.6.17. A G-gapped filtered A∞ algebra is unobstructed if and only if it is unobstructed over G for some G ⊇ G. The proof of the lemma is obvious. Now the main result of this subsection is the following. Theorem 3.6.18. Let G = {(λi , μi ) | i = 0, 1, 2, · · · } such that λi ≤ λj for i < j and λ0 = μ0 = 0 as before and (C, m) be a G-gapped filtered A∞ algebra. Then there are sequences of cocycles ok and cochains bk with the following properties: 1−μk

(3.6.19.1) [ok ] ∈ H 2−μk (C, m1 ) and bk ∈ C . (3.6.19.2) The cocycle ok is defined if bj and oj for j with λj  λk are defined. The cohomology class [ok ] depends on bj and oj for j with λj  λk . (3.6.19.3) The cochain bk is defined if [ok ] = 0. (3.6.19.4) (C, m) is unobstructed over G if and only if there exists a choice of bk inductively such that [ok ] = 0 for all k. We call [ok ] (k = 1, 2, · · · ) the obstruction classes of (C, m).

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Proof. Since (C, m) is a G-gapped filtered A∞ algebra, we can write  μi mk = T λi e 2 mk,i (λi ,μi )∈G

as in Definition 3.2.26. Here mk,i : Bk C[1] → C[1] and mk,0 = mk . To handle the case where some λi ’s coincide, we prepare some notations. When λi is the k-th (k) smallest (non zero) number among the set {λj }, we write λi = λi and put λ(k) = (k) (1) (1) λi . For example, if 0 < λ1 = · · · = λ  λ+1 , we write λ1 = λ1 , · · · , λ = λ (2) and λ+1 = λ+1 . We are going to construct oi and bi inductively on (k). (1)

(1)

If λ1 = · · · = λ

(2)

 λ+1 , we put

oi = m0,i (1) ∈ C[1]1−μi = C

2−μi

i = 1, · · · , .

,

Noting m0,0 = 0, we calculate Tλ

(1)

 

e

μi 2

m1,0 (oi ) =

i=1

 

T λi e

μi 2

m1,0 m0,i (1)

i=1 (1)

≡ m1 (m0 (1)) mod T λ Λ+ 0,nov which is zero by the A∞ formula. Since μi = μi for 1 ≤ i = i ≤  and m1,0 = m1 , we have m1 (oi ) = 0 for all i = 1, · · · , . Therefore oi define the cohomology classes [oi ] ∈ H 2−μi (C, m1 ),

i = 1, · · · , .

If [oi ] = 0 for all i (1 ≤ i ≤ ), we can choose bi ∈ C

1−μi

such that

m1,0 (bi ) + oi = 0. If we put b(1) =

 i=1

(3.6.20) m(eb(1) ) ≡

T λi e

μi 2

bi ∈ C 1 , we have

    μi μi (1) T λi e 2 m0,i (1) + T λi e 2 m1,0 (bi ) = 0 mod T λ Λ+ 0,nov . i=1

Now we assume that we have chosen bi for i with λi < λ(k) inductively so that m(eb(k−1) ) ≡ 0 mod T λ

(3.6.21)

(k−1)

Λ+ 0,nov ,

where (3.6.22)

b(i) =

∗i 

T λj e

μj 2

bj ∈ C 1 ,

∗i = max{j | λj < λ(i+1) }.

j=1 (k)

(k)

(k+1)

Definition 3.6.23. When λn = · · · = λn+m  λn+m+1 (in this case ∗k−1 = n − 1), we denote the coefficient of T λh e h = n, · · · , n + m.

μh 2

in m(eb(k−1) ) by oh ∈ C

Lemma 3.6.24. m1 (oh ) = 0.

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2−μh

for each

3.6. BOUNDING COCHAINS AND THE A∞ MAURER-CARTAN EQUATION

113

Proof. By the A∞ formula again, we have  b(k−1) ) 0 = (m ◦ d)(e  n+m n+m  μh  μh (k) λ(k) ≡T m e 2 oh ≡ T λh e 2 m1,0 (oh ) mod T λ Λ+ 0,nov . h=n

h=n

It follows from this that m1,0 (oh ) = 0 for each h = n, · · · , n + m.



We have thus defined the cohomology classes [oh ]. If they are all zero, we can 1−μh choose bh ∈ C satisfying m1,0 (bh ) + oh = 0.

(3.6.25) Lemma 3.6.26. where b(k)

m(eb(k) ) ≡ 0 mod T λ is defined by (3.6.22).

(k)

Λ+ 0,nov ,

Proof. We have m(eb(k−1) ) ≡

n+m 

T λh e

μh 2

oh

mod T λ

(k)

Λ+ 0,nov ,

h=n

since G = {(λi , μi ) | i = 0, 1, 2, · · · } is a monoid. The lemma then follows from (3.6.25).  We have proved (3.6.19.1),(3.6.19.2) and (3.6.19.3) by now. It is easy to see that if all of bk are defined (and all of [ok ] are zero) then  μk (3.6.27) b= T λk e 2 bk = lim b(k) k→∞

is a bounding cochain. In fact, it follows from Lemma 3.6.26 that m(eb ) ≡ 0 mod T λk Λ+ 0,nov  b ) = eb m(eb )eb = 0. for any k hence m(eb ) = 0. Therefore d(e On the contrary, if (C, m) is unobstructed over G, we can find a bounding cochain b as in (3.6.27). Then we find that bk satisfies (3.6.21) and (3.6.25). It means that we can define bk such that all of [ok ] vanishes. The proof of Theorem 3.6.18 is now complete.  Remark 3.6.28. It may happen that (C, m) is obstructed over some G but is unobstructed over some bigger G . This phenomenon is related to the existence of  the rational point of the moduli space M(C). (Compare [Fuk03I]). In case (C, m) = (C(L, Λ0,nov ), m) as in Theorem 3.5.11, we write ok as ok (L). We enumerate the elements of G = G(L) as {(λ(βi ), μ(βi )) | β0 = 0, β1 , β2 , · · · ∈ {β ∈ Π(L)) | M(β) = ∅}} with λ(βi ) ≤ λ(βi+1 ). Here λ(βi ) = ω[βi ] and μ(βi ) = μL (βi ) are the energy (symplectic area) and the Maslov index, respectively. We put o(L; βi ) := oi (L).

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

3.6.3. Weak unobstructedness and existence of Floer cohomology. It turns out [Oh93] Addenda, [Oh96II] that the Floer cohomology for the pair L(0) = L, L(1) = ψ(L) with ψ a Hamiltonian diffeomorphism can be defined under the weaker condition on L than the condition needed for a general pair (L(0) , L(1) ). This fact is important for various applications. In this subsection, we discuss the algebraic counterpart of this phenomenon. See also §3.7.3 below. Definition 3.6.29. (1) Suppose that C has a unit e. An element b ∈ (C[1])0 = C 1 with b ≡ 0 (mod Λ+ 0,nov ) is called a weak bounding cochain, or a weak Maurer-Cartan solution, if it satisfies m(eb ) = ce e,

(3.6.30) +(0)

+(0)

for some c ∈ Λ0,nov , where Λ0,nov is the degree 0 part of Λ+ 0,nov . We denote by weak (C) the set of all weak bounding cochains. We say C is weakly unobstructed M weak (C) = ∅. if M (2) Suppose that C has a homotopy unit. Then as in Definition 3.3.2, we have the associated A∞ algebra (C + , m+ ) = (C ⊕ Λ0,nov e+ ⊕ Λ0,nov f, m+ ) weak (C + ) = ∅ in the sense of with unit e+ . We say C is weakly unobstructed if M (1) above. Here we remind readers that the symbol e is the formal parameter encoding the Maslov index which has degree 2. Remark 3.6.31. When C has a strict unit e, we can also regard it as a weak (C + ) as well as M weak (C). They are homotopy unit. Thus we can consider M related to each other by weak (C) M

weak (C + ). b −→ b + cef ∈ M

In fact, when C has a strict unit, we can take (C + , m+ ) so that hk = 0 for k ≥ 2 in Definition 3.3.2. Therefore we can see that + + m+ (eb+cef ) = m(eb ) + m+ (eb cefeb ) = cee + cem+ 1 (f) = cee + ce(e − e) = cee . +

Conversely, let b+ ∈ C + [1]0 be a solution of m+ (eb ) = cee+ . By the degree reason, (0) we can write as b+ = b + c ef with c ∈ Λ0,nov . (Recall deg em = 2m and deg e = 0.) The same calculation shows that 

b  + cee+ = m+ (eb+c ef ) = m(eb ) + c em+ 1 (f) = m(e ) + c e(e − e).

Comparing the both hand sides, we have c = c and m(eb ) = cee. However, when C just has a homotopy unit (not a strict unit), the above argument does not hold. (Although we might be able to formally consider a set weak (C) of the solutions to the equation m(eb ) = cee on C itself for the homotopy M unit e ∈ C, we never consider the equation on C unless C has a strict unit.) When C has a homotopy unit, every statement related to weak unobstructedness of C should be interpreted as one for C + . Since C + is homotopy equivalent to C (see Remark 3.3.7 (3) and Lemma 4.2.55 in Section 4.2), we define weak (C) := M weak (C + ) M

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3.6. BOUNDING COCHAINS AND THE A∞ MAURER-CARTAN EQUATION

115

when C has only a homotopy unit. We warn the readers not to get confused that the following lemma may hold for the solutions of the formal equation m(eb ) = cee on C itself when e is not a strict unit. weak (C) then δb,b ◦ δb,b = 0, where δb,b is as in Lemma 3.6.32. If b ∈ M Definition 3.6.6. Proof. By using (3.2.19) and (3.6.7), we have  b xeb ) 0 = (m ◦ d)(e  b )xeb ) + (−1)deg x+1 m((eb xd(e  b )) = m(eb δb,b (x)eb ) + m(d(e = (δb,b ◦ δb,b )(x) + ce m(eb eeb xeb ) + (−1)deg x+1 ce m(eb x eb eeb ) = (δb,b ◦ δb,b )(x) + ce x + (−1)deg x+1 (−1)deg x ce x = (δb,b ◦ δb,b )(x).  weak (C) we define Definition 3.6.33. (1) For b ∈ M HF (C, b) =

Ker(δb,b : C + → C + ) . Im(δb,b : C + → C + )

weak (C) −→ Λ+(0) by the equation (2) We define a function PO : M 0,nov m(eb ) = PO(b)ee. We call this function a potential function. Remark 3.6.34. (1) The potential function we have defined here is closely related to the superpotential introduced in the physics literature ([KKLM00, HoVa00]). It has been partially calculated in some cases [ChOh03]. See also Subsection 3.7.7 for the simplest case. We will use the potential function in §3.7.3. (See also [FOOO08I,08II].) weak (C) = (2) When C has homotopy unit δb,b : C + → C + is defined for b ∈ M +  Mweak (C ). Lemma 3.6.32 below holds also in this case. If we assume the following condition in addition  m(x0 ⊗ f ⊗ x1 ⊗ · · · ⊗ xk−1 ⊗ f ⊗ xk ) ∈ C (k = 0, xi ∈ BC[1]), (3.6.35) m1 (f) − e+ ∈ C, then δb,b maps C to C and so its restriction δb,b : C → C is defined. Moreover the natural inclusion (C, δb,b ) → (C + , δb,b ) is a chain homotopy equivalence. Lemma 3.6.36. Let f : C1 → C2 be a filtered A∞ homomorphism. (1) For any b ∈ C1 [1]0 with b ≡ 0 mod Λ+ 0,nov , define (3.6.37) Here f(eb ) =



f∗ (b) := f(eb ) = f0 (1) + f1 (b) + f2 (b, b) + · · · .  ∞ (eb ). Then it induces a map f∗ : C1 [1]0 → C2 [1]0 and f k k=0

f∗ (b) ≡ 0 mod Λ+ 0,nov . (2) Suppose Ci has a unit and let f : C1 → C2 be a unital filtered A∞ howeak (C1 ), we have momorphism. (See Definition 3.3.11.) Then for any b ∈ M b   weak (C2 ). f∗ (b) = f(e ) ∈ Mweak (C2 ). Thus f induces a map f∗ : Mweak (C1 ) → M

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

 1 ) → M(C  2 ) by putting When Ci is unobstructed, we also have the map f∗ : M(C c = 0 in the proof below. weak (Ci ) → Λ(0) be the (3) Moreover under the situation (2), let POi : M 0,nov potential function. Then we have PO2 ◦ f∗ (b) = PO1 (b). Proof. (1) Since f preserves the shifted degree, it is obvious that f∗ (b) ∈ C2 [1]0 . Since f0 = 0 and f preserves the energy filtration (see (3.2.27.1)) and b ≡ 0 + mod Λ+ 0,nov , we have f∗ (b) ≡ 0 mod Λ0,nov . (2) Let f be the coalgebra homomorphism induced by f. See (3.2.28). Then applying Lemma 3.6.2, we can write f(eb ) as an exponential. In fact, we can check that f(eb ) = ef(eb ) .

(3.6.38)

A straightforward calculation gives rise to  f(eb ) ) = d(  f(eb )) = f(d(e  b )) = cef(eb eeb ) = ceef(eb ) eef(eb ) . d(e On the other hand, we also have  f(eb ) ) = ef(eb ) m(ef(eb ) )ef(eb ) . d(e b

Comparing the two, we prove m(ef(e ) ) = cee as required. The assertion (3) also follows from the above calculation.  When C is a homotopy-unital filtered A∞ algebra, we can define potential function on weak (C) = M weak (C + ). M in the same way. Next we study the analog to Theorem 3.6.18 in the weakly unobstructed case. We work under the following assumption. Assumption 3.6.39. We also assume that H k (C, m1 ) = 0 for k < 0. Moreover H (C, m1 ) is generated by [e] the equivalence class of the unit. 0

Let G ⊂ R≥0 × 2Z be a submonoid satisfying Condition 3.1.6 and assume that C is G-gapped. We write G = {(λi , μi ) | i = 0, 1, 2, · · · } such that i < j ⇒ λi ≤ λj as before. Definition 3.6.40. A G-gapped A∞ algebra C is weakly unobstructed over G, if there exists a weak bounding cochain b, which can be written as  μi b= T λi e 2 bi (λi ,μi )∈G

for some bi ∈ C[1]−μi = C

1−μi

.

Now we put (3.6.41)

Gμ≤0 = {(λ, μ) ∈ G | μ ≤ 0}.

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3.6. BOUNDING COCHAINS AND THE A∞ MAURER-CARTAN EQUATION

117

By omitting i’s with μi > 0 from the given enumeration of G, we find a nondecreasing function j : Z≥0 → Z≥0 with j(0) = 0 such that (3.6.42)

Gμ≤0 = {(λj(i) , μj(i) ) | i = 0, 1, 2, · · · }.

Theorem 3.6.43. Let (C, m) be a G-gapped filtered A∞ algebra. Under Assumption 3.6.39, we have ok (weak) and bi (weak) with the following properties: 1−μi

. (3.6.44.1) [ok (weak)] ∈ H 2−μk (C, m1 ) and bi (weak) ∈ C (3.6.44.2) The cocycle ok (weak) is defined if [oi (weak)] and bi (weak) for i < k are defined and if μk ≤ 0. [ok (weak)] depends on [oi (weak)] and bi (weak) for i < k. (3.6.44.3) When μk ≤ 0, bk (weak) is defined if [ok (weak)] is defined and is zero. If μk > 0, bk (weak) exists if bi (weak) exists for i < k. (3.6.44.4) (C, m) is weakly unobstructed over G if and only if there exists a choice of bi inductively such that [ok (weak)] for μk ≤ 0 are all zero. In other words, the obstruction classes [ok (weak)] can be defined if we assume that the cocycle oi (weak) is defined and [oi (weak)] = 0 in H ∗ (C, m1 ) only for those with μi ≤ 0, i < k. ok (weak) is defined by the same formula as ok (Definition 3.6.23), and bk (weak) is defined by ⎧ ⎪ ⎨ ok (weak) μk ≤ 0, (3.6.45) −m1,0 (bk (weak)) = ok + ck e μk = 2, ⎪ ⎩ μk > 2. ok Note that the existence of such bk (weak) is automatic for μk > 0. (ok is defined in the same way as Definition 3.6.23.) Since we will prove a more general result (Theorem 3.8.50) later, the proof is omitted here and left to the readers. In case (C, m) = (C(L; Λ0,nov )+ , m+ ), where (C(L; Λ0,nov ), m) is as in Section 3.5 and + is as in Section 3.3, we write ok (L; weak) in place of ok (weak).  3.6.4. The superpotential and M(C). Recall the reason why we assume + b ≡ 0 mod Λ0,nov : we want the formal series eb = 1 + b + b ⊗ b + · · · to converge. However, if there exists k(λ) ∈ Z with limλ→∞ k(λ) = ∞ such that if k > k(λ)  b ) will converge for any b ∈ C 0 [1]. This is then mk ≡ 0 mod T λ already, then d(e indeed the case for the A∞ algebra that appeared in Chekanov’s work [Chek02]. In our case such a k(λ) do not seem to exist. (See Conjecture 3.6.53 below.) However, we still expect: Conjecture 3.6.46. We can choose our filtered A∞ algebra (C(L), m) so that  b ) converges for any b ∈ C(L)[1]0 . d(e Compare this conjecture with Conjecture T in introduction. Remark 3.6.47. Note that the Conjecture 3.6.46 means that  mk,β (b, · · · , b) k

converges for any fixed β and b ∈ C(L; Q). A priori, obtaining this convergence should be easier than having the convergence of operators mk after substituting

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

T = e−1 , for example. Namely, Conjecture 3.6.46 appears much easier than proving the convergence of the series  e−ω[β] mk,β (b, · · · , b) k,β

where we need to sum up over all different β. (See [Fuk02I, KoSo01] etc.) Now we discuss how the Maurer-Cartan equation is expected to behave when L ⊂ M is a special Lagrangian submanifold (or more generally Lagrangian submanifold with zero Maslov class) in a Calabi-Yau 3-fold. Firstly, we remark that we can take C(L) = H(L; Q) ⊗ Λ0,nov . This is a consequence of Theorem 5.4.2, which asserts existence of the canonical model of the filtered A∞ algebra. See Definition 5.4.3. We also take C(L) = H(L; Q) as a differential graded algebra. Moreover we can forget the 0-th and 3-rd cohomologies. (See Subsection 5.4.6.) Hence we are left only with H 1 (L; Q) and H 2 (L; Q). Secondly, there is a conjecture which is widely believed to be true but not yet proven. Conjecture 3.6.48. We can take an A∞ structure on (C(L), m) so it satisfies the following: There exists ·, · : C(L) ⊗ C(L) → Λ0,nov induced by the Poincar´e duality pairing in homology which satisfies the cyclic symmetry condition x0 , mk (x1 , · · · , xk ) = (−1)deg



xk ×(deg x0 +···+deg xk−1 )

xk , mk (x0 , x1 , · · · , xk−1 ) .

If one could ignore the delicate problems of transversality and sign, Conjecture 3.6.48 could be easily seen. (Achieving the transversality while keeping the required symmetry is a difficult task. See however Section A3 and [Fuk05II].) Under these hypotheses, we choose a basis ei , i = 1, · · · , m of H 1 (L; Q). We define a function Ψ : Λ0,nov × · · · × Λ0,nov → Λ0,nov by Ψ(x1 , · · · , xm ) =

(3.6.49)

 k



1 b, mk (b, · · · , b)

k+1

where b = xi ei . (See [Laza01, Tom01].) This function seems to be widely known in the physics literature and is called the superpotential. Conjecture 3.6.46 is needed for the right hand side of (3.6.49) to be well-defined for b ∈ C(L; Λ0,nov ).  Proposition 3.6.50. Assume Conjectures 3.6.46 and 3.6.48. Then b = xi ei  b ) = 0 if and only if (x1 , · · · , xm ) is a critical point of Ψ. satisfies d(e Proof. Using Conjecture 3.6.48, we find  ∂ Ψ(x1 , · · · , xm ) = ei , mk (b, · · · , b) = ei , m(eb ) . ∂xi k

The proposition follows.



In a similar way, we can strengthen the statement of this proposition. For this + b purpose, we define mbi1 ,··· ,ik ,i0 ∈ Λ+ 0,nov , and li1 ,··· ,ik ,i0 ∈ Λ0,nov by (3.6.51.1)

mbi1 ,··· ,ik ,i0 = ei0 , m(eb ei1 eb · · · eb eik eb )

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119

and lbi1 ,··· ,ik ,i0 =

(3.6.51.2)



mbiσ(1) ,··· ,iσ(k) ,i0 =

σ∈Sk

1 k+1



mbiσ(1) ,··· ,iσ(k) ,iσ(0)

σ∈Sk+1

where Sk in the symmetric group of order k!. Then lbi1 ,··· ,ik ,i0 are the structure constants of the L∞ structure obtained by symmetrizing m. (See Section A3). Proposition 3.6.52. We assume Conjectures 3.6.46 and 3.6.48. Then we have: ∂ k+1 Ψ (b) = lbi1 ,··· ,ik ,i0 . ∂xi1 · · · ∂xik ∂xi0  0  Proof. If b0 = xi ei , b = xi ei then we have Ψ(x1 + x01 , · · · , xm + x0m )  1 = b0 + b, mk (b0 + b, · · · , b0 + b)

k+1 .  1   = b, mk+ i (b00 , b, b01 , · · · , b0k−1 , b, b0k ) k + 1 + i k,0 ,··· ,k .  1   b0 , mk+ i (b00 , b, b01 , · · · , b0k−1 , b, b0k ) + k + 1 + i k,0 ,··· ,k .  1 ∗  b, mk+ i (b00 , b, b01 , · · · , b0k−1 , b, b0k ) = k+1 k,0 ,··· ,k .  1  1 b0 , mk (b0 , · · · , b0 ) + b, mbk0 (b, · · · , b) . = k+1 k+1 k

k

∗

For the equality = above, we used the cyclic symmetry (Conjecture 3.6.48) in the following way: We consider, for k and , the sum .   b, mk+ (b00 , b, b01 , · · · , b0k−1 , b, b0k ) . 0 +1 +···+k =

We take its cyclic permutations of variables. Then among  + k + 1 of them, k + 1 are of the form .   b, mk+ (b00 , b, b01 , · · · , b0k−1 , b, b0k ) 0 +1 +···+k =

and  of them are of the form 

.  b0 , mk+ (b00 , b, b01 , · · · , b0k , b, b0k+1 ) .

0 +1 +···+k+1 =−1 ∗

= follows. Therefore, replacing mk by mbk0 , it suffices to prove the proposition at b = 0. Then   ∂ k+1 Ψ ∂ k+1 1 (0) = b, mk (b, · · · , b)  = li1 ,··· ,ik ,i0 , ∂xi1 · · · ∂xik ∂xi0 k + 1 ∂xi1 · · · ∂xik ∂xi0 b=0 as required.

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Now we consider (M, L) a pair of Calabi-Yau 3-fold M and its Lagrangian submanifold L ⊂ M with zero Maslov class. By taking an appropriate almost complex structure J, we may assume that all somewhere injective holomorphic discs are isolated. Enumerate them by ui : (D2 , ∂D2 ) → (M, L) and denote " u∗i ω, ai,j = [∂ui ] ∩ xj . λi = D2

Conjecture 3.6.53. Let Ψ be the superpotential defined as above. Then ⎛ ⎞ ∞ ∞  m   Ψ(x1 , · · · , xm ) = c +

i,d nd T dλi exp ⎝d ai,j xj ⎠ . i=1 d=1

j=1

Here exp : C → C is the usual exponential and nd is the number depending only on d and describing how to count d fold cover of the disc ui . (In some literature [OoVa96, KatLi01] there are some results which suggest nd = d−2 .) i,d = ±1 is the sign to count the d fold covering of the disc ui . c ∈ Λ0,nov is independent of (x1 , · · · , xm ) but may depend of L, J and the perturbation. We expect to be able to prove some of the conjectures stated of this subsection in a near future using the technique of [Fuk05II]. Remark 3.6.54. In the toric case the relationship between superpotential and the Floer cohomology is studied in [ChOh03, Cho04II, FOOO08I]. A formula similar to the one in Proposition 3.6.52 was proved by Cho [Cho04II]. In the toric case we can identify the Landau-Ginzburg potential with our potential function by putting T = exp(−1). The relationship between the superpotential and filtered A∞ structure in the Calabi-Yau case has a different form that of the toric case: In the Calabi-Yau case (which we discussed in this section) the first derivative of superpotential is related to m0 (1). Moreover the k-th derivative of the superpotential is related to the symmetrization of mk−1 . In the toric case, m0 (1) is the Landau-Ginzburg potential function itself and its first derivative corresponds to m1 . 3.7. A∞ bimodules and Floer cohomology In this section, we deform the standard Floer’s ‘coboundary’ operator using the constructions given in the previous sections. We first prepare the algebraic frameworks for the filtered A∞ bimodule structures in Subsection 3.7.1 and for filtered A∞ bimodule homomorphisms in Subsection 3.7.2. In Subsection 3.7.3, we provide an algebraic definition of our deformed coboundary operator on a filtered A∞ bimodule when the A∞ algebras are (weakly) unobstructed. After that, we construct the geometric realization associated to a relatively spin pair of Lagrangian submanifolds (L(0) , L(1) ) and its Floer cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) in Subsection 3.7.4 when L(0) is transversal to L(1) , and in Subsection 3.7.5 when L(0) intersects L(1) cleanly. In Subsection 3.7.6, we give a simple example of calculations of Floer cohomologies and discuss the (weak) unobstructedness condition in the examples. In Subsection 3.7.7 a product structure on Floer cohomology is defined. 3.7.1. Algebraic framework. In this subsection, we introduce some algebraic notions without involving Lagrangian submanifolds. Let (C0 , m0 ), (C1 , m1 )

m be two filtered A∞ algebras over Λ0,nov . Let m∈Z D be a graded free filtered

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121

Λ0,nov module and F λ Dm its filtration defined similarly as in (3.2.12). We suppose that the filtration F λ Dm satisfies

the same conditions as (3.2.12.1)-(3.2.12.5). We denote by D the completion of m∈Z Dm with respect to this filtration. We consider a family of operations nk1 ,k0 : Bk1 (C1 [1]) ⊗Λ0,nov D[1] ⊗Λ0,nov Bk0 (C0 [1]) −→ D[1] of degree +1 satisfying + nk1 ,k0 F λ1 (C1 [1])m1 ⊗ · · · ⊗ F λk1 (C1 [1])mk1 ⊗ F λ0 (D[1])m0      ⊗ F λ1 (C0 [1])m1 ⊗ · · · ⊗ F λk0 (C0 [1])mk0 (3.7.1) 

⊆ F Σλi +λ0 +Σλi (D[1])



mi +m0 +



mi +1

.

We now use condition (3.7.1) to extend nk1 ,k0 to the completion  0 [1])  1 [1]) ⊗  Λ0,nov D[1] ⊗  Λ0,nov B(C B(C

Bk1 (C1 [1])⊗Λ0,nov D[1]⊗Λ0,nov Bk0 (C0 [1]). For this purpose we first remark  1 [1]) and right B(C  0 [1])  0 [1]) is a left B(C   Λ0,nov D[1] ⊗  Λ0,nov B(C that B(C1 [1]) ⊗ (formal) bi-comodule. We then extend nk1 ,k0 to a bi-coderivation;

of

k1 ,k0

 0 [1])  1 [1]) ⊗  Λ0,nov D[1] ⊗  Λ0,nov B(C d :B(C  1 [1]) ⊗  0 [1]), Λ Λ → B(C D[1] ⊗ B(C 0,nov

0,nov

which is defined by  1,1 ⊗ · · · ⊗ x1,k ⊗ y ⊗ x0,1 ⊗ · · · ⊗ x0,k ) d(x 1 0  deg x1,1 +···+deg x1,k −k +k1 −k1 1 1 = (−1) k1 ≤k1 ,k0 ≤k0

x1,1 ⊗ · · · ⊗ x1,k1 −k1 ⊗ nk1 ,k0 (x1,k1 −k1 +1 , · · · ,

(3.7.2)

y, · · · , x0,k0 ) ⊗ x0,k0 +1 ⊗ · · · ⊗ x0,k0 + d1 (x1,1 ⊗ · · · ⊗ x1,k1 ) ⊗ y ⊗ x0,1 ⊗ · · · ⊗ x0,k0 + (−1)Σ deg x1,i +deg y+k1 +1 x1,1 ⊗ · · · ⊗ x1,k1 ⊗ y ⊗ d0 (x0,1 ⊗ · · · ⊗ x0,k0 ).

Here di is defined by mi (i = 0, 1). When we use the convention (Conv.2) (1.36), we can simply rewrite (3.7.2) as

(3.7.3)

 1 ⊗ y ⊗ x0 ) d(x    = (−1)deg x1,a x1,a ⊗ n(x1,a ⊗ y ⊗ x0,b ) ⊗ x0,b a,b   + d1 x1 ⊗ y ⊗ x0 + (−1)deg x1 +deg y x1 ⊗ y ⊗ d0 x0 .

Here deg is the degree after shifted as in Section 3.3. Then (3.7.1) implies that  1 [1]) ⊗  0 [1]).  Λ0,nov D[1] ⊗  Λ0,nov B(C (3.7.2) is extended to the completion B(C Since D is a free Λ0,nov module, we have a free R module D and an isomorphism D∼ = D ⊗R Λ0,nov .

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

−1 ∼ Since D/Λ+ ], the operation nk1 ,k0 induces a homomorphism 0,nov = D[e, e

nk1 ,k0 : Bk1 (C 1 [1]) ⊗R D[1] ⊗R Bk0 (C 0 [1]) ⊗R R[e, e−1 ] −→ D[1] ⊗R R[e, e−1 ]. We assume that nk1 ,k0 is induced by an operation Bk1 (C 1 [1]) ⊗R D[1] ⊗R Bk0 (C 0 [1]) −→ D[1]. We denote this operation by the same symbol nk1 ,k0 . An analog to Condition 3.2.18 is the following. Condition 3.7.4. For an element ei ∈ Ci [1] (i = 0, 1), (3.7.4.1)

nk1 ,k0 (x1 ⊗ · · · ⊗ e1 ⊗ · · · ⊗ xk1 −1 ⊗ y ⊗ x1 ⊗ · · · ⊗ xk0 ) = 0

for k1 + k0 ≥ 2 and (3.7.4.2)

nk1 ,k0 (x1 ⊗ · · · ⊗ xk1 ⊗ y ⊗ x1 ⊗ · · · ⊗ e0 ⊗ · · · ⊗ xk0 −1 ) = 0

for k1 + k0 ≥ 2. And (3.7.4.3)

n1,0 (e1 , y) = (−1)deg y n0,1 (y, e0 ) = y.

Definition 3.7.5. (1) We say that nk1 ,k0 defines a left (C1 , m1 ) and right (C0 , m0 ) filtered A∞ bimodule structure on D, or simply D is a (C1 , C0 ) filtered A∞  d = 0. bimodule, (or filtered A∞ bimodule over (C1 , C0 )), if it satisfies (3.7.1) and d◦ i (2) Let (Ci , m ) have a unit ei . We say that {e1 , e0 } acts as a unit if Condition 3.7.4 is satisfied. In this case we sometimes say that D is a unital filtered A∞ bimodule. (3) When (Ci , mi ) has a homotopy unit ei , we say that {e1 , e0 } acts as a homotopy unit if there exists a left (C1+ , m1 ) and right (C0+ , m0 ) filtered A∞ bimodule + + + structure on D such that {e+ 1 , e0 } acts as a unit. Here Ci = Ci ⊕Λ0,nov ei ⊕Λ0,nov fi i i as in (3.3.1) and e+ i = m1 (fi ) + ei − h2 (1 ⊗ 1) as in (3.3.6). We also say that D is a homotopy-unital A∞ bimodule in this case. (4) For unfiltered A∞ algebras (C i , mi ) (i = 0, 1) over R, a (C 1 , C 0 ) A∞ bimodule (D, n) over R is defined in a similar way. An unfiltered unital (homotopy-unital) A∞ bimodule is also defined as in (2),(3) above. If D is a (C1 , C0 ) filtered A∞ bimodule, then D is a (C 1 , C 0 ) A∞ module. We call it the R reduction of D. (5) Let Gi ⊂ R≥0 × 2Z be a monoid satisfying Condition 3.1.6 and (Ci , mi ) Gi -gapped filtered A∞ algebras. Let G ⊂ R≥0 × 2Z be another monoid satisfying Condition 3.1.6 such that G ⊇ G1 , G0 . We say that a filtered A∞ bimodule (D, nk1 ,k0 ) is G-gapped, if there exists nk1 ,k0 ,β : Bk1 C 1 [1] ⊗ D[1] ⊗ Bk0 C 0 [1] → D[1] for each k1 , k0 and β ∈ G such that  nk1 ,k0 = T λ(β) eμ(β)/2 nk1 ,k0 ,β . β∈G

As in Remark 3.6.15, we note that if G is a monoid satisfying Condition 3.7.4 and containing G above, then every G-gapped filtered A∞ bimodule is G -gapped. When we do not necessarily specify G, we say that (D, nk1 ,k0 ) is a gapped filtered A∞ bimodule for simplicity.

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123

Example 3.7.6. Let (C, m) be a filtered A∞ algebra. We can regard C as a left (C, m) and right (C, m) filtered A∞ bimodule. Namely we put nk1 ,k0 (x ⊗ y ⊗ z) = mk1 +k0 +1 (x, y, z). Here x ∈ Bk1 C[1], y ∈ C[1], z ∈ Bk0 C[1]. If C has a unit (resp. homotopy unit), it acts as a unit (resp. homotopy unit) to C (resp. C + ). If C is gapped as a filtered A∞ algebra, then C is gapped as a filtered A∞ bimodule. We will also use Λnov as a coefficient ring to consider filtered A∞ bimodules. This is because Floer cohomology of a pair of two different Lagrangian submanifolds remains isomorphic under the Hamiltonian isotopy only over the Λnov -coefficient. (See Section 5.3.) We define the filtered A∞ bimodule with the Λnov -coefficient to be the tensor product (over Λ0,nov ) of a filtered A∞ bimodule on Λ0,nov and Λnov . (See Subsection 5.2.1.) 3.7.2. A∞ bimodule homomorphisms. Let Ci and Ci be filtered A∞ algebras (i = 0, 1). Let D and D be (C1 , C0 ) and (C1 , C0 ) filtered A∞ bimodules, respectively. Let fi : Ci → Ci be filtered A∞ algebra homomorphisms. Definition 3.7.7. A filtered A∞ bimodule homomorphism D → D over the pair (f1 , f0 ) is a family of Λ0,nov -module homomorphisms ϕ = {ϕk1 ,k0 }, which is simply written as ϕ : D → D ,  Λ0,nov D[1]⊗  Λ0,nov Bk0 (C0 [1]) −→ D [1] ϕk1 ,k0 : Bk1 (C1 [1])⊗ with the following two properties (1) (2) (3): (1) It respects the energy filtration. Namely + ,  λ D[1]⊗F  λ0 Bk0 (C0 [1]) ⊆ F λ1 +λ+λ0 D [1]. ϕk1 ,k0 F λ1 Bk1 (C1 [1])⊗F       [1]⊗B(C  (2) Let ϕ  : B(C1 [1])⊗D[1] ⊗B(C 0 [1]) → B(C1 [1])⊗D 0 [1]) be the comodule homomorphism induced by ϕk1 ,k0 and fi . Namely, it is defined by  f1 (x ) ⊗ ϕ(x ⊗ y ⊗ x ) ⊗ f0 (x ) (3.7.8) ϕ(x  1 ⊗ y ⊗ x0 ) = 1,a 1,a 0,b 0,b a,b

for xi ∈ B(Ci [1]) and y ∈ D[1], where fi is induced by fi as in (3.2.28) and we use convention (Conv.2) (1.36). Then the following diagram commutes:  1 [1]) ⊗  0 [1]) −−−d−→ B(C  1 [1]) ⊗  0 [1])  D[1] ⊗  B(C  D[1] ⊗  B(C B(C ⏐ ⏐ ⏐ ⏐ ϕ  ϕ  d   [1]) −−−   [1]) ⊗   [1]).   [1]) ⊗  D [1] ⊗  B(C  D [1] ⊗  B(C −→ B(C B(C 1 0 1 0 

Diagram 3.7.1     is taken over Here d and d are defined as in (3.7.2) and the tensor product ⊗ Λ0,nov . (3) By the property (1) ϕk1 ,k0 induces 

ϕk1 ,k0 : Bk1 (C 1 [1]) ⊗R D[1] ⊗R Bk0 (C 0 [1]) ⊗R R[e, e−1 ] −→ D [1] ⊗R R[e, e−1 ].

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We assume that it is induced by 

Bk1 (C 1 [1]) ⊗R D[1] ⊗R Bk0 (C 0 [1]) −→ D [1] and denote it by the same symbol ϕk1 ,k0 .



We define an unfiltered A∞ bimodule homomorphism ϕ : D → D in a similar  ◦ . way so that it satisfies ϕ d = d ◦ ϕ We next discuss the gapped condition. Definition 3.7.9. We assume that Ci , Ci are gapped filtered A∞ algebras and f(i) : Ci → Ci are Gi -gapped filtered A∞ homomorphisms. (See Definitions 3.2.26 and 3.2.29). Let G ⊆ R≥0 × 2Z be a monoid satisfying Condition 3.1.6 and containing G1 , G0 . Let D (resp. D ) be G-gapped filtered C1 ,C0 bimodule (resp. C1 ,C0 bimodule). Then a filtered A∞ bimodule homomorphism ϕ = {ϕk1 ,k0 } : D → D over (f1 , f0 ) is called G-gapped, if there exist 

ϕk1 ,k0 ,i : Bk1 C 1 [1] ⊗ D[1] ⊗ Bk0 C 0 [1] → D [1] such that ϕk1 ,k0 =



T λi eμi /2 ϕk1 ,k0 ,i .

i

Here we identify D = D ⊗R Λ0,nov etc. If G is another monoid satisfying Condition 3.1.6 and contains G, then every G-gapped filtered A∞ bimodule homomorphism is G -gapped. When we do not necessarily specify G, we say that ϕ is gapped for simplicity. We can define compositions of the filtered A∞ bimodule homomorphisms in an obvious way. As we mentioned at the end of the previous subsection, when we consider Floer cohomology of a pair of Lagrangian submanifolds, we will also use the ring Λnov as the coefficient ring to study the invariance property of the Floer cohomology. In this case we will introduce the notion of the weakly filtered A∞ bimodule homomorphism and a certain gapped condition for the weakly filtered A∞ bimodule homomorphisms. See Subsection 5.2.1 in Chapter 5. Definition 3.7.10. Suppose that Ci and Ci have unit ei and ei , respectively, and {e1 , e0 } acts on D as a unit and {e1 , e0 } acts on D as a unit. Moreover, let fi : Ci → Ci be a unital A∞ homomorphism in the sense of Definition 3.3.11, and ϕ : D → D a filtered A∞ bimodule homomorphism over (f1 , f0 ). We say ϕ preserves unit, or call it a unital A∞ bimodule homomorphism if it satisfies (3.7.11)

ϕ(x1 ⊗ e1 ⊗ x2 , v, y) = ϕ(x, v, y1 ⊗ e0 ⊗ y2 ) = 0,

where v ∈ D, x, x1 , x2 ∈ BC1 [1], y, y1 , y2 ∈ BC0 [1]. Note that (3.7.11) includes the case when x1 , x2 , x, y1 , y2 , or y is equal to 1. For example, ϕ(e1 , v, y) = ϕ(1, v, y1 ⊗ e0 ⊗ y2 ) = 0, ϕ(e1 , v, 1) = ϕ(1, v, e0 ) = 0. The unfiltered version can be defined in the same way. We next assume that Ci and Ci have homotopy units ei and ei , and they act on D and D as homotopy unit, respectively. Moreover, let fi : Ci → Ci be

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a homotopy-unital A∞ homomorphism and fi,+ : Ci+ → Ci+ its extension as in + Definition 3.3.13, which preserves units e+ i and ei . Definition 3.7.12. Under the situation above, we say a filtered A∞ bimodule homomorphism ϕ : D → D over (f1 , f0 ) preserves homotopy unit, or call it a homotopy-unital A∞ bimodule homomorphism, if it extends to a filtered A∞ bimodule homomorphism ϕ+ over (f1,+ , f0,+ ), which preserves unit in the sense of Definition 3.7.10. The unfiltered version can be also defined in the same way. 3.7.3. Weak unobstructedness and deformations. In Section 3.6, we defined the coboundary operator δb1 ,b0 : C[1] −→ C[1] for an unobstructed filtered A∞ algebra C associated to each given pair (b1 , b0 )  with bi ∈ M(C) (Definition 3.6.6). When C is weakly unobstructed, we similarly weak (C) (Lemma 3.6.33). In defined the coboundary operator δb,b for a given b ∈ M this subsection, we will discuss the A∞ bimodule version. Let D be a left (C1 , m1 ) and right (C0 , m0 ) filtered A∞ bimodule. The A∞ bimodule structure is given by nk1 ,k0 . First, we deform the filtered A∞ bimodule structure. Definition-Lemma 3.7.13. (1) For any bi ∈ Ci [1]0 with bi ≡ 0 mod Λ+ 0,nov (i = 0, 1), we define a family of operators b1 b0 nk1 ,k0 (x1,1 , · · · , x1,k1 , y, x0,1 , · · · , x0,k0 ) = n(eb1 x1,1 eb1 · · · x1,k1 eb1 , y, eb0 x0,1 eb0 · · · x0,k0 eb0 )

for x1,1 ⊗ · · · ⊗ x1,k1 ∈ Bk1 C1 [1], y ∈ D[1] and x0,1 ⊗ · · · ⊗ x0,k0 ∈ Bk0 C0 [1]. Then it defines a ((C1 , m1,b1 ), (C0 , m0,b0 )) filtered A∞ bimodule structure on D. (See Definition 3.6.9 for the deformed A∞ algebras.) We denote it by b1 nb0 and call it a deformed A∞ bimodule structure of n by b1 , b0 . We also denote by b1 db0 the induced operator by b1 nb0 as in (3.7.3). (2) In particular, we put for y ∈ D[1]  nk1 ,k0 (b1 , · · · , b1 , y, b0 , · · · , b0 ). δb1 ,b0 (y) = b1 nb0 (y) = n(eb1 yeb0 ) = k1 ,k0

The proof of that b1 nb0 defines a filtered A∞ bimodule structure is similar to that of Proposition 3.6.10 so we omit it. Later in Subsection 5.2.2, we will discuss deformations of the filtered A∞ bimodule homomorphisms. Now we assume that (C1 , m1 ) and (C0 , m0 ) are unobstructed. Let b1 and b0 be bounding cochains of (C1 , m1 ) and (C0 , m0 ) respectively. We deform the A∞ bimodule structure by b1 , b0 , so that we obtain a coboundary operator δb1 ,b0 : D → D as follows. This corresponds to a deformation of the standard Floer coboundary operator in our geometric context. Lemma 3.7.14. For any bounding cochains b1 , b0 of (C1 , m1 ), (C0 , m0 ) given respectively, we have δb1 ,b0 ◦ δb1 ,b0 = 0. The proof of Lemma 3.7.14 is the same as that of Lemma 3.6.8. So we omit the proof.

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Next, we assume that C1 and C0 have units e1 , e0 respectively, and {e1 , e0 } acts as a unit. Then as in Definition-Lemma 3.7.13, we can deform the filtered A∞ bimodule structure by b1 , b0 . It is easy to see that {e1 , e0 } acts as a unit on the deformed filtered A∞ bimodule as well. If Ci is weakly unobstructed, we have the potential function weak (Ci ) −→ Λ POi : M 0,nov +(0)

(3.7.15)

weak (Ci ), we define for i = 0, 1 defined as in Definition 3.6.33 (2). For each bi ∈ M δb1 ,b0 : D → D by δb1 ,b0 (y) = n(eb1 yeb0 ).

(3.7.16)

Proposition 3.7.17. For y ∈ D, we have δb1 ,b0 ◦ δb1 ,b0 (y) = (−PO1 (b1 ) + PO0 (b0 ))ey. In particular, if PO1 (b1 ) = PO0 (b0 ), we can define the cohomology of (D, δb1 ,b0 ). Proof. A straightforward computations gives δb1 ,b0 (δb1 ,b0 (y)) = n(eb1 n(eb1 yeb0 )eb0 )  b1 yeb0 )) − n(d(e  b1 )yeb0 ) + (−1)deg y n(eb1 y d(e  b0 )) = n(d(e = (−PO1 (b1 ) + PO0 (b0 ))ey. 

At the last equality we used (3.7.4.3).

Remark 3.7.18. When Ci has a homotopy unit ei which acts as a homotopy unit and Ci is weakly unobstructed in the sense of Definition 3.6.29 (2), the argument above also holds by considering Ci+ = Ci ⊕ Λ0,nov e+ i ⊕ Λ0,nov fi as in (3.3.1). (Note δb1 ,b0 : D → D is defined in this case. Compare Remark 3.6.33 (1).) In Subsection 3.7.6, we will give a geometric example of Proposition 3.7.17. 3.7.4. The filtered A∞ bimodule C(L(1) , L(0) ; Λ0,nov ). The main result of this subsection is Theorem 3.7.21. In this subsection we put R = Q. Let L(0) , L(1) be a relatively spin pair of Lagrangian submanifolds of M . We will first define a Λ0,nov module C(L(1) , L(0) ; Λ0,nov ) and construct a filtered A∞ bimodule structure on it. We do not need to assume L(i) is unobstructed for this purpose. In this subsection, we assume that they are transverse to each other. In the next subsection, we will treat the case where two Lagrangian submanifolds intersect cleanly. Let (C(L(0) ; Λ0,nov ), m(0) ) and (C(L(1) ; Λ0,nov ), m(1) ) be the filtered A∞ algebras constructed in Theorem 3.5.11. We define / CF (L(1) , L(0) ; 0 ) = CF (L(1) , L(0) ; 0 ; Q) = Q[p , w] CF (L(1) , L(0) ) =

[p ,w]∈Cr(L(0) ,L(1) ;0 )



CF (L(1) , L(0) ; 0 )

0 ∈π0 (Ω(L(1) ,Ω(L(0) ))

=

/

Q[p , w]

[p ,w]∈Cr(L(0) ,L(1) )

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3.7. A∞ BIMODULES AND FLOER COHOMOLOGY

127

where Cr(L(0) , L(1) ; 0 ), Cr(L(0) , L(1) ) are the sets of the equivalence classes [p , w] which are critical points of our functional A (see Subsection 2.2.2) and the sum in the second line is taken over all connected components of Ω(L(0) , L(1) ). The symbol  denotes the completion of the direct sum. The Γ-equivalence relation is given in ⊕ Definition 2.2.4, and p the constant path to an intersection point p ∈ L(1) ∩ L(0) . We define / (L(1) , L(0) ; Λnov ) := CF (L(1) , L(0) ) ⊗ ˆ Q Λnov CF / (L(1) , L(0) ; Λnov ) as follows: Say and an equivalence relation ∼ on CF 



T λ eμ [p , w] ∼ T λ eμ [p , w ]   / (L(1) , L(0) ; Λnov ), if and only if the following for T λ eμ [p , w], T λ eμ [p , w ] ∈ CF conditions are satisfied:

(3.7.19.1) (3.7.19.2)

"



ω=λ +

λ+ w

(3.7.19.3)

p = p

" w

ω

2μ + μ([p , w]) = 2μ + μ([p , w ]).

Here μ([p , w]) is the Maslov-Morse index given in Subsection 2.2.2. It is easy to see that these conditions are compatible with the conditions put on the Γ-equivalence / (L(1) , L(0) ; Λnov ). (Definition 2.2.12) and so ∼ defines an equivalence relation on CF (1) (0) / Furthermore we define the energy filtration on CF (L , L ; Λnov ) by the following. / (L(1) , L(0) ; Λnov ). We say Let T λ eμ [p , w] be an element of CF  / (L(1) , L(0) ; Λ0,nov ) T λ eμ [p , w] ∈ F λ CF

' if λ + w ω ≥ λ . / (L(1) , L(0) ; Λnov )/∼. This filtration obviously induces an energy filtration on CF (1) (0) Now, we define C(L , L ; Λ0,nov ) as follows. Definition 3.7.20. We denote by C(L(1) , L(0) ; Λ0,nov ) the non-negative en/ (L(1) , L(0) ; Λnov )/ ∼ with respect to the energy ergy part of the completion of CF filtration. Namely C(L(1) , L(0) ; Λ0,nov ) = F 0 CF (L(1) , L(0) ; Λnov ). Giving the grading of an element T λ eμ [p , w] by 2μ + μ([p , w]), it becomes a filtered graded free Λ0,nov module. It is easy to see that C(L(1) , L(0) ; Λ0,nov ) satisfies the conditions (3.2.12.1) (3.2.12.5). We also note that C(L(1) , L(0) ; Λ0,nov ) is isomorphic to the completion (with respect to the filtration on Λ0,nov ) of the free Λ0,nov module generated by the intersection points p ∈ L(1) ∩ L(0) . Namely / Λ0,nov [p] C(L(1) , L(0) ; Λ0,nov ) ∼ = p∈L(1) ∩L(0)

as a Λ0,nov module. Let {Jt }t = {Jt }0≤t≤1 be a t-dependent family of almost complex structures as before. The following is the main theorem of this subsection.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Theorem 3.7.21. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian submanifolds which transversely intersect. Then we have a left (C(L(1) ; Λ0,nov ), m(1) ) and right (C(L(0) ; Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ), which is G(L(1) , L(0) )-gapped. Moreover the pair {e1 , e0 } of the homotopy units of (C(L(1) ; Λ0,nov ), m(1) ) and (C(L(0) ; Λ0,nov ), m(0) ) acts as a homotopy unit. We will give the definition of G(L(1) , L(0) ) in the course of the proof. See (3.7.43). Proof. We need to study the transversality problem precisely like as in Theorem 3.5.11. The proof of this point is a minor modification of one in Section 3.5. The discussion on the homotopy unit is similar to the one in Section 7.3. See Section 8.10 for the signs. Let [p , w1 ], [q , w2 ] ∈ Cr(L(0) , L(1) ). To construct the filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ), we need to define the relevant moduli space of marked stable broken Floer trajectories Mk1 ,k0 ([p , w1 ], [q , w2 ]) = Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) which are now in order. The special case M0,0 ([p , w1 ], [q , w2 ]) will coincide with the moduli space M([p , w1 ], [q , w2 ]; {Jt }t ) defined in Section 2.4.2. Let Σ be R × [0, 1] possibly with a finite number of tower of sphere components attached (at points on R × (0, 1)). (The way how to handle sphere bubble using the fiber product is the same as [FuOn99II]. So we do not repeat its detail here.) Let (0) (1) u : Σ → M be a smooth map and τj ∈ R (j = 1, · · · , k0 ), τj ∈ R, (j = 1, · · · , k1 ). We put (i) (i) τ (i) = (τ1 , · · · , τki ), (i = 0, 1) and τ = (τ (0) , τ (1) ). We consider the following conditions for them. (See Figure 3.7.1.):

(1)

(1)

τ1

τk1

L(0) L (1)

(0)

τ k0

τ (0) 1 Figure 3.7.1 (3.7.22.1)

On R × [0, 1], u satisfies ∂u ∂u + Jt = 0. ∂τ ∂t

If a bubble tree of spheres is rooted at (τ, t), all the spheres in this bubble tree are Jt -holomorphic. (3.7.22.2) u(R × {0}) ⊂ L(0) , u(R × {1}) ⊂ L(1) . (3.7.22.3) limτ →−∞ u = p and limτ →∞ u = q.

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3.7. A∞ BIMODULES AND FLOER COHOMOLOGY (0)

(0)

(1)

129

(1)

(3.7.22.4) τj < τj+1 , j = 1, · · · , k0 − 1, τj > τj+1 , j = 1, · · · , k1 − 1. (3.7.22.5) w1 #u ∼ w2 . (3.7.22.6) ((Σ, τ ), u) is stable, i.e., the set of automorphisms ϕ : Σ → Σ with (i) (i) ϕ(τj ) = τj , u ◦ ϕ = u is finite. In (3.7.22.5), we use the notation of Definition 2.2.4 for the equivalence relation ∼ and w1 #u is the obvious concatenation of w1 and u (see Section 2.2.1). Remark 3.7.23. (1) We would like to remark that we give the structure of a left (C(L(1) ; Λ0,nov ), m(1) ) and right (C(L(0) ; Λ0,nov ), m(0) ) filtered A∞ bimodule to C(L(1) , L(0) ; Λ0,nov ). At first sight, it might seem natural to construct a right (C(L(1) ; Λ0,nov ), m(1) ) and left (C(L(0) ; Λ0,nov ), m(0) ) structure. The reason the left and right is as in Theorem 3.7.21 is as follows: We consider the space of path from L(0) to L(1) . Hence the natural boundary condition for u : Σ → M is as in (3.7.22.2). (See Figure 3.7.2.) z k1 +1 z2 (1)

(1)

τk1

τ1

z1

z0 (0)

(0)

τk0

τ1

zk0 +k1 +1

∼ =

zk1 +2

z2

z k1 +1

z1

z0 zk0 +k1 +1

zk1 +2

Figure 3.7.2 But the standard way of putting the orientation on the circle as a boundary of the disc is the counter clockwise order. (i) Therefore, the natural order of the marked points (τj , i) (i = 0, 1; j = (1)

(1)

(0)

(0)

1, · · · , ki ) and −∞ is (τ1 , 1), · · · , (τk1 , 1), −∞, (τ1 , 0), · · · , (τk0 , 0). Therefore, if we define n using the space Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) as below, the chains of L(1) should come first and then [p , w1 ] and finally the chains of L(0) . (2) When we describe the map u as a map from the unit disc D2 instead of R × [0, 1], the points −∞ and +∞ correspond to the points −1 and +1 on the boundary of the unit disc respectively. In the later argument, these points will be assigned to the first and 0-th marked points z1 , z0 respectively. See Sections 8.7 and 8.8. The order of the marked points on the boundary is important when we

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130

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

consider the orientation problem. We like to keep the notations z0 and z1 as +1 (1) (1) and −1. In Sections 8.1 and 8.8 we put z2 = (τ1 , 1), · · · , zk1 +1 = (τk1 , 1) and (0)

(0)

zk0 +2 = (τ1 , 0), · · · , zk0 +k1 +1 = (τk0 , 0). See also Figure 3.7.2 above. We consider the R-action defined by u(·, ·) → u(· + τ0 , ·) for τ0 ∈ R which induces a natural equivalence relation ∼R among the set of solution u’s of (3.7.22). Definition 3.7.24. We denote by ◦

◦

Mk1 ,k0 ([p , w1 ], [q , w2 ]) = Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) the totality of the ∼R equivalence classes of ((Σ, τ ), u) satisfying (3.7.22). We have the evaluation map (1)

(1)

(0)

(0)

ev = (ev (1) , ev (0) ) = ((ev1 , . . . , evk1 ), (ev1 , . . . , evk0 ))

(3.7.25)

◦

k1

: Mk1 ,k0 ([p , w], [q , w ]) → (L(1) )

k0

× (L(0) )

which is defined by (i)

(i)

evj ((Σ, τ ), u) = u(τj , i) ∈ L(i) ,

(i = 0, 1).

To state the next proposition we need to explain some notations. We recall that the Kuranishi structure on Mk (β) was constructed for β ∈ Π(M ; L) in Proposition 3.4.2. (Π(M ; L) is defined in Definition 2.4.17.) The moduli space Mmain (β) k together with its Kuranishi structure depends on the choice of L, β as well as a compatible almost structure J. To specify the choice we write Mmain (L; β; J). k Proposition 3.7.26. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian submanifolds which are transversal to each other, and [p , w1 ], [q , w2 ] be elements of Cr(L(1) , L(0) ). ◦

Then Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) has a compactification which we denote by Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]). The evaluation map (3.7.25) is extended to the compactification so that it is weakly submersive and strongly continuous and smooth. Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) has an oriented Kuranishi structure, which is compatible to each other and with the Kuranishi structures of the moduli spaces Mmain (L(0) ; β (0) ; J0 ) and Mmain (L(1) ; β (1) ; J1 ) constructed in Proposition 3.4.2. k k The compatibility in the statement of Proposition 3.7.26 means the compatibility at the end of the moduli spaces via fiber product. (See Subsection 7.1.1). Proof. We here only describe the compactification Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]). Construction of the Kuranishi structure thereon is the same as the other cases discussed in Section 7.1. To describe compactification of the moduli space, we need to prepare some notations. Definition 3.7.27. Let l0 , l1 , m0 , m1 and m0,1 , · · · , m0,l0 , m1,1 , · · · , m1,l1 be nonnegative integers such that ki = mi +

li 

mi,a ,

a=1

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3.7. A∞ BIMODULES AND FLOER COHOMOLOGY

131

for i = 0, 1. We consider a subset Ai ⊆ {1, · · · , mi + li },

#Ai = li ,

and put Ai = {σi (1), · · · , σi (li )}, σi (a) < σi (a + 1). Let πi,1 : (L(i) )ki +li → (L(i) )li be the projection πi,1 (x1 , · · · , xki +li ) = (xσi (1) , · · · , xσi (li ) ). Let πi,2 : (L(i) )ki +li → (L(i) )ki be the projection to the other factors. We denote by A the data, l0 , l1 , m0 , m1 , m0,1 , · · · , m0,l0 , m1,1 , · · · , m1,l1 and A0 , A1 . We put MA k1 ,k0 ([p , w1 ], [q , w2 ])  ◦ Ml1 +m1 ,l0 +m0 ([p , w1 ], [q , w2 ]) = w2 #

(3.7.28) 



β1,a #



β0,b =w2 0 (π1,1 ,π0,1 )◦ev ×ev

l1 0

(1) Mmain ; β1,a ; J1 ) m1,a +1 (L

×

a=1



(0) Mmain ; β0,b ; J0 ) m0,b +1 (L

,

b=1

(where ev  0 = (ev0 , · · · , ev0 )) and (3.7.29)

l0 0

Munbr k1 ,k0 ([p , w1 ], [q , w2 ]) =



MA k1 ,k0 ([p , w1 ], [q , w2 ]).

A

Here ‘unbr’ stands for ‘unbroken’. Elements of Munbr k1 ,k0 ([p , w1 ], [q , w2 ]) are called unbroken Floer trajectory with marked points on the boundary. See Figure 3.7.3. (0) (1) (We remark that τj , τj in the figure are positions of the marked points of the ◦

Ml1 +m1 ,l0 +m0 ([p , w1 ], [q , w2 ]) factor of elements in (3.7.28).)

β1,1 (1)

(1)

τ σ1(2) (0)

τ1

τσ1(1) (0)

τσ0(1) β0,1

(0)

τσ0(2) β0,2

Figure 3.7.3 MA k1 ,k0 ([p , w1 ], [q , w2 ]) (1) (0)

there are k1 (resp. k0 ) marked For each element of points which are mapped to L (resp. L ) after taking fiber product.  In fact m1 (resp. m0 ) of them are on the principal component and m1,j (resp. m0,j ) of them are on the bubbles.

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132

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

We renumber them according to the counter clockwise order and use it to define evaluation maps. Thus we defined an evaluation map (1) k1 ev : MA ) × (L(0) )k0 . k1 ,k0 ([p , w1 ], [q , w2 ]) → (L

(3.7.30) We next put

Mk1 ,k0 ([p , w1 ], [q , w2 ]) 1 1 1 =

1

1

K w(1),··· ,w(K) p(0),··· ,p(K) k1,1 ,··· ,k1,K k0,1 ,··· ,k0,K

(3.7.31)

K−1 0

Munbr k1,a ,k0,a ([p(a) , w(a)], [p(a+1) , w(a + 1)]).

a=1

Here the disjoint union in the right hand side is taken over all K, w(·), p(·), k0,∗ , k1,∗ such that p(0) = p, p(K) = q, [p(a) , w(a)] ∈ Cr(L(0) , L(1) ), k1,1 + · · · + k1,K = k1 , k0,1 + · · · + k0,K = k0 .

(3.7.32.1) (3.7.32.2) (3.7.32.3)

p

=

=

q p(i)

p(0)

p(K)

Figure 3.7.4 The map (3.7.30) induces the evaluation map (3.7.33)

ev : Mk1 ,k0 ([p , w1 ], [q , w2 ]) → (L(1) )k1 × (L(0) )k0

in an obvious way. We can define a topology on Mk1 ,k0 ([p , w1 ], [q , w2 ]) and prove that it is Hausdorff and compact in the same way as Subsection 7.1.4. The construction of the Kuranishi structure is also similar to the argument in Section 7.1 given for other cases. The orientation on Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]) is defined in Section 8.7. Proposition 3.7.26 is proved.  Now as in Section 3.5, let S k (L(0) ; Q) (resp. S k (L(1) ; Q)) be the free Q module generated by codimension k smooth singular simplices on L(0) (resp. L(1) ). We consider the submodules C(L(i) ; Q) (i = 0, 1) of S(L(i) ; Q) generated by the (i) (i) countable set X1 (L(i) ) respectively, as mentioned in Section 3.5. Let (Pj , fj ) ∈ X1 (L(i) ), j = 1, · · · , ki and consider the fiber product

(3.7.34)

Mk1 ,k0 (L(1) ,L(0) ; [p , w1 ], [q , w2 ]) k  k0 1 0 0 (1) (0) . Pi × Pi ev ×f (1) ×···×f (0) 1

k0

i=1

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i=1

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133

(i) (i) We put P (i) = (P1 , · · · , Pki ) and denote (3.7.34) by

Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) ).

(3.7.35)

It has a Kuranishi structure by Proposition 3.7.26. We write Mk (L(i) ; β (i) ; Ji ; P (i) ), Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) ; {Jt }t ) in place of Mk (β (i) ; P (i) ), Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) ), respectively, in case we need to specify the choice of Ji etc. We remark that we have chosen a system of multisections of Mk (L(i) ; β (i) ; Ji ; P (i) ) (as well as the countable set X1 (L(i) )) in Section 3.5 to define our filtered A∞ algebra associated to L(i) . We denote by sβ (i) ,P (i) the multisections determined there. Proposition 3.7.36. In the situation above, there exists a system of multisections s[p ,w1 ],[q ,w2 ],P (1) ,P (0) on the moduli space Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) ) with the following properties: (3.7.37.1)

The virtual dimension of the moduli space Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )

is given by (3.7.38)

μ([q , w2 ]) − μ([p , w1 ]) −

k0 

(0)

deg Pi

i=1 (0)

(1)

−

k1 

(1)

deg Pi

+ k0 + k1 − 1.

i=1 (0)

(1)

Here deg Pi and deg Pi are the degree of Pi and Pi as cochains before shifted. (3.7.37.2) The multisection s[p ,w1 ],[q ,w2 ],P (1) ,P (0) are transversal to 0. (3.7.37.3) The multisection s[p ,w1 ],[q ,w2 ],P (1) ,P (0) are compatible with other multisections s[  ,w ],[  ,w ],P (1) ,P (0) and with sβ (0) ,P (0) , sβ (1) ,P (1) at the boundary. p

1

q

2

Proof. The proof is the same as the case of filtered A∞ algebras (Proposition 3.5.2) whose details are given in Section 7.2. The dimension can be calculated in the way similar to the one in Section 2.3 and in Proposition 3.5.2.  Definition 3.7.39. We denote

−1  Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )s := s[p ,w1 ],[q ,w2 ],P (1) ,P (0) (0). As for the orientation of the moduli space in the above formula, we refer to Definition 8.7.1 in Chapter 8.  −1 We remark that s[p ,w1 ],[q ,w2 ],P (1) ,P (0) (0) is a space with triangulation together with rational weights on each simplex of top dimension. (See Section 6 [FuOn99II] or Definition A1.28.) If the dimension (3.7.38) is 0, it consists of a finite number of points with a weight ∈ Q assigned at each point. When dimension (3.7.38) is 1, it is an (oriented) graph with a weight ∈ Q assigned at each edge.

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134

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Remark 3.7.40. We do not take any particular evaluation map on the perturbed moduli space Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )s in this subsection. So it is not regarded as a singular chain in this subsection. However it is more natural to regard it as a singular chain on the zero dimensional space {q} ⊂ L(1) ∩ L(0) . This point becomes clearer in the next subsection where we deal with the case when L(1) ∩ L(0) is a union of finite dimensional manifolds. Using the spaces Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )s , we now define a family of operators nk1 ,k0 which will define the structure of filtered A∞ bimodules on C(L(1) , L(0) ; Λ0,nov ). Definition 3.7.41. Consider the case of the dimension (3.7.38) being 0. Let (0)

(0)

(0)

T λi eμi Pi

(1)

∈ C(L(0) , Λ0,nov ),

(1)

(1)

T λi eμi Pi

∈ C(L(1) , Λ0,nov )

and Then we define nk1 ,k0 (1)

T λ eμ [p , w1 ] ∈ C(L(1) , L(0) ; Λ0,nov ). by the following formula.

(1)

(1)

nk1 ,k0 (T λ1 eμ1 P1

(1)

(0)

(1)

(0)

(0)

T λ1 eμ1 P1

(0)

(0)

(0)

⊗ · · · ⊗ T λk0 eμk0 Pk0 )

  # Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )s



=

(1)

⊗ · · · ⊗ T λk1 eμk1 Pk1 ⊗ T λ eμ [p , w1 ]⊗

[q ,w2 ]∈Cr(L(0) ,L(1) ) 



T λ eμ [q , w2 ], where λ =

k0 

(0)

λi

+λ+

k1 

i=1

μ =

(1)

λi ,

i=1

k0 

(0)

μi

+μ+

i=1

k1 

(1)

μi .

i=1

Here # is the sum of weights assigned to each points. We note that for a pseudoholomorphic map u : Σ → M satisfying (3.7.22), we have " " " ω− ω= u∗ ω ≥ 0 w2

Σ

w1





by (3.7.22.5). Thus we find that the energy of T λ eμ [q , w2 ] is (3.7.42)



" ω≥

λ + w2

k0  i=1

(0) λi

 +

k1  i=1

(1) λi



"  + λ+

 ω ,

w1

which is non negative, since each of the summands is non negative. So it defines an element of C(L(1) , L(0) ; Λ0,nov ). Hence the maps nk1 ,k0 define nk1 ,k0 : Bk1 (C(L(1) , Λ0,nov )[1]) ⊗Λ0,nov C(L(1) , L(0) ; Λ0,nov )[1] ⊗Λ0,nov Bk0 (C(L(0) , Λ0,nov )[1]) −→ C(L(1) , L(0) ; Λ0,nov )[1]. Since the sum in Definition 3.7.41 is taken over [q , w2 ] such that (3.7.38) is equal to 0, it follows that the degree of nk1 ,k0 is +1. Moreover from (3.2.12.1) the inequality (3.7.42) implies that the maps nk1 ,k0 satisfy the condition (3.7.1).

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135

Let us prove the gapped condition. We denote by G(L(1) , L(0) )0 ⊂ R≥0 × 2Z the submonoid generated by the set {(ω[B], μ(p, q; B)) | p, q ∈ L(0) ∩ L(1) , B ∈ π2 (p, q), M(p, q; B) = ∅}. See Subsection 2.3.1 for the notations. We define (3.7.43)

G(L(1) , L(0) ) = {β (1) + α + β (0) | β (i) ∈ G(L(i) ), α ∈ G(L(1) , L(0) )0 }.

Then G(L(1) , L(0) ) ⊇ G(L(1) ), G(L(0) ) is also a submonoid and by Gromov’s compactness theorem G(L(1) , L(0) ) satisfies Condition 3.1.6. Our operator is obviously G(L(1) , L(0) )-gapped by definition. Now let d be the map obtained from the nk1 ,k2 by the formula (3.7.2). To prove Theorem 3.7.21 it suffices to show the following. Lemma 3.7.44. The homomorphism d defined in (3.7.2) satisfies d ◦ d = 0. Proof. Consider the moduli space Mk1 ,k0 (L(1) , L(0) ; [p , w1 ], [q , w2 ]; P (1) , P (0) )s when its dimension is 1. The boundary of this moduli space is described by one of the following Figures 3.7.5-3.7.7. (1)

Pk1

(1)

P1

L(0) q

p (0)

(0) P1

Pk0

L(1)

(0)

P2

(0)

P∗

(0)

P∗

Figure 3.7.5 (1)

(1)

P∗

P∗

(1)

(1)

Pk1

P1

L(0) q

p

(0) Pk0 L(1)

(0)

P1

Figure 3.7.6

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136

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD (1)

Pk1 L(0) p L(1)

(1)

(1) P∗

P∗

P

(1) 1

(0)

(0) P1

P∗

(1)

Pk0

(1)

P∗

L(1) q L(0)

Figure 3.7.7 The terms corresponding to Figure 3.7.5 are   (1) (1) (0) (0) n d(1) (P1 ⊗ · · · ⊗ Pk1 ) ⊗ [p , w1 ] ⊗ P1 ⊗ · · · Pk0 . Here d(1) is defined in (C(L(1) , Λ0,nov ), m(1) ). The terms corresponding to Figure 3.7.6 are   (1) (1) (0) (0) n P1 ⊗ · · · ⊗ Pk1 ⊗ [p , w1 ] ⊗ d(0) (P1 ⊗ · · · Pk0 ) . Here d(0) is defined in (C(L(0) , Λ0,nov ), m(0) ). The terms corresponding to Figure 3.7.7 are   (1) (1) (0) (0) n P1 ⊗ · · · · · · n(· · · ⊗ Pk1 ⊗ [p , w1 ] ⊗ P1 ⊗ · · · ) · · · ⊗ Pk0 . Combing the argument in Sections 8.5 and 8.7, we can check the signs in a similar way. Therefore we find that these three terms cancel each other. Hence the proof of Lemma 3.7.44 is complete.  Thus we have finished the construction of a left (C(L(1) ; Λ0,nov ), m(1) ) and right (C(L(0) ; Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ) and hence the proof of Theorem 3.7.21.  Now, we assume that (C(L(0) ; Λ0,nov ), m(0) ) and (C(L(1) ; Λ0,nov ), m(1) ) are unobstructed. Then by Definition-Lemma 3.7.13 and Lemma 3.7.14 in the previous subsection, we can define the coboundary operator δb1 ,b0 : C(L(1) , L(0) ; Λ0,nov ) −→ C(L(1) , L(0) ; Λ0,nov )  (i) ; Ji ; si ). (Definition 3.6.14.) This gives rise for any bounding cochains bi ∈ M(L to a deformation of the standard Floer coboundary operator. Thus the following theorem is an immediate consequence of Lemma 3.7.14 and Theorem 3.7.21. Theorem 3.7.45. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian sub (0) ; J0 ; s0 ) and b1 ∈ manifolds which are transversal to each other. Let b0 ∈ M(L  (1) ; J1 ; s1 ). Then there exists a cohomology M(L HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) =

Ker δb1 ,b0 , Im δb1 ,b0

which we call the Floer cohomology of (L(0) , L(1) ) deformed by b0 , b1 . Of course, when (C(L(0) ; Λ0,nov ), m(0) ) and (C(L(1) ; Λ0,nov ), m(1) ) are weakly unobstructed (with respect to the homotopy unit), we can apply Proposition 3.7.17

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137

weak (L(i) ; Ji ; si ) which satisfy the and define Floer cohomology deformed by bi ∈ M condition PO0 (b0 ) = PO1 (b1 ). Remark 3.7.46. We define C(L(1) , L(0) ; 0 ; Λnov ) = (C(L(1) , L(0) ; 0 ; Q) ⊗ Λnov )/ ∼ in the same way as in Definition 3.7.20 and put C(L(1) , L(0) ; 0 ; Λ0,nov ) = F 0 (C(L(1) , L(0) ; 0 ; Λnov )). Then, it is easy to see from the construction that the decomposition  C(L(1) , L(0) ; 0 ; Λ0,nov ) C(L(1) , L(0) ; Λ0,nov ) = [0 ]∈π0 (Ω(L(0) ,L(1) ))

is the completion of the direct sum of filtered A∞ bimodule. It follows that we have a decomposition of the Floer cohomology

(3.7.47)

HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov )  ∼ HF ((L(1) , b1 ), (L(0) , b0 ); 0 ; Λ0,nov ) = [0 ]∈π0 (Ω(L(0) ,L(1) ))

into the completed direct sum over π0 (Ω(L(0) , L(1) )). The same remark applies in the Bott-Morse case discussed in the next subsection. 3.7.5. The Bott-Morse case. In this subsection, we generalize the results in the previous subsection to the case of clean intersection (the Bott-Morse analogue of Lagrangian intersection Floer theory). (See Subsection 2.3.3 for earlier works on Bott-Morse theory in Floer homology.) We start with the following situation. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian submanifolds. We assume that they intersect cleanly in the following sense. Definition 3.7.48. We say that L(0) and L(1) intersect cleanly if the following holds. Let L(1) ∩L(0) = h∈π0 (L(0) ∩L(1) ) Rh be the decomposition into the connected components of L(1) ∩ L(0) . We assume that Rh is a smooth submanifold and NL(1) /Rh ⊕ NL(0) /Rh ⊕ T Rh = (T L(1) + T L(0) )|Rh for each component Rh , where NL(0) /Rh and NL(1) /Rh are normal bundles of Rh in L(0) and L(1) , respectively. We note that the case L(0) = L(1) is a special case of clean intersections. Let dh be the dimension of Rh . If L(1) ∩ L(0) is disconnected, we need to fix some additional data on each component. We first fix a base point ph on each Rh . We next consider the map w : [0, 1] × [0, 1] → M such that (3.7.49.1) (3.7.49.2) (3.7.49.3)

w(0, t) = 0 (t). w(s, 0) ∈ L(0) , w(s, 1) ∈ L(1) for all 0 ≤ s ≤ 1. w(1, t) ≡ ph .

Here 0 is a based path that we fixed for each connected component of the path space Ω(L(0) , L(1) ). (See Section 2.2.) We define the equivalence relation on w’s as in Definition 2.2.4. Let π 0 (L(0) ∩ L(1) ) be the set of all equivalence classes of pairs [h, w].

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Definition 3.7.50. We put π 0 (L(0) ∩ L(1) )h0 =

)

 *  [h, w] ∈ π 0 (L(0) ∩ L(1) )  h = h0 ,

 ) *  0 (L(0) ∩ L(1) )h0 . Λh0 = SpanQ [h, wi ]  [h, wi ] ∈ π

and

In the Bott-Morse theory of finite dimension, we should take local systems on critical submanifolds into account as follows. Let f : X → R be a Bott-Morse function on a finite  dimensional manifold X such that its critical point set is decomposed as Cr(f ) = Rh . Then there exists a spectral sequence i+j H j (Rh , Θ− (X; Z). E2ij = Rh ) ⇒ H dim Rh =i

Θ− j

Here is the determinant (real) line bundle of the unstable bundle over Rj . (This is Theorem 2.1 [Fuk96II] in which the first named author overlooked putting Θ− Ri there. This statement is in principle known to Bott. See Subsection 7.2.2 also.) In our case where we need to replace the space X by its Novikov covering  (0) , L(1) ) and f by the action functional A, we need a similar data as the local Ω(L (1) is considered as a cross section of the cotangent bundle system Θ− j above. If L (0) of L , then the same local system as in the Bott-Morse case is what we need. In general, we define the local system as follows. Define a vector bundle Vh on Rh by (3.7.51)

Vh = {(T L(1) + T L(0) )/(T L(1) + T L(0) )⊥ω }|Rh ,

which is a symplectic vector bundle. Denote by U and U  the image of T L(0) and T L(1) in Vh which form Lagrangian subbundles of Vh . We consider a path λ of Lagrangian subspaces in Vh from U to U  so that the path λ ⊕ T Rh is a path of oriented Lagrangian subspaces in T M from Tp L(0) to Tp L(1) . Then, we consider the index of the Dolbeault operator on the half disc with Lagrangian boundary condition, which we will discuss more in Chapter 8. See Proposition 8.8.1 of Subsection 8.8. In this way, we get a family of operators on the space of paths λ. The orientation bundle of the family index of these operators descends to a local system on Rh , which we denote by Θ− Rh . We will define the Maslov index μ([h, w]) ∈ Z for [h, w] ∈ π 0 (L(0) ∩ L(1) ) in Definition 3.7.62 below. Let S ∗−μ([h,w]) (Rh ; Θ− Rh ) be a cochain complex of smooth singular simplices − with coefficients in the local system Θ− Rh ⊗ det T Rh . Namely if ΘRh is induced by the representation ρh : π1 (Rh ) → {±1}, then using the notation of Section A2 Definition A2.1, we put ∗ S ∗ (Rh ; Θ− Rh ) = S (Rh ; ρh ; Q).

(Note in our situation Rh may not be orientable. So we need det T Rh to regard a chain as a cochain.) We will take a countably generated subcomplex C(Rh ; Θ− Rh ) thereof later in Proposition 3.7.59. (We assume that it is independent of w up to degree.) / BM (L(1) , L(0) ; Λnov ) to be a completion of We define CF   C(Rh ; Θ− Rh ) ⊗Q Λh ⊗Q Λnov , h

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139

where Λh is defined in Definition 3.7.50. / kBM (L(1) , L(0) ; Λnov ) with S ∈ C  (Rh ; Θ− ), we For S ⊗ [h, w] ⊗ T'λ eμ ∈ CF Rh define the energy by w ω + λ and the degree by k =  + μ([h, w]) + 2μ. For a codimension  chain S in Rh labelled by [h, w], we define deg S =  + μ([h, w]). In a way similar to (3.7.19) in Subsection 3.7.4, we can define an equivalence / BM (L(1) , L(0) ; Λnov ) by using the energy and the degree. Furtherrelation ∼ on CF more, we can similarly define the energy filtration thereon which also induces the / BM (L(1) , L(0) ; Λnov )/ ∼. filtration on CF Definition 3.7.52. We denote by C(L(1) , L(0) ; Λ0,nov ) the non-negative en/ BM (L(1) , L(0) ; Λnov )/ ∼ with respect to the energy ergy part of the completion of CF filtration. Giving the grading as above, it becomes a filtered graded free Λ0,nov module and satisfies (3.2.12.1) - (3.2.12.5). Remark 3.7.53. If S = (|S|, f, ξ), where f : |S| → Rh is an immersion of a simplex and ξ is a trivialization of det NRh /S ⊗ Θ− Rh , then we obtain an element ∗ (1) (0) [S, w] in C (L , L ; Λ0,nov ). (We take the Poincar´e dual of the local system Θ− Rh coefficient.) Next we will define a ((C(L(1) ; Λ0,nov ), m(1) ), (C(L(0) ; Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ). In order to do so, we will introduce the following moduli space of pseudo-holomorphic maps. Let Rh , Rh be connected components of L(1) ∩L(0) and {Jt }t = {Jt }0≤t≤1 a one parameter family connecting J0 and J1 , and S ∈ C(Rh ; Θ− h ). Let Σ be R × [0, 1] plus sphere bubbles as in Subsection 3.7.4. Let u : Σ → M , (i) (i) (i) τj ∈ R (i = 0, 1) and we put τ = (τ (0) , τ (1) ) with τ (i) = (τ1 , · · · , τki ). We consider the following conditions for them. u satisfies

(3.7.54.1)

∂u ∂u + Jt = 0, ∂τ ∂t on R × [0, 1]. If the bubble tree of spheres is rooted at (τ, t), all the spheres are Jt -holomorphic. (3.7.54.2) u(R × {0}) ⊂ L(0) , u(R × {1}) ⊂ L(1) . (3.7.54.3) u converges to a point in Rh as τ → −∞ and u converges to a point in Rh as τ → +∞. (0) (0) (1) (1) (3.7.54.4) τj < τj+1 , j = 1, · · · , k0 − 1, τj > τj+1 , j = 1, · · · , k1 − 1.  (3.7.54.5) w#u(Σ) ∼ w . Here # is an obvious concatenation. (3.7.54.6) ((Σ, τ ), u) is stable. We consider the obvious R-action of τ -translations and denote by ∼R the corresponding equivalence relation. We denote by ◦

◦

Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; {Jt }t ) = Mk1 ,k0 ([h, w], [h , w ]) = {The ∼R equivalence classes of ((Σ, τ ), u) satisfying (3.7.54)}.

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140

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

We have the evaluation map (1)

(1)

(0)

(0)

ev = (ev (1) , ev (0) )) =((ev1 , . . . , evk1 ), (ev1 , . . . , evk0 )) ◦

k1

: Mk1 ,k0 ([h, w], [h , w ]) → (L(1) )

k0

× (L(0) )

which is defined by (i)

(i)

evj ((Σ, τ ), u) = u(τj , i) ∈ L(i) ,

(i = 0, 1).

We also define ◦

◦

ev−∞ : Mk1 ,k0 ([h, w], [h , w ]) → Rh , ev+∞ : Mk1 ,k0 ([h, w], [h , w ]) → Rh , by ev±∞ ((Σ, τ ), u) = limτ →±∞ u(τ, t). Then we can prove the following. Proposition 3.7.55. We can compactify the moduli space ◦

Mk1 ,k0 ([h, w], [h , w ]; {Jt }t ) to

Mk1 ,k0 ([h, w], [h , w ]; {Jt }t ). It has a Kuranishi structure compatible to each other and to the Kuranishi structures on Mmain (L(0) ; β (0) ; J0 ) and on Mmain (L(1) ; β (1) ; J1 ) in Proposition 3.4.2. k k Moreover we can extend the evaluation map to (ev−∞ , ev, ev+∞ ) : Mk1 ,k0 ([h, w], [h , w ]) → Rh × (L(1) )

k1

× (L(0) )

k0

× Rh

which is weakly submersive.   Proof. We first define Munbr k1 ,k0 ([h, w], [h , w ]) in a way similar to (3.7.29), ◦

using Mk1 ,k0 ([h, w], [h , w ]). Then we define Mk1 ,k0 ([h, w], [h , w ]) 1 1 1 =

1

1

K w(1),··· ,w(K) h(0),··· ,h(K) k1,1 ,··· ,k1,K k0,1 ,··· ,k0,K

(3.7.56)

Munbr k1,1 ,k0,1 ([h(1), w(1)], [h(2), w(2)]) ×Rh(2) Munbr k1,2 ,k0,2 ([h(2), w(2)], [h(3), w(3)]) ×Rh(3) · · · ×Rh(i) · · · ×Rh(K−1) Munbr k1,K−1 ,k0,K−1 ([h(K − 1), w(K − 1)], [h(K), w(K)])

where the disjoint union in (3.7.56) is taken over all K, h(·), w(·), ki,· such that (3.7.57.1) (3.7.57.2) (3.7.57.3)

h(1) = h, h(K) = h , [h(i), w(i)] ∈ Cr(L(0) , L(1) ), k1,1 + · · · + k1,K = k1 , k0,1 + · · · + k0,K = k0 .

We remark that in (3.7.56) we take fiber product over Rh(i) . (In (3.7.33) we took direct product.) We can define a topology on Mk1 ,k0 ([h, w], [h , w ]) and can prove that it is Hausdorff and compact in the same way as Section 7.1. The construction of Kuranishi structure is also similar to the discussion in Section 7.1. 

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141

We remark that the moduli space Mk1 ,k0 ([h, w], [h , w ]) may not be oriented even when (L(0) , L(1) ) is a relatively spin pair. See Lemma 3.7.70 below for the precise statement on this issue and we will discuss the details on the orientation problem for the Bott-Morse case in Section 8.8. We next consider submodules C(L(0) ; Q), C(L(1) ; Q) of S(L(0) ; Q), S(L(1) ; Q) generated by the countable sets X1 (L(0) ), X1 (L(1) ) in Section 3.5 respectively. Let ORh be the orientation (real line) bundle of the tangent bundle of Rh . (Namely ORh = det T Rh .) We will also take (in the course of the proof of the next proposition) a countable set X1 (Rh ) of smooth chains S = (|S|, f ) equipped with − trivialization of f ∗ ORh ⊗ f ∗ Θ− Rh . An element S of X1 (Rh ) represents a ΘRh -valued cochain in Rh . We denote it by the same symbol S. They generate a submodule − C(Rh ; Θ− Rh ) of S(Rh ; ΘRh ). Remark 3.7.58. The countable set of smooth singular simplices X1 (L(i) ) was already taken and fixed at the time when we define filtered A∞ algebras. Here in the course of the proof of Proposition 3.7.59, we make a choice of X1 (Rh ) to define C(Rh ; Θ− Rh ). The argument for it is similar to one in Section 7.2. (i)

(i)

Let (Pj , fj ) ∈ X1 (L(i) ), j = 1, · · · , ki and S = (|S|, f ) ∈ X1 (Rh ). We take the fiber product Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ])

(ev (1) ,ev−∞ ,ev (0) )

⎛

×f (1) ×···f (1) ×f ×f (0) ···×f (0) ⎝ 1

k1

1

k0

k1 0 j=1

(1)

Pj

×S×

k0 0

⎞ (0) Pj ⎠ ,

j=1

which we denote by Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ). (i) (i) Here we put P (i) = (P1 , · · · , Pki ). By Proposition 3.7.55, Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ) has a Kuranishi structure. Moreover the evaluation map

ev+∞ : Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ) → Rh is weakly submersive. As usual, the dimension will be provided by the index of certain Cauchy-Riemann type operator with a suitable boundary condition which is obtained by linearizing the equation (3.7.54). Postponing its topological definition shortly, we first state the following proposition. Proposition 3.7.59. In the above situation, we assume that (L(0) , L(1) ) is a relatively spin pair in addition. Then for each h ∈ π0 (L(0) ∩ L(1) ) there exist a countable set of smooth singular simplices X1 (Rh ) and a set of integers μ([h, w]) ∈ Z assigned to each [h, w] so that we can find a system of multisections s[h,w],[h ,w ],P (1) ,S,P (0) on Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ), that satisfy the following properties:

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142

(3.7.60.1)

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

The (virtual) dimension of the fiber product Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) )

is given by dim S + dh − dh + μ([h , w ]) − μ([h, w]) −

k0 

(0)

deg Pj

j=1 (0)

−

k1 

(1)

deg Pj

+ k0 + k1 − 1.

j=1

(1)

(0)

(1)

Here dh = dim Rh and deg Pj and deg Pj are the degrees of Pj and Pj as cochains before shifted. (3.7.60.2) The multisection s[h,w],[h ,w ],P (1) ,S,P (0) is transversal to 0. (3.7.60.3) The multisection s[h,w],[h ,w ],P (1) ,S,P (0) is compatible with other multisections s[h ,w ],[h ,w ],P (1) ,S  ,P (0) and with sβ (0) ,P (0) , sβ (1) ,P (1) . (3.7.60.4) ((s[h,w],[h ,w ],P (1) ,S,P (0) )−1 (0), ev+∞ ) is an element of the chain complex C(Rh ; Θ− Rh ). We remark that to make the statement (3.7.60.4) more precise we need to take a triangulation and an order of its vertices to regard the virtual fundamental chain ((s[h,w],[h ,w ],P (1) ,S,P (0) )−1 (0), ev+∞ ) as a singular chain similarly to what we did just after (3.5.3.4). Since this is the same as what did for (3.5.3.4) we do not repeat it here. The proof of this proposition is similar to that of Proposition 3.7.36. There are, however, three new points to discuss. One is a choice of X1 (Rh ), which can be handled in the same way as in Section 7.2. The other two are the issues of the degree μ([h, w]) and of the orientation(flat) bundle Θ− Rh . We will explain these two issues below. To give the dimension formula, we need to provide the description of the linearized operator of (3.7.54.1). We first give the description of this linearization problem. We recall from Subsection 2.2.2 that we fixed a base path 0 : [0, 1] → M (where 0 (0) ∈ L(0) , 0 (1) ∈ L(0) ) of the space of paths Ω(L(0) , L(1) ). We also took t → λ0 (t) such that λ0 (t) is an oriented Lagrangian linear subspace of T0 (t) M and λ0 (0) = T0 (0) L(0) ,

λ0 (1) = T0 (1) L(1)

as oriented linear spaces. For ph ∈ Rh ⊆ L0 ∩ L1 , we consider a path space of Lagrangian subspaces PRh (Tph L0 , Tph L1 ) = {λ : [0, 1] → Λori (Tph M ) | λ(0) = Tph (L0 ), λ(1) = Tph (L1 )}. Here Λori (Tph M ) is the oriented Lagrangian Grassmanian of Tph M . Using the above path λ0 and the map w as in (3.7.49) we will define a (homotopy class) of an element of PRh (Tph L0 , Tph L1 ) as follows. Since [0, 1]2 is contractible we have an isomorphism w∗ T M ∼ = [0, 1]2 × Tp M . Then for (s, t) ∈ ({0} × [0, 1]) ∪ ([0, 1] × {0, 1})

homeo

∼ =

we define λ(0, t) = λ0 (t),

λ(i, s) = Tw(i,s) L(i) .

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[0, 1]

3.7. A∞ BIMODULES AND FLOER COHOMOLOGY

143

We denote this path by λw,λ0 . Tw(

)L

(1)

λ0 (t)

Tw(

)L

(0)

Figure 3.7.8 We put Z− = {z ∈ C | |z| ≤ 1} ∪ {z ∈ C | Rez ≥ 0, |Imz| ≤ 1} Z+ = {z ∈ C | Rez ≤ 0, |Imz| ≤ 1} ∪ {z ∈ C | |z| ≤ 1}.

−1 −1 −

−1

Z− −1 1 −

−1

Z+

Figure 3.7.9

√ We put z = τ + −1t. For each λ ∈ PRh (Tph L(0) , Tph L(1) ), we are going to define a Fredholm operator as follows. Let Wλ1,p (Z+ ; Tph M ), (resp. Wλ1,p (Z− ; Tph M )) be the Banach space consisting of locally L1,p maps ζ+ : Z+ → Tph M (resp. ζ− : Z− → Tph M ) such that (3.7.61.1) ζ+ (τ, i) ∈ Tph L(i) . (resp. ζ− (τ, i) ∈ Tph L(i) ). Here i = 0, 1. (3.7.61.2) √ ζ+ (z+ (t)) ∈ λ(t), (resp. ζ− (z− ) ∈ λ(t) (i = 0, 1)) where we put z± (t) = eπ −1(−1/2±t) ∈ ∂Z± . (3.7.61.3) " eδ|τ | (|∇ζ± |p + |ζ± |p ) dτ dt < ∞. Z±

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Let Lp (Z± ; Tph M ⊗ Λ0,1 (Z± )) be the Banach space of locally Lp sections ζ± of the bundle Tph M ⊗ Λ0,1 (Z± ) such that " eδ|τ | |ζ± |p dτ dt < ∞. Z±

The Dolbeault operator induces a bounded linear map ∂ λ,Z± : Wλ1,p (Z± ; Tph M ) → Lp (Z± ; Tph M ⊗ Λ0,1 (Z± )). It is by now well-known that ∂ λ,Z± is a Fredholm operator. Definition 3.7.62. We denote μanal λ0 ([h, w]) = Index ∂ λw,λ0 ,Z− . In Lemma 3.7.69, we will prove that this analytical index coincides with the generalized Maslov-Morse index μ([h, w]; λ0 ). Now we are ready to give the proof of Proposition 3.7.59. Proof of Proposition 3.7.59. We set μ([h, w]) = μanal λ0 ([h, w]). With this choice made, we first verify (3.7.60.1). Let u ∈ M(L(1) , L(0) ; [h, w], [h , w ]). (u(τ, t) : R × [0, 1] → M is a pseudo-holomorphic map.) We consider the Banach space Wδ1,p (R × [0, 1]; u∗ T M, u∗ T L(1) , u∗ T L(0) ) consisting of all sections ζ of u∗ T M of locally L1,p class such that (3.7.63.1) (3.7.63.2)

ζ(τ, i) ∈ Tph L(i) (i = 0, 1). " R×[0,1]

eδ|τ | (|∇ζ|p + |ζ|p ) dτ dt < ∞.

Let Lp (R × [0, 1]; u∗ T M ⊗ Λ0,1 (R × [0, 1])) be the Banach space of all locally Lp sections ζ of the bundle u∗ T M ⊗ Λ0,1 (R × [0, 1]) such that " eδ|τ | |ζ|p dτ dt < ∞. R×[0,1]

The Dolbeault operator induces a bounded linear map ∂ u :Wδ1,p (R × [0, 1]; u∗ T M, u∗ T L(1) , u∗ T L(0) ) → Lp (R × [0, 1]; u∗ T M ⊗ Λ0,1 (R × [0, 1])). Since w#u ∼ w , a standard result (the index sum formula) implies: Lemma 3.7.64. Index ∂ λw,λ0 ,Z− + Index ∂ u = Index ∂ λw ,λ0 ,Z− + dh . Proof. When τ → +∞ the operator ∂ λw,λ0 ,Z− is a product type (that is of ∂ the form ∂τ + Pτ ) and the number of zero eigenvalues of limτ →+∞ Pτ is exactly dh = dim Rh . The operator Index ∂ u when τ → −∞ has a similar form and the number of zero eigenvalues is dh . Using an obvious modification of Theorem 3.10 [APS75], the lemma follows from these facts. 

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Lemma 3.7.65. dim M(L(1) , L(0) ; [h, w], [h , w ]) = Index ∂ u + dh + dh − 1. Proof. The operator ∂ u has the same symbol as the linearization operator Du ∂ of nonlinear Cauchy-Riemann equation at u. Since we assumed (3.7.63.2) the section ζ goes to zero in the exponential order as |τ | → ∞. Therefore the index of ∂ u is the virtual dimension of the moduli space of pseudo-holomorphic strips which converges to fixed points ph ∈ Rh , ph ∈ Rh as τ → −∞, +∞. Note that the boundary condition for v ∈ M(L(1) , L(0) ; [h, w], [h , w ]) at τ → ±∞ is that v converges to some points in Rh , and in Rh as τ → −∞ and τ → +∞, respectively. Also we divide the moduli space by the R action. Hence dim M(L(1) , L(0) ; [h, w], [h , w ]) is as asserted. (See Subsection 7.1.2 for the Fredholm theory in the case Rh = L(0) = L(1) . Its generalization to the case of arbitrary pair (L(0) , L(1) ) of clean intersection, which we are studying here, is straightforward.)  Now we are ready to wrap-up the proof of (3.7.60.1). The moduli space Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ) is obtained from M(L(1) , L(0) ; [h, w], [h , w ]) by taking fiber products with several chains. The process taking a fiber product with S decreases the dimension by (j) dh − dim S. The process taking a fiber product with Pi decreases the dimension (j) by deg Pi . We put k1 + k0 boundary points, and so we have extra dimension k1 + k0 . Therefore dim Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ) = dim M(L(1) , L(0) ; [h, w], [h , w ]) + dim S − dh −

k0  j=1

(0)

deg Pj

−

k1 

(1)

deg Pj

+ k0 + k1 .

j=1

Lemmas 3.7.64 and 3.7.65 imply dim M(L(1) , L(0) ; [h, w], [h , w ]) = μλ0 ([h , w ]) − μλ0 ([h, w]) + dh − 1. Hence (3.7.60.1) follows. (We like to remark that all the dimensions above are the virtual dimensions, that is, by definition, the dimensions in the sense of Kuranishi structure.) Remark 3.7.66. Formula (3.7.60.1) determines the integers μ([h, w]) modulo the addition by a constant. These integers depend on the connected component of the path space Ω(L(0) , L(1) ). (In the above discussion the connected component was determined by 0 .) This ambiguity of constant will not become an issue as long as we define them to provide the dimension formula (3.7.60.1). However, we emphasize that we will need to use a canonical choice of μ([h, w]) i.e., an absolute degree (depending only on λ0 ) for the construction of bimodule, especially for the study of the orientations of relevant moduli spaces. The above analytical index μanal λ0 ([h, w]) given in Definition 3.7.62 will provide one such canonical choice. For example, in Formula (3.7.2), deg y (that is deg S (see (8.8.8)) in our geometric situation) appears in the third term of the right hand side. It does not appear

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146

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

in the second term. If we change μ([h, w]) by overall constant then deg S = deg y (i) changes. But deg xi,j = deg Pj that appears in the second term does not change. As a consequence, (3.7.2) will no longer hold unless we suitably normalize the choice of integers μ([h, w]). In fact in Chapter 8, the orientations that we put on the moduli space M(L(1) , L(0) ; [h, w], [h , w ]) will depend on the choice of the reference Lagrangian path λ0 and so the signs appearing in the operators n depend on λ0 . For the case of transversal pairs of Lagrangian submanifolds, we stated in Section 2.2 that the analytical index coincides with the topological index μ([h, w]; λ0 ) (Definition 2.2.12) which we call the Maslov-Morse index. We next generalize its definition to the Bott-Morse case and prove the coincidence in this generalized context. Let us consider the situation of Proposition 3.7.59, where the intersection L(0) ∩ (1) L is clean. We fix an element h ∈ π0 (L(0) ∩ L(1) ) and let ph ∈ L(0) ∩ L(1) be its representative. Let w : [0, 1]2 → M , 0 , λ0 be a smooth map satisfying (3.7.49). For (s, 0), (s, 1) ∈ ∂[0, 1]2 , we denote λw (s, i) = Tw(s,i) L(i) , i = 0, 1. We now complete these Lagrangian paths into a bundle pair. For this purpose, we need to take a Lagrangian path αh in Λ(Tph M ) such that (3.7.67.1)

αh (0) = Tph L(0) ,

αh (1) = Tph L(1) .

We choose αh (t) so that it defines a path of oriented Lagrangian subspace of Tp(h) M that extends the given orientations on the Lagrangian subspaces (3.7.67.1). As in Section 2.1, the homotopy class of such paths is not unique and so we will fix a homotopy class of αh by putting a condition on the path αh similar to (2.1.4.3). For this purpose, we also require the following (3.7.67.2)

αh (t) ∩ Tph L(1) = Tph L(0) ∩ Tph L(1)

for all 0 ≤ t < 1. Then if we identify Tph M = Cn = Cdh ⊕ Cn−dh where Cdh = Tph Rh ⊗ C, we can decompose αh (t) such that it decompose to (3.7.67.3)

αh (t) = Tph Rh ⊕ αh2 (t).

We assume that (3.7.67.4)

αh2 (t) ∈ U0 (αh2 (0)) for small t

where the right hand side is as in (2.1.2) applied to Cn−dh . We define a symplectic bundle pair (w∗ T M, λ0 ,λ0 ,h ). (Definition 2.1.13.) Here λ0 ,λ0 ,h is λ0 (t), αh (t), or λw (s, i) at (0, t), (1, t), (s, i), respectively. We then define: (3.7.68)

μ([h, w]; λ0 ) := μ(w∗ T M, λ0 ,λ0 ,h ).

We remark that this definition reduces to Definition 2.2.12 in the case when L(0) is transversal to L(1) . Now we prove Lemma 3.7.69. Let w be as above and ∂λw,λ0 ;Z− be the Dolbeault operator satisfying (3.7.61). Then we have μ([h, w]; λ0 ) = μanal λ0 ([h, w])(= Index ∂ λw,λ0 ;Z− ).

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Proof. We note that under the trivialization w∗ T M ∼ = [0, 1]2 × Tph M the Lagrangian subbundle λ0 ,λ0 ,h defines a Lagrangian loop, which we still denote by λ0 ,λ0 ,h . The Maslov index of the loop is invariant under the symplectic transformations and under the homotopy. We will consider homotopy of the loop that is fixed on the boundary {(1, t) | t ∈ [0, 1]} ⊂ ∂[0, 1]2 . We then elongate [0, 1) × [0, 1] ∼ = R × [0, 1] by a cut-off function χ : [0, ∞) → [0, 1] as given in Section 2.1. Under this elongation the operator ∂ λw,λ0 ;Z− becomes the product type as τ → ∞ and its index is invariant under the continuous deformations of coefficients and of the boundary conditions fixing the asymptotic condition, by the homotopy invariance of the Fredholm index. Therefore it suffices to consider the following special case (See Appendix [Oh99] for the detailed explanation on this kind of reduction.): √ Tph M = Cn , Tph L(0) = Rn , Tph L(1) = Rdh ⊕ −1Rn−dh and

√

αh (t) = Rdh ⊕ eπ −1t/2 Rn−dh together with the path λw,λ0 : ∂[0, 1]2 \ {(1, t)} → Cn given by λw,λ0 (s, 0) ≡ Rn , λw,λ0 (s, 1) ≡ Rdh ⊕

√ −1Rn−dh

λw,λ0 (0, t) = Rdh ⊕ (e−(l1 +1/2)π

√

−1t

R ⊕ · · · ⊕ e−(ln−dh +1/2)π

√

−1t

R)

with lj ∈ Z. Then we have the Lagrangian loop λ0 ,λ0 ,h = λw,λ0 ∪ αh . Now the index problem splits into one-dimensional problems of the types ⎧ ⎪ ⎨ ∂ζ = 0 √ ζ(τ, 0) ∈ R, ζ(τ, 1) ∈ −1R ⎪ √ ⎩ ζ(0, t) ∈ e−(l+1/2)π −1t R or the one with the boundary condition ζ(∂[0, 1]2 ) ⊂ R for the constant Lagrangian path t → R arising from the Rdh -factor. For the first type, the corresponding Lagrangian loop λ0 ,λ0 ,h defined above defined on ∂[0, 1]2 becomes √

(s, 0) → R, (0, t) → e−(l+1/2)π −1t R √ √ (s, 1) → −1R, (1, t) → eπ −1t/2 R. This loop travelled in the positive direction along ∂[0, 1]2 is√homotopic to the concatenated Lagrangian path that is t ∈ [0, 1] → e(l+1/2)π −1(1−t) R) followed by √ π −1t/2 R. This concatenated path defines a loop which has the t ∈ [0, 1] → e Maslov index l + 1. Summing-up these, we obtain the Maslov index 

n−dj

μ(λ0 ,λ0 ,h ) =

(lj + 1).

j=1

On the other hand for the corresponding Fredholm index, the one dimensional problem can be explicitly solved by the complex one variable Fourier analysis. For example, the index was computed in (A.12) [Oh99] to be l + 1. On the other hand for the constant path t → R coming from Cdh , the obvious constant solutions do not live in the weighted Sobolev space Wδ1,p whose elements are required to have

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148

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

exponential decay due to the weight factor eδ|τ | in the weighted Sobolev space Wδ1,p . And the corresponding cokernel can be easily shown to be zero. This proves 

n−dj

Index ∂ λw,λ0 ,Z− =

(lj + 1).

j=1



Comparing the last two index formulae we have finished the proof.

We next discuss orientation (flat) bundle Θ− Rh which we need to prove (3.7.60.4). We will discuss this in detail in Section 8.8. We remark that Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) ) may not be orientable in the sense of Kuranishi structure. This is the reason why we need orientation bundle Θ− Rh . Consider the fiber product ⎛ ⎞ k1 k0 0 0 (1) (0) Pj × Pj ⎠ , Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]) ev ×(f (1) ×···×f (0) ) ⎝ 1

k0

j=1

j=1

which we denote by Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ). The signs in the A∞ -structure and for the boundary operators in the Floer theory are determined by the orientation bundle of the moduli space Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ), and the local systems {Θ− Rh }. More precisely we have the following. Put ORh = det T Rh . Lemma 3.7.70. The Kuranishi structure of Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ) has a tangent bundle. We consider the Kuranishi structure of the moduli space Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ). Let Up = Vp /Γp and Ep be as in the definition of Kuranishi structure (Definition A1.1). Then, ev±∞ extends to Up as submersions (ev±∞ are weakly submersive). Then there exists an isomorphism − ∗ ∗ ∗ Θ− det(Ep∗ ) ⊗ det(T Vp ) ∼ = ev+∞ Rh ⊗ ev+∞ ORh ⊗ ev−∞ ΘRh ,

which is compatible with the isomorphism det(Ep∗ )|Vpq ⊗ det(T Vp )|Vpq ∼ = det(Eq∗ ) ⊗ det(T Vq ) for q ∈ ψp (s−1 p (0)) obtained from the existence of a tangent bundle. Here ev−∞ : Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ) → Rh ev+∞ : Mk ,k (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ) → Rh 1

0

are defined similarly as before. Note that det(Ep∗ ) ⊗ det(T Vp ) is the orientation bundle O Mk

1 ,k0

 (1) ,P  (0) ) (L(1) ,L(0) ;[h,w],[h ,w ];P

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149

of the space Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]; P (1) , P (0) ) in the sense of Kuranishi structure. Moreover, since M is oriented, it follows that there is a canonical isomorphism as {±1}-flat bundles − ∼ Θ+ Rh = ORh ⊗ ΘRh .

A detailed argument will be given in Section 8.8 (Proposition 8.8.6, Definition 8.8.11). (In Section 8.8, we study the fiber product of M(L(1) , L(0) ; [h, w], [h , w ]) and the singular simplex S with coefficients in ORh ⊗ Θ− Rh . The orientation on Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], S, [h , w ]) is given in Definition 8.8.11. We can also interpret this moduli space as the fiber product of Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]) and S by an appropriate sign change rule concerning the exchange of factors. Then we can formulate the result as in Lemma 3.7.70.) We use Lemma 3.7.70 to study the orientation in (3.7.60.4) as follows. See the proof of Proposition 8.8.7 for detail. Let (φ : S → Rh ) ∈ X1 (Rh ) equipped with a flat section s of ORh ⊗ Θ− Rh ⊗ OS . Here OS = det T S. We find that − ∗ ∗ det(Ep∗ ) ⊗ det(T (Vp ×Rh S)) ∼ Θ+ = (−1)rh ev+∞ Rh ⊗ ev−∞ ΘRh ⊗ ORh ⊗ OS .

Here rh = dim Rh . (See also the proof of Proposition 8.8.7 for the reason why the factor (−1)rh appears.) Thus we have ∗ ∗ Θ+ ev+∞ Rh ⊗ (det Ep ) ⊗ det(T (Vp ×Rh S))

∗ ∼ (ORh ⊗ Θ− = (−1)rh ·(μ(h,w)+1) ev−∞ Rh ) ⊗ O S .

+ ∗ Hence the flat section s of φ∗ (ORh ⊗Θ− Rh )⊗OS induces a flat section of ev+∞ ΘRh ⊗ ∗ (det Ep ) ⊗ det(T (Vp ×Rh S)). This implies that

((s[h,w],[h ,w ],P (1) ,S,P (0) )−1 (0), ev+∞ ) ∼ + is a chain in Rh with coefficients in ORh ⊗ Θ− Rh = ΘRh . This is the orientation part of the proof of (3.7.60.4). Proposition 3.7.59 is now proved modulo the points deferred to later chapters. We now put Mk1 ,k0 ,(+∞) (L(1) , L(0) ;[h, w], [h , w ]; P (1) , S, P (0) )s := ((s[h,w],[h ,w ],P (1) ,S,P (0) )−1 (0), ev+∞ ). We refer to Definition 8.8.11 for the orientation of the above moduli space. (We need to take triangulation and an order of its vertices to regard the virtual fundamental chain (the right hand side) as a singular chain. See Definition A1.28.) Definition 3.7.71. For S ∈ X1 (Rh ) we write S ⊗ [h, w] for the corresponding element in C(L(0) , Q). Let (0)

(0)

(0)

T λi eμi Pi

∈ C(L(0) , Λ0,nov ),

(1)

(1)

(1)

T λi eμi Pi

∈ C(L(1) , Λ0,nov )

and T λ eμ [S; h, w] ∈ C(Rh ; Θ− Rh ) ⊗ Λ0,nov .

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

We define nk1 ,k0 by (1)

(1)

(1)

nk1 ,k0 (T λ1 eμ1 P1

(1)

(0)

=



(1)

(1)

⊗ · · · ⊗ T λk1 eμk1 Pk1 ⊗ T λ eμ (S ⊗ [h, w]) (0)

(0)

⊗ T λ1 eμ1 P1 

(0)

(0)

(0)

⊗ · · · ⊗ T λk0 eμk0 Pk0 )



T λ eμ Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) )s ⊗ [h , w ],

[h ,w ]

where λ =

k0  i=1

(0)

λi

+λ+

k1 

(1)

λi ,

μ =

k0 

i=1

(0)

μi

i=1

+μ+

k1 

(1)

μi .

i=1

In case (k1 , k0 ) = (0, 0). In case (k1 , k0 ) = (0, 0) we add extra term (−1)dh +μ([h,w]) ∂S ⊗ [h, w]. We remark that the sum in Definition 3.7.71 contains the case when [h, w] = [h , w ], λ = λ , μ = μ , (k1 , k0 ) = (0, 0). In such a case the corresponding term is (1)

mk1 +1+k0 ((P1

(1)

(0)

∩ Rh ), · · · , (Pk1 ∩ Rh ), S, (P1

(0)

∩ Rh ), · · · , (Pk0 ∩ Rh ))

where mk1 +1+k0 is an A∞ deformation of the cup product on Rh . In fact the moduli space we use to define such a term consists of constant maps to Rh . We also remark that we can choose our countable set of chains on L(0) , L(1) so that they are transversal to Rh . (Otherwise the above equality does not hold. However for the transversality of our moduli space we do not need to choose so.) (i) (i) We also remarks that deg S = dh − dim S, deg Pj = n − dim Pj . Hence (3.7.60.1) implies dh − dim Mk1 ,k0 ,(+∞) (L(1) , L(0) ; [h, w], [h , w ]; P (1) , S, P (0) )s   (0) (1) = deg Pj + deg Pj + μ([h, w]) − μ([h , w ]) + deg S + 1. Hence nk1 ,k0 increase the (shifted) degree by 1. Now we can show the following. Theorem 3.7.72. The nk1 ,k0 above defines a left (C(L(1) , Λ0,nov ), m(1) ) and right (C(L(0) , Λ0,nov ), m(0) ) filtered A∞ bimodule structure on C(L(1) , L(0) ; Λ0,nov ). It is G(L(0) , L(1) )-gapped. Furthermore the pair of the homotopy units {e1 , e0 } of (C(L(1) ; Λ0,nov ), m(1) ) and (C(L(0) ; Λ0,nov ), m(0) ) acts as a homotopy unit. The proof is a straight-forward generalization of the proof of Theorem 3.7.21 in Subsection 3.7.4 and hence is omitted. Now we assume that the A∞ algebras (C(L(0) , Λ0,nov ), m(0) )

and

(C(L(1) , Λ0,nov ), m(1) )

are unobstructed. Let b0 and b1 be bounding cochains of (C(L(0) ; Λ0,nov ), m(0) ) and (C(L(1) ; Λ0,nov ), m(1) ) respectively. Then by Lemma 3.7.14 and Theorem 3.7.72 we obtain a cochain complex (3.7.73)

(C(L(1) , L(0) ; Λ0,nov ), δb1 ,b0 )

by the same way as in Subsection 3.7.4.

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Definition 3.7.74. We call the cohomology of (3.7.73) the Floer cohomology deformed by b0 , b1 and write it as HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ). This cochain complex (3.7.73) coincides with the one defined in Subsection 2.1.2 if L(0) is transverse to L(1) . When L(0) = L(1) we have the following. Proposition 3.7.75. Assume that L = L(0) = L(1) . We also assume that Jt is independent of t. Then, the filtered A∞ bimodule defined in Theorem 3.7.72 can be taken to be the same as the one we obtain from Example 3.7.6. Proof. Note that there is only one component L(1) ∩ L(0) , which is L. The orientability of L and M implies that the local system Θ− L is trivial in this case. Then, by inspecting the proofs, we may take X1 (Rh ) = X1 (L), where X1 (L) is as in Section 3.5 and X1 (Rh ) is as in Proposition 3.7.55. Moreover we can take (3.7.76)

Mk1 ,k0 ,(+∞) (L(1) , L(0) ;[h, w], [h, w ]; P (1) , S, P (0) )  (1) , S, P (0) ), = Mk +k +1 (L; β; P 1

0

where the left hand side is an in Proposition 3.7.55 and the right hand side is as in Proposition 3.7.59 and w = w#β. Then we can take multisections in Proposition 3.5.2 which are identified with those in Proposition 3.5.2 via (3.7.76). The equality nk1 ,k0 (x ⊗ y ⊗ z) = mk1 +k0 +1 (x, y, z) now follows from construction. Our conventions and definitions concerning the orientation ensure that the orientations on the both hand sides of (3.7.76) indeed coincide. See Sections 8.5 and 8.8. Thus the above equality holds with signs.  Thus Proposition 3.7.75 yields that (3.7.77)

HF (L, b; Λ0,nov ) ∼ = HF ((L, b), (L, b), Λ0,nov ),

where the left hand side is defined by Definition 3.6.11 or Definition 3.6.33 (1) (weakly unobstructed case) from the filtered A∞ algebra (C(L; Λ0,nov ), m) and the right hand side is defined from the filtered A∞ bimodule as in this subsection. Furthermore, Proposition 3.7.75 together with independence of HF ((L(1) , b1 ), (L(0) , b0 ); Λnov ) := HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) ⊗Λ0,nov Λnov under the Hamiltonian isotopy (which we prove as Theorem 5.3.14 and Theorem 4.1.5 in Section 5.3) enable us to use spectral sequence (which we establish in Chapter 6) to study Floer cohomology of L and its Hamiltonian deformation. In particular, if L(1) = ψ 1 (L(0) ) is a Hamiltonian deformation of L(0) , we will see that (3.7.78)

HF (L, b; Λnov ) ∼ = HF ((ψ 1 (L), ψ∗1 (b)), (L, b), Λnov ).

See examples in the next subsection. 3.7.6. Examples. In this subsection, we give simple but instructive examples of calculations of Floer cohomology and explain how we can geometrically see the relation between the existence of Floer cohomology and the potential function PO discussed in Subsections 3.6.3 and 3.7.3. Moreover we will exhibit how the choice of the coefficient rings Λ0,nov and Λnov affects the invariance of or the dependence of Floer cohomology under the Hamiltonian isotopy. More examples are given in [FOOO08I,08II,09I,09II].

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Consider an embedded circle L in S 2  CP 1 . There are two simple holomorphic discs bounding the circle. We denote them by Du and D . (Actually there are multiple covers of them. However we can show that they do not contribute to the Floer cohomology by the degree reason.) In case they have the same area, e.g., when L is an equator, they belong to the same class β in Π(L) = Π(S 2 ; L) ∼ = Z. See Definition 2.4.17 for the notation. Otherwise they belong to two different classes in Π(S 2 ; L) ∼ = Z ⊕ Z. We denote the classes by βu and β respectively. It is easy to see that (3.7.79)

μL (β) = μL (βu ) = μL (β ) = 2.

1 1 1 for the circle such that Π(Seq ) = Z and Sneq for the circle such We write by Seq 1 that Π(Sneq ) = Z ⊕ Z. From (3.7.79), we note that any Lagrangian circle L in S 2 is weakly unobstructed. The obstruction classes live only in H 0 (L; Q).

Case I. The case where L(1) is a Hamiltonian deformation of L(0) . We denote by ψ1 a Hamiltonian isotopy such that L(1) = ψ1 (L(0) ). 1 . Case I-a. The case for L(0) = Seq 1 Firstly, we calculate the first obstruction class [o1 (Seq )]. As we mentioned above, we have two holomorphic discs Du and D with the same area. There is an 1 anti-symplectic involution τ on S 2 such that τ fixes Seq and τ (Du ) = D . When we 1 1 ; τ∗ β) which define the obstruction class, we consider both M1 (Seq ; β) and M1 (Seq are the moduli spaces of pseudo-holomorphic discs with one marked point. Therefore, to calculate the obstruction class precisely, we need to analyze how τ respects the orientation on the moduli spaces of pseudo-holomorphic discs with marked 1 points. They are studied in [FOOO09I]. There we find that τ∗ : M1 (Seq ; β) → 1 M1 (Seq ; τ∗ β) preserves the orientation, because of the fact that μL (β) = 2 and the number of the marked point is one. Therefore the two holomorphic discs Du 1 1 and D have the same contribution to the obstruction class [o1 (Seq )] = [o(Seq ; β)]. 1 (Here β = [β] = [τ∗ β] ∈ Π(Seq ).) Hence we have

(3.7.80)

1 1 [o(Seq ; β)] = 2[Seq ]

in H 0 (S 1 ; Q).

Secondly, since there is no holomorphic discs with negative Maslov indices (in fact this is the monotone case), the Floer cohomology can be defined. (Namely 1 1 Seq ⊂ S 2 is weakly unobstructed.) Moreover 0 is an element of Mweak (Seq ). Hereafter we drop 0 from the notation in the rest of this subsection. Thirdly, we calculate HF (L(0) , L(0) ; Λnov ) using the Bott-Morse theory. (Here (0) 1 .) The boundary map m1,β for β = 0 is induced by the moduli space L = Seq (0) M2 (L ; β; J). According to the results from [FOOO09I] we obtain ev∗ ([M2 (L(0) ; β; J)]) = −ev∗ ([M2 (L(0) ; τ∗ β; J)]). since in this case the number of the marked points is two so τ∗ is orientation reversing. Therefore we have (3.7.81)

HF (L(0) , L(0) ; Λ0,nov ) ∼ = Λ0,nov ⊕ Λ0,nov .

(Alternatively we can say that the spectral sequence which will be constructed in Chapter 6 degenerates at E2 level.)

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Let us compare (3.7.81) with HF (L(1) , L(0) ; Λnov ) where L(1) = ψ1 (L(0) ) is Hamiltonian isotopic to L(0) . Let us take a Hamiltonian isotopy such that ψ1 (L(0) )∩ 1 1 ) ∩ Seq ) ≥ 2, in general.) The moduli space of L(0) = {p, q}. (Note that #(ψ1 (Seq connecting orbits from p to q (or q to p) with minimal area has two components, respectively. On the other hand, the invariance of Floer cohomology of Hamiltonian isotopy and by (3.7.81), we find that (3.7.82.1)

1 1 HF (ψ1 (Seq ), Seq ; Λnov ) ∼ = Λnov ⊕ Λnov .

It follows that the two have different contribution cancels each other. (We can also prove it directly by comparing it with the moduli space of gradient trajectories of a Morse function on L(0) = S 1 with two critical points {p, q}.) The contribution of the pseudo-holomorphic strip with non-minimal energy is 0 by degree reason. Thus we have (3.7.82.2)

1 1 HF (ψ1 (Seq ), Seq ; Λ0,nov ) ∼ = Λ0,nov ⊕ Λ0,nov .

1 . Case I-b. The case for L(0) = Sneq 1 In this case, the two holomorphic discs Du and D which bound L(0) = Sneq 1 1 have different symplectic areas. We have [o(Sneq ; βu )] and [o(Sneq ; β )] that are 1 )∼ obstruction classes corresponding to βu and β which generate Π(S 2 ; Sneq = Z⊕Z. We can find that

(3.7.83)

1 1 1 [o(Sneq ; βu )] = [o(Sneq ; β )] = [Sneq ] ∈ H 0 (S 1 ; Q).

1 1 Let us calculate the Floer cohomology HF (ψ1 (Sneq ), Sneq ). We like to emphasize that, in general, the Floer cohomology over Λ0,nov depends on the Hamiltonian diffeomorphism ψ1 but one over Λnov is always independent of the Hamiltonian diffeomorphism. This point will be proved as Theorem 5.3.14 in Section 5.3. Here we will elaborate the phenomena by this example. We first note that there exists a large Hamiltonian isotopy {ψt }0≤t≤1 with 1 1 ψ1 (Sneq ) ∩ Sneq = ∅. In this case, we have 1 1 ), Sneq ; Λ0,nov ) = 0. HF (ψ1 (Sneq

Obviously, it also vanishes over Λnov .

ψ1 (L(0) )

A q

C

L(0)

D p B

Figure 3.7.10

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On the other hand, let us consider the case ψ1 (L(0) ) ∩ L(0) = {p, q}. In this case, the complement CP 1 \ (ψ1 (L(0) ) ∪ L(0) ) consists of 4 components. We denote their symplectic areas by A, B, C and D, respectively. Since ψ1 is a Hamiltonian diffeomorphism, we have C = D or A = B. (Since L(0) is not an equator, the case that A = B and C = D is not our case.) See Figure 3.7.10. We can find that the orientation on the moduli space of the connecting orbit form p to q (resp. from q to p) with the area A (resp. C) is opposite to one corresponding to having the area B (resp. D). We can prove this using the fact that the orientation of the moduli space appearing here is the same as the one in I-a. (Alternatively, we can use the reduction to the Morse (Novikov) theory.) When C = D, the two connecting orbits from p to q are in different orbits under the deck 1 transformation Π(S 2 ; Sneq ), while those from q to p cancels each other. The case A = B is similar. Therefore, the coboundary operator δ is given by (3.7.84.1)

[p] → T A [q] − T B [q],

[q] → 0,

or (3.7.84.2)

[p] → 0,

[q] → T C [p]e − T D [p]e.

(Here we are considering the degree of [q] is one bigger than that of [p] and recall that the indeterminate e has degree 2 in our convention.) The contribution to the coboundary operator from the other components of the moduli space of pseudoholomorphic discs vanish by degree reason. Hence we obtain δ ◦ δ = 0. So the 1 1 Floer cohomology HF (ψ1 (Sneq ), Sneq ; Λ0,nov ) can be also defined, but it is not zero for this case. This is a torsion group. (In fact it is Λ0,nov /T C Λ0,nov if C < 1 1 D.) This example shows that HF (ψ1 (Sneq ), Sneq ; Λ0,nov ) actually depends on the Hamiltonian diffeomorphism. However, when we consider the Floer cohomology over Λnov , we find that (3.7.85)

1 1 ), Sneq ; Λnov ) = 0, HF (ψ1 (Sneq

because T A − T B is invertible in Λnov in the case C = D and T C − T D is invertible in Λnov in the case A = B. In fact, if A < B, then (T A − T B )−1 = T −A (1 − T B−A )−1 = T −A

∞ 

T (B−A)j .

j=0 1 1 ) ∩ Sneq =∅ This result coincides with the Floer cohomology for the case ψ1 (Sneq over Λnov .

Case II. The case where L(1) is not a Hamiltonian deformation of L(0) . Let us consider the case L(1) ∩ L(0) = {p, q}. As in the Case I-b, we can find that the coboundary operator should be [p] → T A [q] − T B [q],

[q] → T C [p]e − T D [p]e.

But since L(1) is not a Hamiltonian deformation of L(0) , A = B and C = D in this case. Therefore we have δ ◦ δ = 0 and hence Floer cohomology is not defined.

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Remark 3.7.86. In Subsections 3.6.1 (unobstructed case) and in Subsections 3.6.3 (weakly unobstructed case), we defined the Floer cohomology HF (L, b; Λ0,nov ) (see Definition 3.6.11 and Definition 3.6.33 (1)). On the other hand, in Definition 3.7.74 Subsection 3.7.5, we defined HF ((L, b), (L, b); Λ0,nov ), as a special case of the Floer cohomology for a pair of Lagrangian submanifolds in the Bott-Morse situation. Then Proposition 3.7.75 says that they are isomorphic over Λ0,nov . However, as we saw in the above example, they are not isomorphic over Λ0,nov in general to the Floer cohomology of the pair ((L, b), (ψ(L), ψ∗ (b)) for a Hamiltonian diffeomorphism ψ = id, while they are isomorphic over Λnov . This will be proved as Theorem 4.1.5 in Section 5.3 Chapter 5. Moreover, the dependence of the Floer cohomology HF ((ψ(L), ψ∗ (b)), (L, b); Λ0,nov ) on ψ appears in the torsion parts of the Floer cohomology. In Subsection 6.5.3, we will study the torsion part and use this in relation to the study of Hofer’s distance of some Hamiltonian isotopies. 3.7.7. The multiplicative structure on Floer cohomology. In this subsection, we prove some purely algebraic results which will be used to prove (B.2) in Theorem B and (G.5) in Theorem G in Subsection 3.8.10.  Proposition 3.7.87. (1) Let C be a filtered A∞ algebra and bi ∈ M(C), then we have an operation (3.7.88)

m2 : H(C, δb0 ,b1 ) ⊗ H(C, δb1 ,b2 ) → H(C, δb0 ,b2 )

which is associative up to the sign. Namely (3.7.89)



m2 (m2 (x, y), z) = (−1)deg x m2 (x, m2 (y, z)).

weak (C) with (2) If C is unital then the same conclusion holds for bi ∈ M PO(b1 ) = PO(b0 ). Proof. Let xij ∈ C. We put (3.7.90)

˜ 2 (xij , xjk ) = m(ebi , xij , ebj , xjk , ebk ). m

It is easy to see from A∞ formula that (3.7.91)

˜ 2 )(xij , xik ) − (δbi ,bk ◦ m ˜ 2 (δbi ,bj xij , xjk ) + (−1)deg =m



xij

˜ 2 (xij , δbj ,bk xjk ). m

Therefore (3.7.90) induces (3.7.88). We next put ˜ 3 (xhi , xij , xjk ) = m(ebh , xhi , ebi , xij , ebj , xjk , ebk ). m Then we can prove the following formula if δbu ,bv (xuv ) = 0. ˜ 3 )(xhi , xij , xik ) =m ˜ 2 (m ˜ 2 (xhi , xij ), xjk ) −(δbh ,bk ◦ m + (−1)deg



xij

˜ 2 (xhi , m ˜ 2 (xij , xjk )). m

(3.7.89) follows from this. The proof of ‘weak’ version is the same. We remark that (3.7.89) induces product on H(C, δb,b ) defined by (3.7.92)

x ∪ y = (−1)deg x(deg y+1) m2 (x, y).

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Proposition 3.7.93. (1) Let C1 , C0 be the filtered A∞ algebra and D is be a  i ). Then H(D, δb ,b ) is a H(C, δb ,b )(C1 , C0 ) filtered A∞ bimodule. Let bi ∈ M(C 1 0 1 1 H(C, δb0 ,b0 ) bimodule. weak (Ci ) (2) If Ci are unital and D is unital then the same holds for bi ∈ M with PO(b1 ) = PO(b0 ). ˜ 1,0 (x, v) = n(eb1 xeb1 , v, eb0 ). It is Proof. We define n ˜ 1,0 : C1 ⊗ D → D by n easy to see that n ˜ 1,0 induces a left H(C, δb1 ,b1 ) module structure on H(D, δb1 ,b0 ). The other part of the proof is similar.  3.8. Inserting marked points in the interior In this section, we will construct the operators p,q, and r considering the moduli spaces of pseudo-holomorphic discs with some marked points in the interior. They are useful to study the relationship of the obstruction class or of the Floer cohomology to the homomorphism i∗ : H k (M ) → H k (L) and the Gysin homomorphism i! : H k (L) → H n+k (M ). The results in this section will be also used to study the differential in the spectral sequence to be constructed in Chapter 6 and in the proofs of the Theorems H, I. 3.8.1. The operator p. We recall that the Gysin homomorphism is defined by i! = P DM ◦ i∗ ◦ P DL : H k (L) → H n+k (M ), where P D’s are the Poincar´e duality maps in M and L, and i : L → M is the natural inclusion. Let (C(L; Λ0,nov ), m) be the filtered A∞ algebra constructed by Theorem 3.5.11 and (S ∗ (M ; Z), δ) be the smooth singular chain complex of M . Identifying a codimension k chain as a degree k cochain, we regard S ∗ (M ; Z) as a cochain complex. We put S ∗ (M ; R) = S ∗ (M ; Z) ⊗ R where R is a commutative ring. The Gysin homomorphism is induced by the cochain map i! : S k (L; R) → S k+n (M ; R), given by i! (|P |, f ) = (|P |, i ◦ f ). To precisely state the main technical result (Theorem 3.8.9), we need to recall a piece of Gromov-Witten theory. Let 0 = β˜ ∈ π2 (M ). We consider the moduli ˜ consisting of the set of all ((S 2 , (z + , z + )), w) where w : S 2 → M (reg (M ; β) space M 0 1 0,2 is a pseudo-holomorphic map whose homotopy class is β˜ and (z0+ , z1+ ) is a pair ˜ by the action of P SL(2; C) and of distinct points on S 2 . We divide M0,2 (M ; β) ˜ the compactify the quotient via the stable maps. Let us denote by M0,2 (M ; β) ˜ stands for the resulting moduli space. The suffix 0 in the notation M0,2 (M ; β) genus of the domain of pseudo-holomorphic maps. We have two evaluation maps at the marked points z1+ , z0+ : ˜ → M 2. ev = (ev1 , ev0 ) : M0,2 (M ; β) We consider a chain (3.8.1)

˜ ev ×M L) ev0∗ (M0,2 (M ; β) 1

in M . it has a Kuranishi structure as constructed in [FuOn99II]. We denote its virtual fundamental chain by ˜ GW0,1 (M ; β)(L) ∈ S ∗ (M ; Q).

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157

˜ + 2. Here the degree ∗ (= the codimension) is calculated by ∗ = n − 2c1 (M )[β] ˜ ev ×M L and an order of Precisely speaking, we fix a triangulation on M0,2 (M ; β) 1 the vertices to identify (3.8.1) as a smooth singular chain in M . (See Definition A1.28.) We then define a Λ0,nov -cochain  ˜ ˜ ˜ T ω[β] ec1 (M )[β] GW0,1 (M ; β)(L) ∈ S n+2 (M ; Λ0,nov ). (3.8.2) GW0,1 (M )(L) = β˜

We next introduce a cyclic bar complex Bkcyc C[1]. Let C be a filtered A∞ algebra and define an automorphism cyc : Bk C[1] → Bk C[1] by 

k−1



cyc(x1 ⊗ · · · ⊗ xk ) = (−1)deg xk ×( i=1 deg xi ) xk ⊗ x1 ⊗ · · · ⊗ xk−1 . It induces a Zk action on Bk C[1]. Let Bkcyc C[1] be the invariant set of cyc and

B cyc C[1] =  k Bkcyc C[1] the completed direct sum of them. Remark 3.8.3. The notion of the cyclic bar complex which sometimes appear in the literature (for example in [GeJo90]) is the symmetrization with respect to the symmetric group Sk acting on Bk C[1]. Here we symmetrize it only by the cyclic group Zk . We denote the symmetrization by Sk action by Ek C. (See Definition 3.8.30.)  the Lemma & Definition 3.8.4. Let (C, m) be an A∞ algebra and (BC[1], d) cyc  associated bar complex. Then B C[1] is preserved under the map d. Thus we define  B cyc C[1] . dcyc := d| Proof. Let x1 ⊗ · · · ⊗ xk ∈ Bk C[1] and consider the element [x1 ⊗ · · · ⊗ xk ] =

(3.8.5)

k 

cyci (x1 ⊗ · · · ⊗ xk ) ∈ Bkcyc C[1].

i=1

Then we have  1 ⊗ · · · ⊗ xk ]) d([x (3.8.6)

=

k  

(−1)deg



x +···+deg x+i−2 +α

i≤j+1 =1



x ⊗ · · · ⊗ mj−i+1 (x+i−1 , · · · , x+j−1 ) ⊗ · · · ⊗ x−1 ,

where α = a 0, p1 ◦ m0 (1) + δM ◦ p0 (1) + GW0,1 (M )(L) = 0.

We can further extend pk to cyc + ∗ p+ k : Bk C(L; Λ0,nov ) [1] −→ S (M ; Λ0,nov )

incorporating the homotopy unit so that the same formula as (3.8.10.2) holds after replacing p, dˆcyc by p+ , dˆ+cyc , respectively. (Here dˆ+cyc is defined from m+ in the same way as dˆcyc .) Moreover we have (3.8.10.4) (3.8.10.5) (3.8.10.6)

+ p+ k [e , x1 , · · · , xk−1 ] = 0 for k = 1, 2, + + p1 (e ) = P D([L]), + p+ 2 (e , x) = x.

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Here C(L; Λ0,nov )+ = C(L; Λ0,nov ) ⊕ Λ0,nov e+ ⊕ Λ0,nov f as in Section 3.3 and [L] is the fundamental cycle of L. The proof of Theorem 3.8.9 will be given in Subsection 3.8.3 and Section 7.4. (To show (3.8.10.4)-(3.8.10.6) we also need to use the argument in Section 7.3.) As for the orientation problem related to (3.8.10.3), see Proposition 8.10.6 in Subsection 8.10.2. 3.8.2. Applications to vanishing of the obstruction classes ok (L). In this subsection, we discuss some applications of Theorem 3.8.9. Let [GW0,1 (M )(L)] ∈ H ∗ (M ; Q) ⊗ Λ0,nov be the cohomology class of GW0,1 (M )(L). In the next theorem P DM denotes the Poincar´e duality P DM : H∗ (M ; Q) → H 2n−∗ (M ; Q). Theorem 3.8.11. Let L ⊂ M be a relatively spin Lagrangian submanifold. (3.8.12.1) (3.8.12.2) the Gysin (3.8.12.3) (3.8.12.4)

 If [GW0,1 (M )(L)] = 0, then M(L) = ∅. If [GW0,1 (M )(L)] = 0, then the obstruction [oi (L)] lies in the kernel of homomorphism i! : H 2−μi (L; Q) → H n+2−μi (M ; Q). weak (L) = ∅. If [GW0,1 (M )(L)] is not of the form c P DM [L] then M If [GW0,1 (M )(L)] = c P DM [L] then oi (L; weak) ∈ Ker(i! : H 2−μi (L; Q) → H n+2−μi (M ; Q)).

weak (L), and [L] = 0 in (3.8.12.5) Suppose [GW0,1 (M )(L)] = c P DM [L], b ∈ M H(M ; Q). Then, we have c = PO(b). Proof. We recall that G(L) is a monoid generated by G0 (L) = {(ω[β], μ(β)) | β ∈ Π(L), M(L; β) = ∅}. Let G ⊃ G(L) be a monoid satisfying Condition 3.1.6. We put G = {(λi , μi ) | i = 0, 1, 2 · · · } with λi ≤ λi+1 , and λi = λi+1 ⇒ μi < μi+1 . For simplicity of notation, we assume that λi < λi+1 for any i. Other cases can be handled after an obvious modification. (See the proof of Theorem 3.6.18 in Subsection 3.6.2). Suppose that [oj (L)] = 0 for j = 1, · · · , i − 1. It means that we can choose b(i−1) =

i−1 

T λj e

μj 2

bj

j=1

such that m(eb(i−1) ) ≡ 0 mod T λi−1 Λ+ 0,nov . We remark that

 b(i−1) ) ≡ T λi oi (L) d(e

mod T λi Λ+ 0,nov .

We also remark that eb(i−1) is invariant under the cyc operation and dcyc (eb(i−1) ) = eb(i−1) ⊗ m(eb(i−1) ) ⊗ eb(i−1) .

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Now we calculate using (3.8.10.2) and (3.8.10.3) to obtain p(dcyc (eb(i−1) )) = −δM (p(eb(i−1) )) − GW0,1 (M )(L). Using (3.8.10.1), we have (3.8.13.1)

T λi i! (oi (L)) ≡ −δM (p(eb(i−1) )) − GW0,1 (M )(L) mod T λi Λ+ 0,nov .

Since GW0,1 (M )(L) defines a Λ0,nov -coefficient cocycle in M , (3.8.13.1) gives us an equality on cohomology classes: (3.8.13.2)

T λi i! ([oi (L)]) ≡ −[GW0,1 (M )(L)] mod T λi Λ+ 0,nov .

Hence if [oi (L)] = 0 for all i then [GW0,1 (M )(L)] ≡ 0 mod T λi Λ+ 0,nov for all i. Therefore we have [GW0,1 (M )(L)] = 0. On the other hand, if [GW0,1 (M )(L)] = 0, it follows from (3.8.13.2) that i! ([oi (L)]) = 0. The statement about [oi (L; weak)] can be proved in a similar way using Formulae (3.8.10.4) and (3.8.10.5). The proof of Theorem 3.8.11 is now complete.  Theorem 3.8.11 yields several concrete corollaries. Let us start with the following lemma. ˜ Lemma 3.8.14. If [GW0,1 (M ; β)(L)] ∈ H ∗ (M ; Q) with ∗ = 2n or 2n − 1, then ˜ = 0. we have [GW0,1 (M ; β)(L)] ˜ + 2 , since dim M0,2 (M ; β) ˜ = Proof. We have degree ∗ = n − 2c1 (M )[β] ˜ − 2. Therefore the virtual dimension of M0,1 (M ; β), ˜ which is 2n + 2c1 (M )[β] ˜ ˜ similarly defined as M0,2 (M ; β), is 2n − 4 + 2c1 (M )[β] = 3n − ∗ − 2. Thus, by the assumption, ˜ = n − 2 or n − 1. dim M0,1 (M ; β) ˜ ev ×M L is empty. We may take the perturHence the fiber product M0,1 (M ; β) 0 ˜ bation of M0,2 (M ; β) compatible with the forgetful map and obtain an evaluation ˜ → M0,1 (M ; β) ˜ and ev0 is compatible with this evaluation map. map M0,2 (M ; β) ˜ = 0. (See Section 23 [FuOn99II].) Therefore we can conclude [GW0,1 (M ; β)(L)] Remark 3.8.15. The argument here is simpler than a similar argument in Section 7.3 (construction of homotopy unit). This is because when we deal with Gromov-Witten invariant, we can work in the cohomology level and do not need to work in the chain level. Corollary 3.8.16. If [oi (L)] is defined and in H n (L; Q), then [oi (L)] = 0. If [oi (L)] is in H n−1 (L), then i! ([oi (L)]) = 0. The same applies to [oi (L; weak)]. Proof. We remark that i! : H n (L) → H 2n (M ) is injective. The result then follows from Theorem 3.8.11 and Lemma 3.8.14.  Corollary 3.8.17. Suppose that dimR M = 6 and c1 (M ) = 0, and let L be a relatively spin Lagrangian submanifold. If i∗ : H∗ (L; Q) → H∗ (M ; Q) is injective, L is unobstructed. The proof is immediate from Theorem 3.8.11 and Lemma 3.8.14. The next corollary also follows from Corollary 3.8.16 and Theorem 3.6.43.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

161

Corollary 3.8.18. Any relatively spin Lagrangian submanifold which is a rational homology sphere is weakly unobstructed. We remark that Theorem 3.8.11 also gives some new information on the relation of Gromov-Witten invariant and homology class of L. For example, we have: Corollary 3.8.19. Let M be a symplectic manifold with c1 (M ) = 0 and L a relatively spin Lagrangian submanifold with vanishing Maslov class. We assume that H 2 (L; Q) = 0. Then [GW0,1 (M )(L)] = 0. The proof is obvious from Theorem 3.8.11. This, for example, implies in case n = 4 that a certain middle dimensional homology class ∈ H4 (M ; Z) can not be realized by a relatively spin Lagrangian rational homology S 4 in 8 dimensional symplectic manifold M with c1 (M ) = 0. To obtain more explicit result we need to calculate Gromov-Witten invariant. Remark 3.8.20. In the preprint 2000 version [FOOO00] of this book, we asserted that L is unobstructed if H∗ (L; Q) → H∗ (M ; Q) is injective. This statement does not seem correct. We overlooked the term GW0,1 (M )(L) which appears in the right hand side of (3.8.10.3). In the next section we will explain why such term appears. However, this trouble is partially overcome by using operators q and p,k which we introduce in Subsections 3.8.4 and 3.8.9 below. (See Corollary 3.8.43, for example.) In particular, ‘Theorem B’ in [FOOO00] (that is Theorem I in this book) is proved as was stated in [FOOO00]. (See Subsection 6.5.2.) In Chapter 6, we will also use operators p to prove that the image of the differential of the spectral sequence in Theorem D lies in the kernel of i! : H ∗ (L; Q) → H ∗+n (M ; Q). 3.8.3 Outline of the construction of the operator p. The main idea of the construction of p is to use the space Mmain k,1 (L; β). Here β ∈ π2 (M, L), (main (L; β) be the set of k = 0, 1, 2, · · · , and Mmain (L; β) is defined as follows. Let M k,1

k,1

(w, z , z + ) = ((Σ, z, z + ), w) where (Σ, z, z + ) ∈ Mmain z = (z1 , · · · , zk ), zi ∈ ∂Σ, z + k,1 ,  is an interior marked point and w : Σ → M is pseudo-holomorphic with w(∂Σ) ⊂ L, [w] = β. (Σ is a disc D2 plus possibly sphere bubbles.) We divide it by a P SL(2; R) action and compactify the quotient by adding stable maps to obtain Mmain k,1 (L; β). See Definition 2.1.24. Mmain (L; β) needs a special attention in its compactification 0,1 as we will explain later in this subsection. Proposition 3.8.21. If k ≥ 1 then Mmain k,1 (L; β) has a Kuranishi structure with corners. We remark that there exists an action of Zk on Mmain k,1 (L; β) given by the cyclic change of the marked points. Namely cyc : (w, (z1 , · · · , zk ), z + ) → (w, (zk , z1 , · · · , zk−1 ), z + ). Proposition 3.8.22. The action cyc preserves the Kuranishi structure. It is orientation preserving (in the sense of Kuranishi structure) if and only if k is odd. The proofs of these two propositions are closely tied to the transversality question. We postpone them till Section 7.4. Constructing a virtual fundamental chain by perturbing the Kuranishi map, while keeping the cyclic symmetry, is a nontrivial matter.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Let k ev = (ev∂ , ev0 ) = (ev1 , · · · , evk , ev0 ) : Mmain k,1 (L; β) → L × M

be the evaluation map defined by ev(w, (z1 , · · · , zk ), z + ) = (w(z1 ), · · · , w(zk ), w(z + )). Let Pi = (|Pi |, fi ) be a singular chain representing one of the generators of C(L). We put main Mmain k,1 (L; β; P1 , · · · , Pk ) = Mk,1 (L; β)

ev∂

×(f1 ×···×fk ) (P1 × · · · × Pk ).

Then Mk,1 (β; P1 , · · · , Pk ) has a Kuranishi structure and we take an appropriate multisection s to define its virtual fundamental chain. We will carry out the detail of this construction at Section 7.4. We then define the map p as follows. Definition 3.8.23. We put

+ , pk,β (P1 , · · · , Pk ) = ev0∗ Mmain k,1 (L; β; P1 , · · · , Pk ) ,

and define (3.8.24)

pk =



T ω[β] e

μ(β) 2

pk,β .

β

The convergence of the right hand side of (3.8.24) follows from Gromov’s compactness in the usual way. Lemma 3.8.25. We may choose a multisection s so that the following cyclic property of pk,β holds: pk,β (P1 , · · · , Pk ) = (−1)deg



Pk ×(

k−1 i=1

deg Pi )

pk,β (Pk , P1 , · · · , Pk−1 ).

In this book we do not prove a similar cyclic symmetry for our operator mk (see Remark 3.8.8). The proof of Lemma 3.8.25 will be given at Section 7.4 with the relevant orientation problem analyzed in Subsection 8.10.2. Now we prove (3.8.10). The proof of (3.8.10.1) is easy. In fact the moduli space Mmain 1,1 (L; β0 ) (where β0 = 0) is transverse and identified with L itself. Therefore we do not need to perturb it and so p1,β0 (P ) = P , which implies (3.8.10.1). We next prove (3.8.10.2). To see the term δM ◦ pk , we consider the bound(m) (m) (m) ary of Mmain , z (m) , z +(m) ), (p1 , · · · , pk ), z +(m) ) is k,1 (L; β; P1 , · · · , Pk ). If ((w a diverging sequence, then one of the following occurs: (m)

(3.8.26.1) Some pi goes to the boundary of Pi . (3.8.26.2) w(m) : Σ → M goes to a map from union of two genus zero bordered Riemann surfaces w(∞) : Σ1 ∪ Σ2 → M glued at boundary. We remark that there may be, of course, a sphere bubble. But this does not contribute to the boundary since sphere bubble occurs in codimension two. (When we are studying stable maps, sphere bubble always occurs in real codimension two). (3.8.26.1) corresponds to pk,β (P1 , · · · ,∂Pi , · · · , Pk ), where ∂ is the classical boundary operator on C(L), which satisfies m1,0 = (−1)n ∂. (See Definition 3.5.6.)

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3.8. INSERTING MARKED POINTS IN THE INTERIOR (∞)

163

(∞)

Let us study (3.8.26.2). Let ((p1 , · · · , pk ), z +(∞) ) be the limit points of marked points. We may number Σ1 and Σ2 so that z +(∞) ∈ Σ2 . There are several cases. (∞)

(∞)

(3.8.26.2.1) There exists a ≤ b such that pa , · · · , pb are on ∂Σ1 and other (∞) pi are on ∂Σ2 . (∞) (∞) (∞) (∞) (3.8.26.2.2) There exists a ≤ b such that pb+1 , · · · , pk , p1 , · · · , pa−1 are on (∞)

are on ∂Σ2 . ∂Σ1 and other pi (∞) (3.8.26.2.3) The case there are no pi on ∂Σ1 . See Figures 3.8.1 - 3.8.3. Then (3.8.26.2.1) (Figure 3.8.1) corresponds to p(P1 , · · · , m(Pa , · · · , Pb ), · · · , Pk ), (3.8.26.2.2) (Figure 3.8.2) corresponds to p(Pa , · · · , Pb , m(Pb+1 , · · · , Pk , P1 , · · · , Pa−1 )), and (3.8.26.2.3) (Figure 3.8.3) corresponds to p(P1 , · · · , Pi , m0 (1), Pi+1 , · · · , Pk ). Taking the cyclic symmetry (Lemma 3.8.25) into account, (3.8.10.2) follows from the argument similar to the proof of Theorem 3.5.11. 

pa





pa−1

pa+ 1



p1 +( )

z



pb−1

pk

  pb+1

pb 31

32

Figure 3.8.1 

pb +1 p b



pk p 1

+( )

z pa −1 p a 31 Figure 3.8.2

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32

164

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

p i

+( )

z

p1 pk

 pi+1

32

31

Figure 3.8.3 Now we prove (3.8.10.3). For this purpose, we need to point out the following: Consider the exact sequence i

∂

∗ H2 (M ) −→ H2 (M, L) −→ H1 (L).

We need to include unstable maps to compactify M0,1 (L; β) when ∂β = 0 ∈ H1 (L). ˜ for some β˜ ∈ H2 (M ). There could Suppose ∂β = 0. Then we have β = i∗ (β) ˜ ˜ We consider the element (w, (z + , z + )) ∈ M0,2 (M ; β) be more than one such β. 0

1

such that w(z0+ ) ∈ L. (We like to note that the reader should not be confused ˜ with M0,2 (L; β). The suffix 0, 2 in M0,2 (M ; β) ˜ with the notation M0,2 (M ; β) denotes the genus (= 0) of the domain (closed Riemann surface) and the number of (interior) marked points (= 2). The suffix 0, 2 in M0,2 (L; β) is the number of boundary marked points (= 0) and the number of interior marked points (= 2).) We consider the map constw(z+ ) : D2 → M such that constw(z+ ) (x) ≡ w(z0+ ). 0 0 We glue D2 with S 2 at 0 ∈ D2 and z0+ ∈ S 2 to obtain Σ. Then we consider ((Σ, z1+ ), w# constw(z+ ) ). This is not a stable map in the standard sense, since 0 Aut(D2 , 0) ∼ = U (1) and so Aut(D2 , 0) is not finite. However, this object appears as a limit of the element of M0,1 (L; β). (See the proof of Proposition 3.8.27 given later in Section 7.4.) We remark that such ((Σ, z1+ ), w# constw(z+ ) ) corresponds 0 ˜ ×M L. Summing up we have the following: one to one to M0,2 (M ; β) Proposition 3.8.27. We consider  M0,1 (L; β) ∪

˜ ×M L). (M0,2 (M ; β)

˜ β˜ ; i∗ (β)=β

We can define a topology on it so that it is compact and Hausdorff. It then has ˜ ×M L lies in its an oriented Kuranishi structure with boundary. Here M0,2 (M ; β) ˜ ×M L is compatible with the boundary and the Kuranishi structure of M0,2 (M ; β)  ˜ ×M L). Kuranishi structure of M0,1 (L; β) ∪ β˜ (M0,2 (M ; β) We will prove this proposition in Subsection 7.4.1. The essential point is that ˜ ×M L) occur in codimension one sphere bubbles (which corresponds to M0,2 (M ; β) in this case. This is because the constant disc with the sphere bubble in question has automorphism group of dimension 1.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

165

From now on we write M0,1 (L; β) ∪ (M0,2 (M ; (i∗ )−1 (β)) ×M L) in place of  ˜ ×M L) to simplify the notation. We use this moduli M0,1 (L; β) ∪ β˜(M0,2 (M ; β) space to define p0,β in the same way as Definition 3.8.23. Now we are ready to prove of (3.8.10.3). We note that the boundary of M0,1 (L; β) ∪ (M0,2 (M ; (i∗ )−1 (β)) ×M L) is the union of the fiber product M0,2 (M ; (i∗ )−1 (β)) ×M L and  M1 (L; β1 ) ×L M1,1 (L; β2 ). β1 +β2 =β

Thus we have δM (p0,β (1)) +

 β1 +β2 =β

p1,β2 (m0,β1 (1)) +



˜ ×M L) = 0. ev0∗ (M0,2 (M ; β)

˜ ˜ ∗ (β)=β β;i

(3.8.10.3) follows. The proofs of (3.8.10.4), (3.8.10.5) are combination of the arguments in Sections 7.2 and 7.3.  3.8.4. The operator q. We have constructed the operator p by moving the last (0-th) boundary marked point into the interior. We will construct operator q by moving 1-st, . . . , k-th boundary marked points into the interior. This construction enables us to relax the condition for Floer cohomology to be defined as we will discuss in the next subsection. For example, this construction is used to prove Theorem I stated in the introduction. We remark that [Ono96, Sei02, OzSa04] also use the idea of cutting down the moduli space of pseudo-holomorphic discs via the cycle of the ambient symplectic manifold. To precisely state our main technical result Theorem 3.8.32, we need to prepare some notations. For the rest of this subsection, we put R = Q. Take a countable set X1 (L) of smooth singular simplices on L as in Section 3.5 and let C1 (L; Λ0,nov ) be the cochain complex generated by X1 (L). Our filtered A∞ structure m in Section 3.5 is constructed on this complex. Let S(M ) be the smooth singular chain complex of M , which we regard as a cochain complex. We choose a countable set X (M ) of smooth singular simplices on M . Let C(M ) be the Q submodule generated by X (M ) in S(M ). We put  0,nov . C(M ; Λ0,nov ) = C(M )⊗Λ Assumption 3.8.28. (3.8.29.1) C(M ) is a cochain subcomplex of S(M ). (3.8.29.2) The natural inclusion C(M ) → S(M ) induces an isomorphism of cohomology H(C(M ), δM ) ∼ = H(M ). (3.8.29.3) Each element Q ∈ X (M ) is transversal to L and L ∩ Q has a smooth triangulation. It defines an element of X1 (L). (3.8.29.4) Moreover the triangulation of ∂i Q ∩ L coincides with the restriction of the triangulation of Q ∩ L. Here ∂i Q is the i-th face of Q. We define i : C(M ) → C(L) by i (Q) = L ∩ Q. Here we fix a triangulation of L ∩ Q so that (3.8.29.3) holds. Then L ∩ Q defines an element of C(L). Then (3.8.29.4) implies that i is a cochain homomorphism. Clearly it induces a natural homomorphism H ∗ (M ) → H ∗ (L) (induced by the inclusion).

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166

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

We shift the degree of elements of C(M ; Λ0,nov ) by 2 and consider the shifted complex C(M ; Λ0,nov )[2]. We also consider its associated bar complex BC(M ; Λ0,nov )[2]. We define an action of the symmetric group Sk on Bk C(M ; Λ0,nov )[2] as follows: We put σ(y1 ⊗ · · · ⊗ yk ) = (−1) (σ) yσ(1) ⊗ · · · ⊗ yσ(k)  for σ ∈ Sk , where (σ) = iσ(j) deg yi deg yj . (Here we note that the degree is shifted by 2 and so we do not need to use the shifted degrees.) Definition 3.8.30. Ek C(M ; Λ0,nov )[2] is the subspace of BC(M ; Λ0,nov )[2] consisting of elements which are invariant of the Sk action. We put /  EC(M ; Λ0,nov )[2] = Ek C(M ; Λ0,nov )[2].  ; Λ0,nov )[2]; The coproduct Δ on BC(M ; Λ0,nov )[2] induces one on EC(M (3.8.31)

    EC(M ; Λ0,nov )[2] ⊗ ; Λ0,nov )[2]. Δ : EC(M ; Λ0,nov )[2] → EC(M

 (Namely the restriction of Δ to EC(M ; Λ0,nov )[2] is contained in the right hand side of (3.8.31).) The coboundary operator δM on C(M ; Λ0,nov ) induces a coderivation  δM on EC(M ; Λ0,nov )[2] by δM (y1 ⊗ · · · ⊗ yk ) =

k 

(−1)deg y1 +···+deg yi−1 y1 ⊗ · · · ⊗ δM yi ⊗ · · · ⊗ yk .

i=1

(See Section A3.) Let y ∈ EC(M )[2] and x ∈ BC(L)[1]. We put  y(k;1) ⊗ · · · ⊗ y(k;k) , Δk−1 y = c1 c1 c1

k−1

Δ

x=



x(k;1) ⊗ · · · ⊗ x(k;k) . c2 c2

c2

We use this notation throughout this section. Now the following is the main result of this subsection. Theorem 3.8.32. There exists a countable set of smooth singular simplices X1+ (L) containing X1 (L), and a structure of a filtered A∞ algebra on C1+(L; Λ0,nov ) generated by X1+ (L) such that (C1+ (L; Λ0,nov ), m) satisfies the following properties: (1) The filtered A∞ algebra structure on C1+ (L; Λ0,nov ) extends the filtered A∞ algebra structure on C1 (L;Λ0,nov ). (2) There exists a sequence of operators q,k : E C(M ; Λ0,nov )[2] ⊗ Bk C1+ (L; Λ0,nov )[1] → C1+ (L; Λ0,nov )[1] (k = 0, 1, 2, · · · ,  = 0, 1, 2, · · · ) of degree +1 satisfying the following relations  (2;2)  (3;1)  (3;1) (2;1) (−1)deg yc1 deg xc2 +deg xc2 +deg yc1 q,k (δM (y) ⊗ x) + c1 ,c2 (3.8.33) (3;3) q(y(2;1) ⊗ (x(3;1) ⊗ q(y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 )) = 0.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

If  = 0, then we have



q,k ≡

(3.8.34.1)

i mod Λ+ 0,nov 0 mod

Λ+ 0,nov

167

 = 1 and k = 0, otherwise.

For  = 0, we have (3.8.34.2)

q0,k = mk

on Bk C1+ (L; Λ0,nov )[1]. (3) Furthermore, we can extend q,k to + + q+ ,k : E C(M ; Λ0,nov )[2] ⊗ Bk C1+ (L; Λ0,nov ) [1] −→ C1+ (L; Λ0,nov ) [1]

which satisfies the same formula as (3.8.33) after replacing q by q+ . It also satisfies the following: + q+ ,k (y ⊗ (x1 ⊗ e ⊗ x2 )) = 0,

(3.8.35.1)

≥1

+ q+ 0,k = mk .

(3.8.35.2)

˜ = P D[M ] (the Poincar´e dual to (4) Moreover we have the following. We put e the fundamental class), and e+ ]. C(M ; Λ0,nov )+ = C(M ; Λ0,nov ) ⊕ Λ0,nov [˜f] ⊕ Λ0,nov [˜ ˜+ = 0. We can then further extend q+ Here deg ˜f = −1 and deg e ,k to + + + q++ ,k : E C(M ; Λ0,nov ) [2] ⊗ Bk C1++ (L; Λ0,nov ) [1] −→ C1++ (L; Λ0,nov ) [1]

which satisfies the same formula as (3.8.33) after replacing q with q++ and (3.8.35) after replacing q+ with q++ . It also satisfies ˜+ , x) = 0 q++ +1,k (y ⊗ e

(3.8.36.1) for (, k) = (0, 0) and

(3.8.36.3)

e+ , 1) = e+ , q++ 1,0 (˜ δM (˜f) = e+ − e,

(3.8.36.4)

+ ˜ q++ 1,0 (f, 1) ≡ f mod Λ0,nov .

(3.8.36.2)

We will sketch the proof of Theorem 3.8.32 in Subsection 3.8.6. The detail of the proof is given in Section 7.4 and Subsection 7.3.3. Note q+ ,k is obtained from an extension of h (in Definition 3.3.3) to h+M : EC(M ; Λ0,nov )[2] ⊗ Bk (BC1+ (L; Λ0,nov )[1]) −→ C1+ (L; Λ0,nov )[1] k by (3.8.35.3)

q+ (y ⊗ (x1 ⊗ f ⊗ x2 ⊗ f ⊗ · · · ⊗ f ⊗ xk )) = h+M (y, x1 ⊗ · · · ⊗xk ) k

unless  = 0, k = 2 and x1 , x2 ∈ Λ0,nov . (In the case when  = 0, k = 2 and x1 , x2 ∈ Λ0,nov , we have (3.8.35.4)

+ +M + (1, 1⊗1), q+ 0,1 (1 ⊗ f) = m1 (f) = e − e + h2

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

which is consistent with (3.8.35.2).) Here C1+ (L; Λ0,nov )+ [1], e+ and f as in Section 3.3. (3.8.33) is a rather complicated formula. We introduce several algebraic structures (L∞ structures) in Section 7.4 and use them to rewrite Theorem 3.8.32. It is Theorem Y in the introduction and Theorems 7.4.102. A related algebraic discussion can also be found in [KaSt05] for example. 3.8.5. Bulk deformation of the filtered A∞ structures. We can use Theorem 3.8.32 to define a deformation of the A∞ structure m on C1+ (L; Λ0,nov ). Elements of H 2 (M ; Λ+ 0,nov ) induces an extended deformation of symplectic structure, and this deformation is the one used for the deformation defined in this subsection. In this sense we can call this deformation the bulk deformation. + + 0 Let b ∈ C 2 (M ; Λ+ 0,nov ) = C(M ; Λ0,nov )[2] . We assume that b ≡ 0 mod Λ0,nov . We put k times

∞

   e = b ⊗ · · · ⊗ b ∈ EC(M ; Λ0,nov )[2]. b

k=0

We remark (see Lemma 3.6.2) Δ(eb ) = eb ⊗ eb .

(3.8.37) We remark that

eb =

∞  1 ⊗k b k! k=1

even though we symmetrized Bk C[2] to obtain Ek C[2]. This is justified by (3.8.37). In fact if we choose different convention to define Δ (coproduct) on EC[2], then we will need the coefficient 1/k! in the definition of eb . Actually, instead, we put 1/! in (3.8.68) in the definition of q. + Definition 3.8.38. For b ∈ C 2 (M ; Λ+ 0,nov ) with b ≡ 0 mod Λ0,nov and b ∈ + 1 C1+ (L; Λ0,nov ), we define b b b b b mb,b k (x1 , · · · , xk ) = q(e ⊗ e x1 e · · · e xk e ),

k = 0, 1, 2, · · · .

Similarly we define (x1 , · · · , xk ) = q+ (eb ⊗ eb x1 eb · · · eb xk eb ), m+b,b k

k = 0, 1, 2, · · · .

Lemma 3.8.39. Suppose δM b = 0. Then mb,b k defines a structure of a filtered b b A∞ algebra on C1+ (L; Λ0,nov ). Moreover, mb,b 0 = 0 if and only if q(e ⊗ e ) = 0. +b,b Similarly mk defines a filtered A∞ algebra with unit on C1+ (L; Λ0,nov )+ and +b,b m0 = 0 is equivalent to q+ (eb ⊗ eb ) = 0. Proof. By assumption δM eb = 0. Combining (3.8.33) and (3.8.34) we have    (−1)deg x1 +··· deg xi q(eb ⊗ eb x1 eb · · · eb xi eb 0= 0≤i≤j≤k

q(eb ⊗ eb xi+1 eb · · · eb xj eb )eb xj+1 eb · · · eb xk eb ), +b,b which turns out to be the A∞ relation of mb,b is similar.  k . The version for mk

Now we study the obstruction for the equation mb,b 0 = 0 having a solution.

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Definition 3.8.40. (1) We say L is unobstructed after bulk deformation if b,b  mb,b 0 = 0 has a solution. We write Mdef (L) the set of all (b, b) such that m0 = 0. For a fixed b with δM b = 0 we put b (L) = {b ∈ C 1 (L; Λ+ ) | (b, b) ∈ M def (L)}. M 0,nov (2) We say L is weakly unobstructed after bulk deformation if (1) = cee+ m+b,b 0 (0)

with c ∈ Λ0,nov (the degree zero part) has a solution. Here m+ defines the filtered A∞ structure on C1+ (L; Λ0,nov )+ and e+ is its unit. We write weak,def (L) = {(b, b) | m+b,b = cee+ }. M 0 b (L) in a similar way. We define M weak (3) Furthermore we define the potential function weak,def (L) −→ Λ(0) PO : M 0,nov by the equation (1) = PO(b, b)ee+ . m+b,b 0 An analog of Lemma 3.6.36 will be given later. See Theorem 4.6.52 and Corollary 4.6.53 for the precise statement. b (L) is the set of bounding cochains of the filtered A∞ We remark that M b (L) = M((C(L;  algebra (C(L; Λ0,nov ), m(b,0) ). Namely M Λ0,nov ), m(b,0) )). Now we have the following analog to Theorem 3.6.18. We recall G(L) is a monoid generated by G0 (L) = {(ω[β], μL (β)) | β ∈ Π(L), M(L; β) = ∅}. Let G ⊃ G(L) be a monoid satisfying Condition 3.1.6. We enumerate as G = {(λi , μi ) | i = 0, 1, 2 · · · } with λi ≤ λi+1 , and λi = λi+1 ⇒ μi < μi+1 . Theorem 3.8.41. There exists a sequence of obstruction classes [ok (L; def)] ∈

H 2−μk (L; Q) , Im(i∗ : H 2−μk (M ; Q) → H 2−μk (L; Q))

k = 1, 2, · · ·

and bk ∈ C 1−μk (L; Q), bk ∈ C 2−μk (M ; Q), (δM bk = 0 and μk is the Maslov index which is even), with the following properties: (3.8.42.1) The cocycle ok (L; def) is defined if bj , bj and oj for j with λj  λk are defined. [ok (L; def)] depends on bj , bj and oj for j with λj  λk . (3.8.42.2) bk and bk are defined if [ok (L; def)] = 0. (3.8.42.3) L is unobstructed after bulk deformation if all the obstruction classes [ok (L; def)] are defined and are zero. The next corollary follows immediately from Theorem 3.8.41. Corollary 3.8.43. If i∗ : H 2k (M ; Q) → H 2k (L; Q) is surjective for all k, then L is unobstructed after bulk deformation.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Proof of Theorem 3.8.41. We will prove the theorem by induction over k. As in the proof of Theorem 3.8.11, we assume that λi < λi+1 for any i to simplify the notation. Other cases can be handled after an obvious modification. (See §3.6.2). We now state the induction hypothesis precisely. Put b(i) =

i 

T λj e

μj 2

bj ,

b(i) =

j=1

i 

T λj e

μj 2

bj

j=1

and assume that bi , bi for i < k satisfy δM bi = 0 and q(eb(i) ⊗ eb(i) ) ≡ 0 mod T λi Λ+ 0,nov

(3.8.44)

for i < k.

Under this assumption, we will construct a cochain ok (L; def) which is indeed a cocycle. Moreover, if the cohomology class [ok (L; def)] vanishes, we will prove that bk and bk are defined and satisfy q(eb(k) ⊗ eb(k) ) ≡ 0 mod T λk Λ+ 0,nov . We now define ok (L; def) by the equation T λk e

(3.8.45)

μk 2

ok (L; def) ≡ q(eb(k−1) ⊗ eb(k−1) )

mod T λk Λ+ 0,nov .

Lemma 3.8.46. ok (L; def) is a cocycle in C(L; Q). Proof. We apply (3.8.33) to eb(k−1) ⊗ eb(k−1) . Then we have q(eb(k−1) ⊗ eb(k−1) q(eb(k−1) ⊗ eb(k−1) )eb(k−1) ) = 0. By (3.8.45) we have T λk e

μk 2

q(eb(k−1) ⊗ eb(k−1) ok (L; def)eb(k−1) ) ≡ 0 mod T λk Λ+ 0,nov .

Using (3.8.34.2) and b(k) ≡ b(k) ≡ 0 mod Λ+ 0,nov , we have m1 (ok (L; def)) = 0 as required.  We define [ok (L; def)] the class induced by ok (L; def) in H 2−μk (L; Q) . Im(i∗ : H 2−μk (M ; Q) → H 2−μk (L; Q)) Now suppose [ok (L; def)] = 0. It means that there exist bk ∈ C21−μk (L; Q) and bk ∈ C 2−μk (M ; Q) such that i bk + m1 (bk ) + ok (L; def) = 0,

(3.8.47)

b(k)

We calculate q(e (3.8.48)

⊗e

b(k)

). We find

q(eb(k) ⊗ eb(k) ) − q(eb(k−1) ⊗ eb(k−1) ) ≡ T λk e

μk 2

(q1,0 (bk ⊗ 1) + q0,1 (1 ⊗ bk )) mod T λk Λ+ 0,nov .

The right hand side is equal to T λk e and (3.8.34.2). Hence (3.8.49)

δM (bk ) = 0.

μk 2

(i bk + m1 bk ) mod T λk Λ+ 0,nov by (3.8.34.1)

q(eb(k) ⊗ eb(k) ) ≡ 0 mod T λk Λ+ 0,nov

as required. Thus we have proved (3.8.42.1) and (3.8.42.2). Then the proof of (3.8.42.3) is now similar to one of (3.6.19.4) in Theorem 3.6.18. 

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

171

Next we generalize Theorem 3.8.41 to the version of the weakly unobstructed case after bulk deformation. This can be regarded as an analog of Theorem 3.6.43. As before, the monoid G ⊃ G(L) is enumerated as G = {(λi , μi ) | i = 0, 1, 2 · · · } with λi ≤ λi+1 , and λi = λi+1 ⇒ μi < μi+1 . Then as in Subsection 3.6.3, we put Gμ≤0 = {(λi , μi ) ∈ G | μi ≤ 0}. By omitting i’s with μi > 0 from the given enumeration of G, we find the nondecreasing function j : Z≥0 → Z≥0 with j(0) = 0 such that Gμ≤0 = {(λj(i) , μj(i) ) | i = 0, 1, 2, · · · }. See (3.6.42). Obviously Assumption 3.6.39 is satisfied in our geometric situation. We note that the proof of Theorem 3.6.43 can be immediately reduced to one given below by ignoring the bulk deformation. Theorem 3.8.50. There exists a sequence of obstruction classes [ok (L; weak,def)] ∈

Im(i∗

:

H 2−μk (L; Q) , → H 2−μk (L; Q))

H 2−μk (M ; Q)

k = 1, 2, · · ·

and bk ∈ C21−μk (L; Q), bk ∈ C 2−μk (M ; Q), (δM bk = 0 and μk is the Maslov index which is even), with the following properties. (3.8.51.1) [ok (L; weak,def)] is defined if oi (L; weak,def) and bi , bi for i < k are defined and μk ≤ 0. [ok (L; weak,def)] depends on oi (L; weak,def) and bi , bi for i < k. (3.8.51.2) When μk ≤ 0, bk and bk are defined if [ok (L; weak,def)] is defined and is zero. If μk > 0, bk and bk are defined if bi and bi exist for i < k. (3.8.51.3) L is weakly unobstructed after bulk deformation if all the obstruction classes [ok (L; weak,def)] for μk ≤ 0 are defined and are zero. Proof. The proof goes in a way similar to that of Theorem 3.8.41. As in the proof of Theorem 3.8.41, we assume that λi < λi+1 for any i to simplify the notation and put b(i) =

i 

T λj e

μj 2

bj ,

b(i) =

j=1

i 

T λj e

μj 2

bj .

j=1

If  ∈ Im(j) for any  ≤ k, then the argument for [ok (L; weak,def)] and bk , bk is exactly the same as in Theorem 3.8.41 and the induction goes to the next step. The main problem to be discussed is the following case. The other cases can be handled similarly. We assume that  ∈ Im(j) for any  < k and k ∈ / Im(j). Hence we have μk ≥ 2 because μk is an even integer. Recall that e = P D[L] is the homotopy unit of our filtered A∞ algebra and consider C1+ (L; Λ0,nov )+ = C1+ (L; Λ0,nov ) ⊕ Λ0,nov e+ ⊕ Λ0,nov f + with deg e+ = 0 and deg f = −1 as in Section 3.3. Note that since m+ 1 (f) ≡ e − e + + + ∼ mod Λ0,nov from (3.3.5.2), we have H(L; Q) = H(C(L; Q) , m1 ). Together with this, the operators q is also extended to q+ by Theorem 3.8.32. Under the situation above, we will prove the following: We assume that we have bi ∈ C1+ (L; Q)+ , bi ∈ C(M ; Q) for i < k satisfying δM bi = 0 and

q+ (eb(i) (weak) ⊗ eb(i) (weak) ) ≡ ci ee+

mod T λi Λ+ 0,nov

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for i < k

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD (0)

with ci ∈ Λ0,nov . Under this assumption we will construct obstruction cochains ok (L; weak,def) and also bk , bk such that ok (L; weak, def) is a cocycle and they satisfy q+ (eb(k) ⊗ eb(k) ) ≡ ck ee+ mod T λk Λ+ 0,nov (0)

for some ck ∈ Λ0,nov . We define ok (L; weak,def) by the equation T λk e

μk 2

ok (L; weak,def) ≡ q+ (eb(k−1) ⊗ eb(k−1) ) mod T λk Λ+ 0,nov

as before. Then we can similarly show the following. 2−μk Lemma 3.8.52. ok (L; weak,def) is an m+ (L; Q)+ . 1 -cocycle in C

Proof. If e = P D[L] is a strict unit, then we do not have to consider the extended ones like C(L; Λ0,nov )+ and q+ , etc. In this case the assertion can be shown by the same argument as Lemma 3.8.46. If P D[L] is a homotopy unit, Definition 3.8.37 and Lemma 3.8.38 imply that (C(L; Λ0,nov )+ , m+b ) is the filtered A∞ algebra with unit for any b ∈ C 2 (M ; Λ+ 0,nov ) with δM b = 0. Then we can prove the assertion in a way similar to the case of strict unit as before.  Firstly we consider the case that μk > 2. Since we assume that L is oriented, we have μk ≥ 4. Then ok (L; weak,def) is automatically cohomologous to zero by degree reason. Namely ok (L; weak,def) is a cycle of (homology) dimension > dim L. (Note that deg f = −1. Moreover even if μk = 3, we find that any element of degree + + −1 in C(L; Q)+ is not a cocycle over Q, because m+ 1 (f) ≡ e − e = 0 mod Λ0,nov from (3.3.5.2).) We put bk = 0 and can find bk such that δbk = ok (L; weak,def). Then obviously we have q+ (eb(k) ⊗ eb(k) ) ≡ ck ee+

mod T λk Λ+ 0,nov

with ck = ck−1 as required. If μk = 2, then we can write ok (L; weak,def) = ck,1 P D[L] + ck,2 e+ ∈ C 0 (L; Q)+ = C 0 (L; Q) ⊕ Qe+ for some ck,i ∈ Q. In this case of μk = 2, we do not assume that [ok (L; weak,def)] = 0 ∈

H 0 (L; Q) . Im(i∗ : H 0 (M ; Q) → H 0 (L; Q))

We define bk = ck,1 f, bk = 0.

(3.8.53.1) (3.8.53.2)

Noticing (3.3.5.2) and (3.8.35.4), the argument similar to that in the proof of Theorem 3.8.41 gives q+ (eb(k) ⊗ eb(k) ) − q+ (eb(k−1) ⊗ eb(k−1) ) ≡ q0,1 (1 ⊗ T λk e

μk 2

bk ) = ck,1 (e+ − e)T λk e

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μk 2

mod T λk Λ+ 0,nov ,

3.8. INSERTING MARKED POINTS IN THE INTERIOR

173

because h+M (1, 1⊗1) ≡ 0 mod Λ+ 2 0,nov . Therefore we obtain q+ (eb(k) ⊗ eb(k) ) ≡ ok (L; weak, def)T λk e

μk 2

+ ck,1 (e+ − e)T λk e

μk 2

≡ (ck,1 + ck,2 )T λk ee+ = ck ee+ mod T λk Λ+ 0,nov with ck = (ck,1 + ck,2 )T λk as required. Therefore the induction works. The other part is similar. In fact, if we put b = lim b(k) , k→∞

b = lim b(k) , k→∞

c = lim ck , k→∞

then they converge in the completion with respect to the energy filtration and they = cee+ . This finishes the proof of Theorem 3.8.50.  satisfy the equation m+b,b 0 We next use (3.8.36) to prove Proposition 3.8.54 below. Here we write q and C(L; Λ0,nov )+ in place of q++ and C++ (L; Λ0,nov )+ respectively. + Proposition 3.8.54. Let b ∈ C 2 (L; Λ+ and c ∈ Λ+ 0,nov ) 0,nov with deg c = 0.  e+ . Then we have the following. We put b = b − ce˜

(3.8.55.1) (3.8.55.2) (3.8.55.3)

b (L). b (L) = M M weak weak  If b is an element of the above set, then mkb ,b = mb,b k for k = 0. b ,b b,b + m0 (1) = m0 (1) − cee . In particular PO(b , b) = PO(b, b) − ce.

Proof. Let x ∈ Bk C(L; Λ0 ) with k = 0. Using (3.8.36.1), (3.8.36.2) we calculate (3.8.56.1)



˜+ , x) + · · · = q(eb , x). q(eb , x) = q(eb , x) − ceq(eb e

On the other hand, 

˜

e+ , 1) − ceq((eb − 1)˜ e+ , 1) q(eb , 1) = q(eb , 1) − ceq(˜ ˜

e+ )2 , 1) + · · · + c2 e2 q(eb (˜

(3.8.56.2)

= q(eb , 1) − cee+ . (3.8.55) follows from (3.8.56) easily.



b (L), there exists some b for which b ∈ Corollary 3.8.57. If b ∈ M weak b  M (L). Proof. We have only to choose b = b − PO(b, b)e˜ e+ .



Now we assume that L is unobstructed after bulk deformation. Let def (L). (b, b), (b, b ) ∈ M (We remark that the first factor b is the same while the second may be different. b (L).) We define a cochain complex CF ((L; (b, b)), (L; (b, b))) as Namely b, b ∈ M follows.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

Definition 3.8.58. (1) When L is unobstructed after bulk deformation, we put CF ((L; (b, b)), (L; (b, b))) = C(L; Λ0,nov ) as a graded module. Its differential δ(b,b),(b,b) is defined by (3.8.59.1)





(x) := q(eb , eb xeb ) δ(b,b),(b,b ) (x) = mb,b,b 1

def (L). for (b, b) ∈ M (2) When L is weakly unobstructed after bulk deformation, we put CF ((L; (b, b)), (L; (b, b))) = C(L; Λ0,nov ) + , (we will sometimes write as δ(b,b),(b,b) as in Section 3.3 and its differential δ(b,b),(b,b) for simplicity, when no confusion can occur), is defined by

(3.8.59.2)





+b,b,b + δ(b,b),(b,b (x) := q+ (eb , eb xeb )  ) (x) = m1

weak,def (L). We remark that the right hand side is contained in for (b, b) ∈ M C(L; Λ0,nov ) (⊂ C(L; Λ0,nov )+ ) by construction. Therefore we can apply Remark 3.6.34 (2). When b = b in both cases, we will simply write as +b,b + b b b (x) = q+ (eb , eb xeb ). δ(b,b) (x) = mb,b 1 (x) = q(e , e xe ), or δ(b,b) (x) = m1

Lemma 3.8.60. (1) If L is unobstructed after bulk deformation, then we have def (L). δ(b,b),(b,b ) ◦ δ(b,b),(b,b ) = 0 for (b, b), (b, b ) ∈ M + (2) If L is weakly unobstructed after bulk deformation, then we have δ(b,b),(b,b ) ◦ +   δ  = 0 for (b, b), (b, b ) ∈ Mweak,def (L), whenever (b,b),(b,b )

PO(b, b) = PO(b, b ).

(2).

Proof. Lemma 3.8.60 follows from Lemmas 3.8.39, 3.6.8 and Remark 3.6.34  Definition 3.8.61. (1) We define the Floer cohomology HF ((L; (b, b)), ((L; b, b )); Λ0,nov )

def (L) to be the cohomology group of the cochain deformed by (b, b), (b, b ) ∈ M complex: HF ((L; (b, b)), ((L; b, b )); Λ0,nov ) = H(C((L; (b, b)), ((L; b, b ))), δ(b,b),(b,b ) ). (2) For the case when L is weakly unobstructed after bulk deformation, we deweak,def (L) with PO(b, b) fine HF ((L; (b, b)), ((L; b, b )); Λ0,nov ) for (b, b),(b, b ) ∈ M = PO(b, b ) in a similar way. When b = b in both cases, we will simply write as HF (L; (b, b); Λ0,nov ). We will study how the Floer cohomology depends on (b, b) in the next chapter. The relation of it to mirror symmetry was mentioned in Section 1.4. We turn to another application of Theorem 3.8.32.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

175

def (L), there exists a homomorphism Theorem 3.8.62. For each (b, b) ∈ M i∗qm,(b,b) : H ∗ (M ; Λ0,nov ) −→ HF ((L; (b, b)), (L; (b, b)); Λ0,nov ). (Here H ∗ (M ; Λ0,nov ) is the usual cohomology group with Λ0,nov coefficient.) The homomorphism above is induced by a cochain homomorphism iqm,(b,b) : C(M ; Λ0,nov ) −→ C1+ (L; Λ0,nov ) such that iqm,(b,b) ≡ ±i

(3.8.63)

mod Λ+ 0,nov .

The same also holds for the case when L is weakly unobstructed after bulk deformation. Proof. Let x ∈ C(M ; Λ0,nov ). We define iqm,(b,b) (x) = (−1)deg x q(eb xeb ⊗ eb ). We plug in eb xeb ⊗ eb to (3.8.33) and take B1 C1+ (L) component of it. We then obtain 

(−1)deg x q(eb xeb ⊗ q(eb ⊗ eb )) + q(eb δM (x)eb ⊗ eb ) + q(eb ⊗ eb q(eb xeb ⊗ eb )eb ) = 0. def (L). Hence we have The first term is zero since (b, b) ∈ M iqm,(b,b) (δM (x)) = ±δ(b,b),(b,b) (iqm,(b,b) (x)).

(3.8.64)

It implies that iqm,(b,b) is a cochain homomorphism. (3.8.63) follows from (3.8.34.1). We can handle weakly unobstructed case by using (3.8.35.1).  A part of Theorem 3.8.62 can be restated in the language of the spectral sequence we will construct in Chapter 6. 3.8.6. Outline of the construction of the operator q. One delicate point of the proof of Theorem 3.8.32 is the choice of a countable set of smooth singular simplices X1+ (L). We postpone this choice to Section 7.4 (which is based on the discussion of Section 7.2). In this subsection, we explain an outline of the construction of the operator q and the proofs of the formulae (3.8.33) and (3.8.34), assuming the existence of such a choice of X1+ (L). Let us consider the moduli space Mmain k+1, (L, β) introduced in Subsection 2.1.2. We denote the (k + 1) boundary marked points by z0 , · · · , zk and the  interior marked points by z1+ , · · · , z+ . We put z = (z1 , · · · , zk ), z + = (z1+ , · · · , z+ ). The evaluation map  k+1 ev = (evint , ev∂ , ev0 ) : Mmain k+1, (L, β) → M × L

is defined by ev(w, (z + , z , z0 )) = (w(z1+ ), · · · , w(z+ ), w(z1 ), · · · , w(zk ), w(z0 )). For given Qi ∈ X (M ) (i = 1, · · · , ) and Pi ∈ X1+ (L) (i = 1, · · · , k), we define the fiber product main   Mmain k+1, (L, β; Q, P ) = Mk+1, (L, β)

evint ,ev∂ ×M  ×Lk (Q1 ×· · ·×Q )×(P1 ×· · ·×Pk ).

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

 = (Q1 , · · · , Q ), P = (P1 , · · · , Pk ).) (Here Q The following propositions are proved in Subsection 7.1.4 and Section 7.4, respectively.   Proposition 3.8.65. Mmain k+1, (L, β; Q, P ) has a Kuranishi structure which is   compatible with the Kuranishi structures of Mmain k +1 (L, β ; P ) (introduced in (3.5.1)) main    and Mk +1, (L, β ; Q , P ) at the boundary. Proposition 3.8.66. We can make a choice of X1+ (L) so that there exists a system of multisections sβ,Q,  P  compatible to sβ  ,P   and compatible to one another at the end and so that (s−1  (0), ev0 ) ∈ C1+ (L). β,Q,P

Here we take an appropriate triangulation and an order of its vertices to regard (s−1  (0), ev0 ) as a smooth singular chain. β,Q,P We remark that we have to use such sβ,P for Pi ∈ X1+ (L) \ X1 (L) also in the proof of Proposition 3.8.65 as well. We denote −1   s Mmain k+1, (L, β; Q, P ) = (sβ,Q,  P  (0), ev0 ).

Lemma 3.8.67. We assume R ⊇ Q. Then we can choose sβ,Q,  P  so that for each σ ∈ S we have P P   sβ,σ(Q),   sβ,Q, = (−1) (σ) Mmain , Mmain k+1, (L, β; σ(Q), P ) k+1, (L, β; Q, P )

where  = (Qσ(1) , · · · , Qσ(k) ), σ(Q)



(σ) =

deg Qi deg Qj .

iσ(j)

We now define ⎧ 1 main ⎪   s ⎪ ⎨ q,k,β ((Q1 , · · · , Q ) ⊗ (P1 , · · · , Pk )) = ! Mk+1, (L, β; Q, P ) ,  (3.8.68) ⎪ T ω[β] eμL (β)/2 q,k,β . q,k = ⎪ ⎩ β∈π2 (M,L)

  We now check (3.8.33). We study the boundary of Mmain k+1, (L, β; Q, P ) for this (m)

(m)

(m)

pint , p∂ )) be a divergent sequence of purpose. Let ((w(m) , z+(m) , z (m) , z0 ), (  P ). Here p(m) = (p(m) , · · · , p(m) ), p(m) ∈ Qi , and p(m) = points of Mmain (L, β; Q, k+1, int int,1 int,i int, ∂ (m)

(m)

(m)

(p∂,1 , · · · , p∂,k ), p∂,i ∈ Pi . Then there are three cases. (3.8.69.1)

(m)

limm→∞ p∂,i ∈ ∂Pi . (m)

(3.8.69.2) limm→∞ pint,i ∈ ∂Qi . (3.8.69.3) w(m) : Σ → M converges to a map from union of two genus zero bordered Riemann surfaces w(∞) : Σ1 ∪ Σ2 → M glued at boundary. See Figure 3.8.4.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

177

P

P

P

Q

Q

Q Q

P

ev0

Q

P

P

P

P

Figure 3.8.4 (∞)

Let us consider the case (3.8.69.3). We may take Σ1 , Σ2 so that z0 ∈ Σ2 . Now for each partition A1 ∪ A2 = {1, · · · , } with A1 ∩ A2 = ∅, we have corresponding components of (3.8.69.3) that is Qi ∈ Σj if and only if i ∈ Aj . We call such   components ∂A1 ,A2 Mmain k+1, (L, β; Q, P ). For i ≤ j we put Pi,j = (Pi , · · · , Pj ). P≤i = (P1 , · · · , Pi ), P>j = (Pj+1 , · · · , Pk ). Let mA1 ,1 < · · · < mA1 ,#A1 be elements of A1 and mA2 ,i (i = 1, · · · , #A2 ) are defined in the same way. We put  Aj )i = Qσ(m ) , σ(Q; Aj ,i  σ(Q; Aj ) = (Qσ(mAj ,1 ) , · · · , Qσ(mAj , Aj ) ), and QAj = (QmAj ,1 , · · · , QmAj , Aj ). Then we have 1  !



  s (−1) (σ) ∂A1 ,A2 Mmain k+1, (L, β; σ(Q), P )

(A1 ,A2 ) σ∈S

=

1  !



σ∈S β1 +β2 =β βj =0





 (−1) (σ) Mmain k−j+i,#A2 (L; β2 ; σ(Q; A2 )

(A1 ,A2 ) 0≤i≤j≤k

  s  ; P≤i , Mmain j−i+1,#A1 (L, β1 ; σ(Q; A1 ); Pi,j ) , P>j ). Using Lemma 3.8.67 we can see that the above formula is equal to the following.      1 (−1) i,j,A1 ,A2 #A1 !#A2 ! (A1 ,A2 ) β1 +β2 =β,βj =0 0≤i≤j≤k σ1 ∈S A1 σ2 ∈S A2

main     s  Mmain k−j+i,#A2 (L; β2 ; σ2 (QA2 ); P≤i , Mj−i+1,#A1 (L, β1 ; σ1 (QA1 ); Pi,j ) , P>j ).

We put  = [Q]



 σ(Q)

σ∈S

 = (Q1 , · · · , Q ). Then we have for Q 1     s (−1) (σ) ∂A1 ,A2 Mmain k+1, (L, β; σ(Q), P ) ! (A1 ,A2 ) σ∈S    (3.8.70)  A ]⊗ = q#A2 ,2 ,k−j+i,β2 ([Q 2 (A1 ,A2 ) β1 +β2 =β,βj =0 0≤i≤j≤k

 A ] ⊗ Pi,j ), P>j ). P≤i , q#A1 ,j−i+1,β1 ([Q 1

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

We remark that  = ΔQ



QA1 ⊗ QA2

A1 0,  = 1 we remark that the support of Mk+1,1 (L, β0 ; Q, P ) is in  mL k+1 (P1 , · · · , Pi , L ∩ Q, Pi+1 , · · · , Pk ). i

On the other hand, deg mL k+1 (P1 , · · · , Pi , L ∩ Q, Pi+1 , · · · , Pk ) = deg q1,k (Q ⊗ (P1 , · · · , Pk )) + 1. Hence we set q1,k,β0 (Q ⊗ (P1 , · · · , Pk )) = 0. The case  > 1 can be handled in the same way. See Remark 7.2.172 (3). The proof of (3.8.35) and (3.8.36) are combination of the above argument with one in Section 7.3. See the end of Section 7.3.  3.8.7. The operator r and the A∞ bimodule. To prove Theorem I, we need to combine the story of Subsection 3.8.5 to that of filtered A∞ bimodules. Let (L(0) , L(1) ) be a relatively spin pair of Lagrangian submanifolds of M . We take X1+ (L(1) ), X1+ (L(0) ), X (M ) as in Subsection 3.8.5. We remark that we use the same X (M ) for both L(0) and L(1) . We assume that L(0) and L(1) intersect cleanly and decompose L(1) ∩ L(0) = ∪h Rh . (We consider the Bott-Morse case only since it includes the transversal case.) We take countable sets X1 (Rh ) of chains and define C(Rh , Λ0,nov ) in the same way as in Subsection 3.7.5, and put C k (L(1) , L(0) ; Λ0,nov ) as in Subsection 3.7.5. Now we have Theorem 3.8.71. There exists a countable set of smooth singular simplices X1+ (Rh ) ⊃ X1 (Rh ) which generates the free Λ0,nov module C1+ (L(1) , L(0) ; Λ0,nov ) on which we can extend the filtered A∞ bimodule structure constructed in §3.7.5. And there exists a sequence of operators r;k1 ,k0 : E C(M ; Λ0,nov )[2] ⊗ Bk1 C1+ (L(1) ; Λ0,nov )[1] ⊗ C1+ (L(1) , L(0) ; Λ0,nov )[1] ⊗ Bk0 C1+ (L(0) ; Λ0,nov )[1] −→ C1+ (L(1) , L(0) ; Λ0,nov )[1]

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

179

of degree +1 such that 0 =r;k1 ,k0 (δM (y) ⊗ x ⊗ v ⊗ z)  (2;2)  (3;1)  (3;1) (2;1) (−1)deg yc1 deg xc2 +deg xc2 +deg yc1 + c1 ,c2 (3;3) r(y(2;1) ⊗ (x(3;1) ⊗ q(y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 ) ⊗ v ⊗ z)  (2;2)  (2;1)  (2;1) (2;1) + (−1)deg yc1 deg xc2 +deg xc2 +deg yc1

(3.8.72)

c1 ,c2 ,c3

+

(2;2) r(y(2;1) ⊗ x(2;1) ⊗ r(y(2;2) ⊗ x(2;2) ⊗ v ⊗ z(2;1) c1 c2 c1 c2 c3 )) ⊗ zc3 )



(2;2)

(−1)(deg yc1

+1)(deg x+deg v+deg z(3;1) )+deg y(2;1) c3 c1

c1 ,c3 (3;3) r(y(2;1) ⊗ (x ⊗ v ⊗ (z(3;1) ⊗ q(y(2;2) ⊗ z(3;2) c1 c3 c1 c3 ) ⊗ zc3 )).

Here δM is defined in Subsection 3.8.4. Moreover we have (3.8.73.1)

r0;k1 ,k0 = nk1 ,k0 ,

(3.8.73.2)

r;k1 ,k0 ≡ 0 mod Λ+ 0,nov

for  = 0.

Furthermore, we can extend r to (1) ; Λ0,nov )+ [1] r+ ;k1 ,k0 : E C(M ; Λ0,nov )[2] ⊗ Bk1 C1+ (L

⊗ C1+ (L(1) , L(0) ; Λ0,nov )[1] ⊗ Bk0 C1+ (L(0) ; Λ0,nov )+ [1] −→ C1+ (L(1) , L(0) ; Λ0,nov )[1]. It satisfies the same formula as (3.8.72) after replacing r,q by r+ ,q+ , respectively. It also satisfies: + r+ ;k1 ,k0 (y ⊗ x1 ⊗ · · · ⊗ e1 ⊗ · · · ⊗ xk1 −1 ⊗ v ⊗ z1 ⊗ · · · ⊗ zk0 ) = 0, + r+ ;k1 ,k0 (y ⊗ x1 ⊗ · · · ⊗ xk1 ⊗ v ⊗ z1 ⊗ · · · ⊗ e0 ⊗ · · · ⊗ zk0 −1 ) = 0 (i) + (i) for k1 + k0 ≥ 2. Here e+ i is the unit in C1+ (L ; Λ0,nov ) = C1+ (L ; Λ0,nov ) ⊕ + Λ0,nov ei ⊕ Λ0,nov fi as in Section 3.3. Moreover + r+ 0;k1 ,k0 = nk1 ,k0 (1) where n+ ; Λ0,nov )+ , C1+ (L(0) ; Λ0,nov )+ bimodule struck1 ,k0 is the filtered C1+ (L (1) (0) ture on C+ (L , L ; Λ0,nov ).

We sketch the proof in the next subsection. The detail of the proof is given in Sections 7.4.8-9. There we reinterpret Theorem 3.8.71 (especially Formula (3.8.72)) using the language of L∞ module homomorphism. See Theorem 7.4.154. Now let b ∈ C 2 (M ; Λ+ 0,nov ) with δM b = 0. We define nbk1 ,k0 : Bk1 C1+ (L(1) ;Λ0,nov )[1] ⊗ C1+ (L(1) , L(0) ; Λ0,nov ) ⊗ Bk0 C1+ (L(0) ; Λ0,nov )[1] −→ C1+ (L(1) , L(0) ; Λ0,nov ) by nbk1 ,k0 (x ⊗ v ⊗ z) = r(eb ⊗ x ⊗ v ⊗ z).

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180

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Moreover, as in Definition-Lemma 3.7.13, for bi ∈ C1+ (L(i) ; Λ0,nov ) with bi ≡ 0 mod Λ+ 0,nov , we define (b,b1 ) (b,b0 )

n

(x ⊗ v ⊗ z)

= r(e ⊗ eb1 ⊗ x1 ⊗ eb1 · · · eb1 ⊗ xk1 ⊗ eb1 ⊗ v ⊗ eb0 b

⊗ z1 ⊗ eb0 · · · eb0 ⊗ zk0 ⊗ eb0 ) with x = x1 ⊗ · · · ⊗ xk1 and z = z1 ⊗ · · · ⊗ zk0 . Lemma 3.8.74. (1) The operators nbk1 ,k0 define a ((C(L(1) ; Λ0,nov ), m(b,0) ), (C(L(0) ; Λ0,nov ), m(b,0) )) filtered A∞ bimodule structure on C1+ (L(1) , L(0) ; Λ0,nov ). (2) The operator (b,b1 ) n(b,b0 ) also defines a left (C(L(1) ; Λ0,nov ), m(b,b1 ) ) and right (C(L(0) ; Λ0,nov ), m(b,b0 ) ) filtered A∞ bimodule structure on C1+ (L(1) , L(0) ; Λ0,nov ). Proof. The proof is a straightforward calculation based on (3.8.72).



Lemma 3.8.74 implies that nbk1 ,k0 defines a filtered A∞ bimodule structure deformed by b. And (b,b1 ) n(b,b0 ) can be regarded as a deformed filtered A∞ bimodule structure after the bulk deformation. In particular, we consider k+1 k δ(b,b1 ),(b,b0 ) : C1+ (L(1) , L(0) ; Λ0,nov ) −→ C1+ (L(1) , L(0) ; Λ0,nov )

defined by δ(b,b1 ),(b,b0 ) (v) = r(eb ⊗ eb1 ⊗ v ⊗ eb0 ). Then we have b (L(0) ) and b1 ∈ M b (L(1) ), we Lemma-Definition 3.8.75. (1) If b0 ∈ M have δ(b,b1 ),(b,b0 ) ◦ δ(b,b1 ),(b,b0 ) = 0. Thus we can define the Floer cohomology group HF ((L(1) , (b, b1 )), (L(0) , (b, b0 )); Λ0,nov ) to be the cohomology of the complex (C1+ (L(1) , L(0) ; Λ0,nov ), δ(b,b1 ),(b,b0 ) ). (2) If L(i) (i = 0, 1) are weakly unobstructed after bulk deformation (see Defiweak,def (L(i) ) satisfying nition 3.8.39), then for any (b, bi ) ∈ M POL(0) (b, b0 ) = POL(1) (b, b1 ), we define + δ(b,b (v) = r+ (eb ⊗ eb1 ⊗ v ⊗ eb0 ). 1 ),(b,b0 ) + + ◦ δ(b,b = 0. Thus we can define the Floer cohomolThen we have δ(b,b 1 ),(b,b0 ) 1 ),(b,b0 ) ogy group HF ((L(1) , (b, b1 )), (L(0) , (b, b0 )); Λ0,nov ) + to be the cohomology of the complex (C1+ (L(1) , L(0) ; Λ0,nov ), δ(b,b ). 1 ),(b,b0 )

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

181

In Lemma-Definition 3.8.75 (2), we can also define the deformed filtered A∞ bimodule structure after bulk deformation in a similar way as follows: (b,b1 ) +(b,b0 )

n

(x ⊗ v ⊗ z)

= r (e ⊗ eb1 ⊗ x1 ⊗ eb1 · · · eb1 ⊗ xk1 ⊗ eb1 +

b

⊗ v ⊗ eb0 ⊗ z1 ⊗ eb0 · · · eb0 ⊗ zk0 ⊗ eb0 ). 3.8.8. Construction of the operator r. In this subsection, we discuss the proof of Theorem 3.8.71. The proof is a straight forward combination of the construction we have already discussed. We take a one parameter family {Jt }t of almost complex structures. Let us consider the set of ((Σ, τ , z+ ), u) such that (3.8.76.1) ((Σ, τ ), u) satisfies the same condition as an element of the moduli space Mk1 ,k0 (L(1) , L(0) ; [h, w], [h , w ]) except it may have unstable component. (3.8.76.2) z + = (z1+ , · · · , z+ ) are interior points on Σ. (3.8.76.3) ((Σ, τ , z+ ), u) is stable. Namely the group of its automorphisms is of finite order. (i)

(i)

Here τ = (τ (0) , τ (1) ) with τ (i) = (τ1 , · · · , τki ) (i = 0, 1) as in Subsection 3.7.5. Let Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]) be the moduli space of such system ((Σ, τ , z + ), u). We define evaluation maps (evint , ev (1) , ev (0) ) : Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]) → M  × (L(1) )

k1

× (L(0) )

k0

by (evint , ev (1) , ev (0) )(((Σ, τ , z+ ), u)) (1)

(0)

= (u(z1+ ), · · · , u(z+ ), u((τ1 ), 1), · · · , u((τk0 ), 0)) and ev−∞ : Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]) → Rh ev+∞ : Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]) → Rh , by ev±∞ (((Σ, τ , z+ ), u)) = lim u(τ, t). τ →±∞

(j)

Let Qi ∈ X (M ), Pi

∈ X1+ (L(j) ) and S ∈ X1+ (Rh ). We define

 P (1) , S, P (0) ) Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]; Q, =Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]) 0 0 (1) 0 (0)  . Pi × S × Pi Qi × (evint ,ev (1) ,ev−∞ ,ev (0) ) × See Figure 3.8.5.

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CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

(1)

Pk1

(1)

Q1

S

Q2

P1

ev+

Q3

(0)

Pk0

(0)

P1

Figure 3.8.5 It has a Kuranishi structure and we can define a multisection s on it compatible at the end. We then put  P (1) , S, P (0) )s = (−1) ev+∞ (s−1 (0)), Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]; Q, which is a chain on Rh . Here the sign (−1) is the same as one just before Definition 3.7.71 in Subsection 3.7.5. We can choose X1+ (Rh ) so that the virtual fundamental  P (1) , S, P (0) )s again is contained in it. chain Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]; Q, Now we take (1)

(1)

(0)

(0)

rk1 ,k0 ; (Q1 ⊗ · · · Q ⊗ P1 ⊗ · · · ⊗ Pk1 ⊗ [S, w] ⊗ P1 ⊗ · · · ⊗ Pk0 )  1  P (1) , S, P (0) )s . Mk1 ,k0 ; (L(1) , L(0) ; [h, w], [h , w ]; Q, = !   [h ,w ]

The proofs of (3.8.72) and (3.8.73) are now straightforward.



3.8.9. Generalization of the operator p. We generalize the operator pk by combining it with q to obtain an operator p,k . (The difference between operators p and q is as follows: For the moduli space used to define p the 0-th marked point is an interior marked point: On the other hand, for the moduli space used to define q, the 0-th marked point is a boundary marked point.) The operator p,k will be used in Chapter 6 to describe the image of the differential of the spectral sequence constructed there. Since the construction of p,k is a straightforward analogue of ones discussed already in this section, we write the conclusion only. We first recall Gromov-Witten ˜ be the moduli space of pseudo-holomorphic invariant a bit more. Let M0,+2 (M ; β) spheres with ( + 2) marked points, whose homology class β˜ ∈ H2 (M ). We can define an obvious evaluation map ˜ → M +2 . ev = (ev1 , · · · , ev+1 , ev0 ) : M0,+2 (M ; β) For each chains Qi ⊂ M , i = 1, · · · ,  + 1 we put ˜ Q) ˜  = M0,+2 (M ; β) M0,+2 (M ; β; and define (3.8.77)

 = GW0,+1 (M )(Q)



(ev1 ,··· ,ev+1 )

× (Q1 × · · · × Q+1 )

˜ ˜ ˜ Q)).  T ω[β] ec1 (M )(β) ev0∗ (M0,+2 (M ; β;

β˜

We remark that (3.8.77) is a generalization of (3.8.2). (We regard the right hand as an element of S ∗ (M ; Λ0,nov ) by taking an appropriate triangulation.)

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

183

We next remark that q induces ˆqcyc : EC(M ; Λ0,nov )[2] ⊗ B cyc C1+ (L; Λ0,nov )[1] → B cyc C1+ (L; Λ0,nov )[1] in the same way as mk induces dˆ cyc in Lemma 3.8.4. Proposition 3.8.78. In case R = Q, there exists a sequence of operators p,k : E C(M ; Λ0,nov )[2] ⊗ Bkcyc C1+ (L; Λ0,nov )[1] −→ S ∗ (M ; Λ0,nov ) (k = 0, 1, 2, · · · , = 0, 1, 2, · · · ) of degree n + 1 with the following properties: (3.8.79.1) (3.8.79.2) (3.8.79.3)

p0,k = pk . p,k ≡ 0 mod Λ+ 0,nov for  > 0. For k > 0 we have p,k (δM (y) ⊗ x) + δM (p,k (y ⊗ x))  p(y(2;1) ⊗ (ˆqcyc (y(2;2) ⊗ x))) = 0. + c1 c1 c1 ,c2

Here the notations are the same as in Theorem 3.8.32.

(3.8.79.4)

p,0 (δM (y) ⊗ 1) + δM (p,0 (y ⊗ 1))  (2;1) (−1)deg yc1 p(y(2;1) ⊗ (q(y(2;2) ⊗ 1))) + c1 c1 c1

1 + GW0,+1 (M )(y ⊗ L) = 0. ! Here the last term on the left hand side is defined as in (3.8.77). Namely, Qi = yi , (i = 1, · · · , ), where y = y1 ⊗ · · · ⊗ y and L = Q+1 . We can furthermore extend p,k to cyc + ∗ p+ ,k : E C(M ; Λ0,nov )[2] ⊗ Bk C1+ (L; Λ0,nov ) [1] −→ S (M ; Λ0,nov )

satisfying (3.8.79.5)

+ p+ 0,k = pk

and a similar compatible condition as in (3.8.35.1) − (3.8.35.3). In particular, (3.8.79.6)

+ p+ ,k (y ⊗ [e , x1 , · · · , xk−1 ]) = 0 for  = 0 and any k.

Here C1+ (L; Λ0,nov )+ = C1+ (L; Λ0,nov ) ⊕ Λ0,nov e+ ⊕ Λ0,nov f as in Section 3.3. The construction of the operator p,k uses the moduli space Mmain k,+1 (L, β) where the last (0-th) marked point is the interior marked point. (Recall that the last marked point in the case of defining p is the interior point, while one in the case of q is the boundary marked point.) See Figure 3.8.6 below. For example Figure 3.8.7 gives p((Q3 ⊗ Q4 ) ⊗ (q((Q1 ⊗ Q2 ) ⊗ (P6 ⊗ P1 ⊗ P2 )) ⊗ P3 ⊗ P4 ⊗ P5 ). It is one of the terms of the last term in the left hand side of (3.8.79.3).

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184

CHAPTER 3. A∞ ALGEBRA TO A LAGRANGIAN SUBMANIFOLD

P3 P2

Q2 P4

Q1 Q

ev0

P5

P1 Pk

p k((Q1 ⊗ · · · ⊗ Q ) ⊗ (P1 ⊗ · · · ⊗ Pk )) 

Figure 3.8.6 P5

P6

Q3

Q2 Q1

Q4 P4

ev0

P1 P2

P3

Figure 3.8.7 The reason we have the term GW0,+1 (M )(y ⊗ L) in (3.8.79.4) is the same as the case of p. The detail of the proof of Proposition 3.8.78 is a combination of the arguments we have already discussed. It will be discussed in Subsection 7.4.11 in detail.  We remark Proposition 3.8.78 will be reinterpreted in Sections 7.4.10-11 using L∞ structure. See Theorem 7.4.192. Example 3.8.80. Let us consider (M, ωM ) = (CP 1 , ω) × (CP 1 , −ω), and L = diagonal, which is oriented so that the first factor projection respects the orientation. We take p = (z0 , w0 ) ∈ M and consider P D[p] ∈ H 2n (M ; Q). We put y = P D[p] in (3.8.79.4) and get (3.8.81)

δM (p1,0 (P D[p] ⊗ 1)) + p1,1 (P D[p] ⊗ m0 (1)) + p0,1 (1 ⊗ q1,0 (P D[p] ⊗ 1)) + GW (M )0,2 (P D[p] ⊗ [L]) = 0.

By dimension counting we have m0 (1) = 0 as chain. Hence (3.8.81) implies (3.8.82)

[p0,1 (1 ⊗ q1,0 (P D[p] ⊗ 1)))] = −[GW0,2 (M )(P D[p] ⊗ [L])].

Let us check the leading term of Formula (3.8.82) directly. We put H 2 (M ; Z) = Z ⊕ Z. We take the complex structure JM = JCP 1 ⊕ −JCP 1 on M . Then β˜1 = (1, 0) and β˜2 = (0, −1) are both represented by pseudoholomorphic spheres of minimal symplectic area. We find M0,3 (M ; β˜1 ) ×M ({p} × L) ∼ = S 2, and its evaluation map image is ev0 (M0,3 (M ; β˜1 ) ×M ({p} × L)) = {(z, w0 ) | z ∈ CP 1 }.

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3.8. INSERTING MARKED POINTS IN THE INTERIOR

185

Hence GW0,2 (M ; β˜1 )(P D[p] ⊗ [L]) = CP 1 × {w0 }.

(3.8.83.1)

On the other hand, taking β˜1 ∩ L = −β˜2 ∩ L into account, we find GW0,2 (M ; β˜2 )(P D[p] ⊗ [L]) = {z0 } × CP 1 .

(3.8.83.2)

Here we note that the orientation of CP 1 on the right hand side of (3.8.83.2) is opposite to that of the pseudo-holomorphic sphere representing β˜2 . Hence (3.8.84)

1

1

[GW0,2 (M )(P D[p] ⊗ [L])] ≡ T ω[CP ] (1, 1) mod T ω[CP ] Λ+ 0,nov .

We next calculate the left hand side of (3.8.82). Let β ∈ H2 (M, L; Z) be the image of β˜i . (β˜1 and β˜2 go to the same element of H2 (M, L; Z).) We identify CP 1 = C∪{∞} and z0 = 0, w0 = ∞. Then Mmain,reg (L; β) 0,1

(3.8.85)

ev

× {p} ∼ = (0, ∞)

and r ∈ (0, ∞) is identified with a holomorphic map ur : D2 → M defined by ur (z) = (rz, r/z). (Note ur (0) = p and ur (∂D2 ) ⊆ L.) (See Figure 3.8.8.) It 1 ∼ follows that Mreg 1,1 (L; β) ev × {p} = R × S and [ev0 (Mmain 1,1 (L; β) Namely

ev

× {p})] = ±[L].

1

1

q1,0 (P D[p] ⊗ 1) ≡ ±T ω[CP ] [L] mod T ω[CP ] Λ+ 0,nov . Thus we have (3.8.86)

1

1

[p0,1 (1 ⊗ q1,0 (P D[p] ⊗ 1))] ≡ ±T ω[CP ] [L] mod T ω[CP ] Λ+ 0,nov .

This is consistent with (3.8.82), (3.8.84). (We do not try to check the sign here.) We remark that the two ends of the moduli space (3.8.85) correspond to the trivial disc plus sphere bubble. Those bubbles correspond to (3.8.83.1) and (3.8.83.2), respectively.

p

L

Figure 3.8.8 Now let b ∈ C

2

(M ; Λ+ 0,nov ),

with δM b = 0. We define

pbk : Bkcyc C1+ (L; Λ0,nov )[1] −→ S ∗ (M ; Λ0,nov ) by pbk (x) =



p,k (b ⊗ x) = p(eb ⊗ x).



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We then have (3.8.87.1) (3.8.87.2) (3.8.87.3)

pb1 ≡ i! mod Λ+ 0,nov . pb ◦ db,cyc + δM ◦ pbk = 0, for k > 0. 1 GW0,+1 (M )(b⊗ ⊗ L) = 0. pb1 ◦ mb0 (1) + δM ◦ pb0 (1) + ! 

This is the same formula as (3.8.10.1),(3.8.10.2),(3.8.10.3). In other words, we can use pbk in place of pk when we study HF ((L, (b, b)), (L, (b, b )); Λ0,nov ). We can generalize Theorem 3.8.11 as follows. We recall from Definition 3.8.40 that b (L) = {b ∈ C(L; Λ+ ) | (b, b) ∈ M def (L)} M 0,nov b (L) is defined in a similar way. and M weak Let [oi (L; b)] be the obstruction classes for Mb (L) being nonempty. (See Theorem 3.6.18.) [oi (L; b; weak)] is defined in a similar way. (See Theorem 3.6.43.) Theorem 3.8.88. Let L ⊂ M be a relatively spin Lagrangian submanifold and b ∈ H 2 (M ; Λ+ 0,nov ). (3.8.89.1) (3.8.90)

If ∞  1 [GW0,+1 (M )(b⊗ ⊗ L)] = 0, ! =0

b (L) = ∅. then M (3.8.89.2) If the left hand side of the formula (3.8.90) is 0, then the obstruction [oi (L; b)] lies in the kernel of the Gysin homomorphism i! : H 2−μi (L; Q) → H n+2−μi (M ; Q). (3.8.89.3) If the left hand side of the formula (3.8.90) is not of the form c P DM [L] b (L) = ∅. then M weak (3.8.89.4) If the left hand side of (3.8.90) is of the form c P DM [L] then oi (L; b; weak) ∈ Ker(i! : H 2−μi (L; Q) → H n+2−μi (M ; Q)). b (L), and [L] = 0 (3.8.89.5) If the left hand side of (3.8.90) is c P DM [L], b ∈ M weak in H(M ; Q), then c = PO(b, b). The proof of Theorem 3.8.88 is the same as the proof of Theorem 3.8.11 using (3.8.87) and is omitted. Remark 3.8.91. We remark that we can not generalize Lemma 3.8.14 to our situation. Namely the left hand side of (3.8.90) may have a nontrivial H ∗ (M ; Q) ⊗ Λ+ 0,nov component with ∗ = 2n, 2n − 1. In fact if b ∈ H n (M ; Q) ⊗ Λ+ 0,nov with b ∩ L = 0 then GW0,2 (M ; β˜0 )(b, L) = P DM (b ∩ L) = 0. Here β˜0 = 0, and P DM : Hd (M ; Q) ⊗ Λ0,nov → H 2n−d (M ; Q) ⊗ Λ0,nov is the Poincar´e duality of M . We can partially generalize Lemma 3.8.14 as follows.

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Lemma 3.8.92. Let Bj ∈ H(M ; Q), j = 1, · · · , . We assume + ∗ ˜ GW0,+1 (M, β)(B 1 , · · · , B , L) ∈ H (M ; Q) ⊗ Λ0,nov

with ∗ = 2n or 2n − 1. We assume, in addition, that one of the following conditions is satisfied. (3.8.93.1) (3.8.93.2)

β˜ = 0 in H2 (M ; Z).  = 1.

˜ Then GW0,+1 (M, β)([B 1 , · · · , B , L]) = 0. Proof. We put dj = deg Bj = 2n − dim Bj . We have ˜ +2+ ∗ = n − 2c1 (M )[β]

 

(dj − 2),

j=1

˜ = 2n + 2c1 (M )[β] ˜ − 2 + 2. This implies that the virtual since dim M0,+2 (M ; β) dimension of the moduli space ˜ ×M  (B1 × · · · × B ) M0,+1 (M ; β) ˜ +  (2 − dj ) = 3n − ∗ − 2. Thus, by the assumption, is 2n − 4 + 2c1 (M )[β] j=1

(3.8.94)

˜ ×M  (B1 × · · · × B ) = n − 2 or dim M0,+1 (M ; β)

n − 1.

Therefore the fiber product ˜ ×M +1 (B1 × · · · × B × L) M0,+1 (M ; β) is empty. We may take the perturbation of (3.8.95)

˜ ×M +1 (B1 × · · · × B × L) M0,+2 (M ; β)

compatible with the forgetful map ˜ ×M +1 (B1 × · · · × B × L) M0,+2 (M ; β) ˜ ×M +1 (B1 × · · · × B × L), → M0,+1 (M ; β) and ev0 is compatible with evaluation map. (See Section 23 [FuOn99II].) Note we use our assumption (3.8.93) here. In fact in the case β˜ = β˜0 = 0 and  = 1, M0,2 (M ; β˜0 ) is empty since constant map from sphere with two marked points is unstable. (On the other hand, M0,3 (M ; β˜0 ) is M and is nonempty.) Since (3.8.95) is an emptyset under the given hypothesis we have ˜ [GW0,+1 (M, β)([B 1 , · · · , B , L])] = 0.  Let Πd : H(M ; Λ0,nov ) → H d (M ; Q)⊗Λ0,nov be the projection. We use Lemma 3.8.92 and (3.8.87.3) to find + , Π∗ pb1 ◦ mb0 (1) + δM ◦ pb0 (1) = Π∗ (P DM ([L] ∩ b)) for ∗ = 2n − 1 or ∗ = 2n. We can use it to prove the following Corollary 3.8.96 which generalize Corollary 3.8.16.

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Corollary 3.8.96. We assume either (a) Πn (P DL (b ∩ L)) = 0, or (b) Πn−1 (P DL (b ∩ L)) = 0. (Here P DL : Hd (L; Q) ⊗ Λ0,nov → H n−d (L; Q) ⊗ Λ0,nov is the Poincar´e duality in L.) (1) We assume (a) and that the obstruction class [oi (L; b)] is defined and in H n (L; Q). Then we have [oi (L; b)] = 0. (2) We assume (b). If [oi (L; b)] ∈ H n−1 (L; Q) then i! (oi (L; b)) = 0. The same applies to [oi (L; b; weak)]. We can use Corollary 3.8.96 to prove the following two corollaries which partially generalize Corollary 3.8.19. Corollary 3.8.97. Let L be a relatively spin Lagrangian submanifold which is a rational homology sphere. Let b∈ H ∗ (M ; Q) ⊗ Λ+ 0,nov . ∗≥2

We assume Π (P DL (b ∩ L)) = 0. We assume in addition that there exists no pseudo-holomorphic disc of nonnegative Maslov index which bounds L. Then n

(3.8.98)

∞  1 [GW0,+1 (M )(b⊗ ⊗ L)] = 0. ! =0

Proof. Since L is a rational homology sphere [oi (L; b)] lies either in H n (L; Q) or in H 0 (L; Q). Non-existence of pseudo-holomorphic disc with nonnegative Maslov index which bounds L and b having degree ≥ 2 imply that mb0 (1) ∈ H ∗ (M ; Q) ⊗ Λ+ 0,nov . ∗≥2

Therefore [oi (L; b)] does not lie in H 0 (L; Q). Then [oi (L; b)] = 0 follows from Corollary 3.8.96. Now Theorem 3.8.88 (1) implies Corollary 3.8.97.  Corollary 3.8.99. Let L be a relatively spin Lagrangian submanifold with 2 vanishing Maslov index. Let b ∈ H 2 (M ; Q) ⊗ Λ+ 0,nov . Assume H (L; Q) = 0. Then (3.8.98) holds. Proof. We can prove that [oi (L; b)] lies in H 2 (M ; Q) by degree counting. b (L) = ∅. Corollary 3.8.99 follows from Theorem 3.8.88 Therefore by assumption M (1).  3.8.10. Proof of parts of Theorems B, C and G. Now we summarize the results so far and see that they imply Theorem B (B.2), Theorem C, and Theorem G (G.1),(G.5) (and the construction of Floer cohomologies in Theorems B and G). The construction of the Floer cohomology of Theorem B is a consequence of Proposition 3.7.17 and Definition 3.8.40. (B.2) then is a consequence of Proposition 3.7.87 (2) and Definition 3.8.38. Theorem C follows from Theorem 3.8.50 and Corollary 3.8.96. Namely the statements of Theorem C except mk < dim L/2 are consequences of Theorem 3.8.50.

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L We will use Corollary 3.8.96 to prove odim (L; def) = 0 as follows. We put n = k dim L. We consider the proof of Theorem 3.8.41. We will prove by induction that we can choose b(i) such that Πn (b(i) ) = 0. Namely the H n (M ; Q) ⊗ Λ+ 0,nov component of b(i) . Suppose we have chosen such b(i) for i < k. Then we can apply Corollary 3.8.96 (1). Therefore if ok (L : weak, def) ∈ H n (L; Q) then it is automatically 0. This implies that when we choose bk such that (3.8.47) is satisfied, we may choose bk = 0 if ok (L : weak, def) ∈ H n (L; Q). Note bk ∈ H n (M ; Q) if and only if ok (L : weak, def) ∈ H n (L; Q). Thus induction works. We proved at the same time that if ok (L : weak, def) ∈ H n (L; Q) then ok (L : weak, def) = 0. This is what we wanted to prove. L Note also that the condition odim (L; def) = 0 is equivalent to the condition k dim L (L; weak, def) = 0 by Proposition 3.8.54. ok The existence of Floer cohomology in Theorem G is also a consequence of Proposition 3.7.17 and Definition 3.8.58. (G.1) follows from Proposition 3.7.75 and its analog which asserts that operation r, r+ coincides with q,q+ in case L(1) = L(0) . (The proof of this fact is similar to the proof of Proposition 3.7.75.) (G.5) is a consequence of Proposition 3.7.93.

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https://doi.org/10.1090/amsip/046.1/04

CHAPTER 4

Homotopy equivalence of A∞ algebras 4.1. Outline of Chapters 4 and 5 In Chapter 3, we have constructed the filtered A∞ algebra (C(L; Λ0,nov ), m) associated to a relatively spin Lagrangian submanifold L and the filtered A∞ bimodule C(L(1) , L(0) ; Λ0,nov ) for a relatively spin pair (L(0) , L(1) ). In this chapter and the next, we study dependence of the filtered A∞ algebra C(L; Λ0,nov ), the filtered A∞ bimodule C(L(1) , L(0) ; Λ0,nov ) and the Floer cohomology on the Hamiltonian deformations, other parameters involved in the definitions: that is, the compatible almost complex structure J, the perturbation (multisection) s, triangulation of s−1 (0), and the choice of countable set of smooth singular simplices. Among them, we define the gauge equivalence relation on the set of the bounding cochains b (the solutions of the Maurer-Cartan equation) and study dependence of the Floer cohomology on the choice of the bounding cochain b. Throughout these two chapters, we assume that a Lagrangian submanifold L is orientable and relatively spin, and that a pair of two Lagrangian submanifolds is relatively spin. (See Definition 3.1.1 in Section 3.1.) First we explain the reason why the Floer cohomology depends on various choices (especially on the bounding cochain b). We recall that the basic trouble being discussed in this book in defining the Floer cohomology of Lagrangian submanifolds lies in the fact that in general the moduli space of pseudo-holomorphic discs has codimension one boundary. It means that the fundamental cycle of the moduli space of pseudo-holomorphic discs may not be well-defined in the usual sense. A similar phenomenon was found by Donaldson [Don86II] in his study of the moduli space of solutions of ASD equation on 4-manifolds with b+ 2 = 1. In that case, Donaldson observed that if we consider one parameter family of perturbations (especially the perturbation of Riemannian metric), then the homology class of the moduli space may jump at some point. This phenomenon is called the wall crossing. We recall how the wall crossing occurs for the case, for simplicity, where the virtual dimension of the moduli space is 0. The order, counted with sign, of the moduli space is the invariant (Donaldson invariant). If we consider one parameter family of perturbations, we have one parameter family Ms of moduli spaces (s ∈ [0, 1]) whose union ∪s {s} × Ms is a one dimensional manifold. Independence of the invariant under the perturbation could be proved by a cobordism argument if ∪s {s} × Ms were compact for a generic one parameter family, in which case its boundary would become −M0 ∪ M1 . However in the case where the bubbling phenomenon is of codimension one, we cannot completely avoid bubbling even for a generic one parameter family and so the parameterized moduli space ∪s {s}×Ms is not compact in general. Therefore the cobordism argument cannot prove independence of the invariant thus constructed on the perturbation involved. 191

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In the gauge theory, Donaldson introduced the “chamber structure” to handle this trouble. In that case, the trouble has a nice feature that we can control where the wall crossing occurs, in terms of the abelian gauge theory which is nothing but the cohomology theory. Namely the wall crossing occurs at the points where there exists a nonzero anti-self-dual harmonic 2 form whose cohomology class is integral. In particular, if we take a perturbation which does not change the abelian gauge theory (especially the perturbation which does not change the Riemannian metric) then the wall crossing does not occur. So the Donaldson invariant is well-defined, for example, if we fix a Riemannian metric. The problem is similar in the case of Seiberg-Witten invariant of 4-manifolds with b+ 2 = 1. In our situation, the problem is more non trivial, because we do not have any control where the wall crossing might occur. So we are forced to consider the whole family of perturbations to obtain something invariant under the perturbations. To carry out this, we will investigate infinitesimal structure of the moduli space of (unobstructed) Lagrangian submanifolds. Let us take and fix a Lagrangian submanifold L of M . (We also fix a compatible almost complex structure J.) We are going to find a formal neighborhood of L in the extended moduli space of Lagrangian submanifolds. For another Lagrangian submanifold L near L, we say that L ∼ L if there exists a Hamiltonian isotopy ψs such that L = ψ1 (L). One can see that the set of ∼ equivalence classes of such Lagrangian submanifolds is isomorphic to a neighborhood of 0 in the first cohomology group H 1 (L; R). (There exists a trouble related to the Hausdorffness of the moduli space of Lagrangian submanifolds. We do not discuss this since we concern only with the local problem here. See Section 1 [Fuk02I].) Because of various reasons (especially because of the presence of relative homology class H2 (M ; L) of nonzero Maslov-index), we want to extend this parameter space to H odd (L; R). Only H 1 (L; R)-direction of this extended space H odd (L; R) has direct geometric meaning. But one can still define deformations of “Floer (co)homology” to other directions. As we will explain below, our construction is similar to that of deformations of the cup product using the genus 0 Gromov-Witten invariant: Let M be a symplectic manifold. The space H 2 (M ; R) parameterizes (locally) the moduli space of symplectic structures of M . The Gromov-Witten theory and the Frobenius structure obtained from it is regarded as a family of deformations of the cup product of M parameterized by H even (M ; R), which is the “extended moduli space of symplectic structures on M ”. The deformations to H 2 (M ; R)direction have obvious geometric meaning, that is the deformation of symplectic forms. This deformation space is extended to H even (M ; R) direction also by using the Gromov-Witten potential. One may extend it to H odd (M ; R) direction by regarding this parameter space as a super space. (See [ASKZ97, Dub96, KoMa94, Man99, Ran96].) The extended moduli space of Lagrangian submanifolds which we mentioned above is a natural analog to the extended moduli space that appeared in the Frobenius structure and Gromov-Witten potentials. We consider the formal neighborhood of a Lagrangian submanifold L in this extended moduli space, which is the zero set of a formal power series expansion of the defining equation of the moduli space or of the Kuranishi map. (The convergence of this formal power series is not known yet.) In other words, we are studying our moduli space as a formal  scheme. The set of bounding cochains M(L) we introduced in the last chapter,

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after dividing it out by appropriate gauge equivalence which we will define in this chapter, will turn out to be this formal scheme M(L). See Sections 1.4 and 5.4. One important fact on the Gromov-Witten potential is that the extended moduli space of deformations of the symplectic structure is not obstructed. Namely the full neighborhood of zero in H even (M, R) locally parameterizes the extended moduli space. The group H odd (M, R) in principle is the obstruction group. However the obstruction map or the Kuranishi map is identically zero for the case of the Gromov-Witten potential. This fact is consistent with the mirror symmetry in the following way: Let M ∨ be the mirror of M . Then a part of H odd (M ; C) will be isomorphic to H 2 (M ∨ , T M ∨ ) which is the vector space where the obstruction to the Kodaira-Spencer deformation theory lives. In other words, this is the space where the Kuranishi map (of the deformation theory of complex structure) takes its value. However the Kuranishi map is automatically 0 for the Calabi-Yau manifold. For the classical direction H 1 (M ∨ , T M ∨ ) this fact was proved by Bogomolov-Tian-Todorov [Bog78, Tia87, Tod89]. For the other extended directions H k (M ∨ , Ωodd−k M ∨ ), this fact was proved by Baranikov-Kontsevich [BaKo98]. On the other hand, our extended moduli space of Lagrangian submanifolds is indeed obstructed in general. The corresponding obstruction map or the Kuranishi map is exactly our obstruction class. Thus, finding a formal neighborhood of extended moduli space of (unobstructed) Lagrangian submanifolds is equivalent to studying the behavior of the obstruction class in a neighborhood of L. Our problem of understanding dependence of the obstruction class and of the Floer cohomology on the perturbation is closely related to this study. The obstruction phenomenon we mentioned above is directly related to the wall crossing phenomenon we mentioned before. Namely wall crossing occurs because the parametrized version of the obstruction class is nonzero. Now we describe the contents of Chapters 4 and 5. Chapter 4 is devoted to studying the homotopy equivalence of filtered A∞ algebras and Chapter 5 to that of filtered A∞ bimodules. These two chapters contain both algebraic and geometric discussions. The algebraic part is a homotopical (homological) algebra of A∞ structures. There are various references about it. (See for example [Fuk02II,03II, Kel01, Lef03, MSS02, Smi00, Sei06].) However we include self contained discussions in this book, since there is no discussion in the case of a filtered A∞ algebra in the literature. Moreover the relation to the gauge equivalence of the bounding cochains is not so transparent in the literature, and we need to include discussion about (homotopy) unit and to handle the case over torsion coefficient for the application in [FOOO09I]. The first four sections Sections 4.2-4.5 are of purely algebraic nature. (In Subweak,def (L). In Chapter 3, when section 4.3.3 we discuss the gauge equivalence on M we introduced the notion of bulk deformation we used the operator q which is defined geometrically. However, the results also holds for any filtered A∞ algebra which has an operator q satisfying the conclusion of Theorem 3.8.32.) In Section 4.2, we introduce the notion of two (filtered) A∞ homomorphisms to be homotopic each other and establish basic properties about homotopy between (filtered) A∞ homomorphisms. This allows us to define two (filtered) A∞ algebras are homotopy equivalent. In Definition 3.2.33, we introduced a notion that a (filtered) A∞ homomorphism is a weak homotopy equivalence. In fact, a (filtered) A∞ homomorphism is a homotopy equivalence if and only if it is a weak homotopy equivalence

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(Theorem 4.2.45 proved in Section 4.5). This gives a nice criterion for an A∞ homomorphism to be a homotopy equivalence. The key result (Theorem 4.2.34) to prove basic properties of homotopy of A∞ homomorphisms is proved in Section 4.4. In Section 4.3, we introduce the notion that two bounding cochains are gauge equivalent, and establish its basic properties. It is an equivalence relation and we  divide the set of bounding cochains M(C) by this gauge equivalence relation to get a set M(C). (Later on in Section 5.4 we will define a structure on M(C).) We show that it is invariant of the homotopy type of the filtered A∞ algebra C. In Section 4.6, we will consider our geometric situation and show that the homotopy type (in the sense of Section 4.3) of our filtered A∞ algebra (C(L; Λ0,nov ), m) associated to a relatively spin Lagrangian submanifold L is independent of various choices involved. More precisely, we will prove the following Theorem 4.1.1. Let (M, ω), (M  , ω  ) be symplectic manifolds and L ⊂ M and L ⊂ M  Lagrangian submanifolds. Let ψ : M → M  be a symplectic diffeomorphism such that ψ(L) = L . We fix various choices we made (compatible almost complex structures, perturbations, triangulation, and countable sets of smooth singular simplices), for M, L and M  , L to define (C(L; Λ0,nov ), m) and (C(L ; Λ0,nov ), m). Theorem 4.1.1. In the above situation, there exists a homotopically unital filtered A∞ homomorphism ψ∗ : (C(L; Λ0,nov ), m) → (C(L ; Λ0,nov ), m) induced by ψ, which is a homotopy equivalence of the filtered A∞ algebras. If we consider the case when M = M  and L = L , Theorem 4.1.1 implies that our filtered A∞ algebra (C(L; Λ0,nov ), m) is invariant of various choices involved up to homotopy equivalence. Moreover we will prove that the homotopy equivalence ψ∗ in Theorem 4.1.1 is independent of various choices involved up to homotopy. Namely we will prove: Theorem 4.1.2. In the situation of Theorem 4.1.1, we take a one parameter family {ψs }s (0 ≤ s ≤ 1) of symplectic diffeomorphisms such that ψs (L) = L for each s. Then ψ0∗ is homotopic to ψ1∗ as homotopically unital filtered A∞ homomorphisms. These two theorems together with algebraic results in sections Sections 4.2-4.5 imply that the set M(L) of the gauge equivalence classes of the bounding cochains is independent of various choices involved up to canonical isomorphism. It also implies that the Floer cohomology HF ((L, b1 ), (L, b0 ); Λ0,nov ) depends only on the gauge equivalence classes of bi . More precisely, we have the following. Theorem 4.1.3. In the situation of Theorem 4.1.1, there exists a bijection ψ∗ : M(L) → M(L ) which depends only of the homotopy class of ψ (in the sense of Theorem 4.1.2).    ) and ψ∗ [bi ] = [b ] ∈ M(L ), then Moreover, if b0 , b1 ∈ M(L), b0 , b1 ∈ M(L i there exists a canonical isomorphism ψ∗ : HF ((L, b1 ), (L, b0 ); Λ0,nov ) → HF ((L , b1 ), (L , b0 ); Λ0,nov ). Theorem B (B.3) and (B.4) in the introduction follows from Theorem 4.1.3 and its ‘def,weak’ version. See Subsection 4.6.5. Chapter 5 deals with the case when there are two Lagrangian submanifolds or (filtered) A∞ bimodules. In Section 5.1, we discuss relations of various Novikov

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rings which appear in the theory of Floer cohomologies. In Section 5.2, we study homotopy between (filtered) A∞ bimodule homomorphisms, in a way similar to the case of (filtered) A∞ algebras. Namely we define homotopy between (filtered) A∞ bimodule homomorphisms and define homotopy equivalence of A∞ bimodules, etc. Section 5.3 is devoted to the geometric part of the construction. We will construct homotopy equivalence between filtered A∞ bimodules associated to a relatively spin pair of Lagrangian submanifolds (L(0) , L(1) ) defined in Section 3.7. It implies that Floer cohomology HF ((L(1) , b(1) ), (L(0) , b(0) )) between two Lagrangian submanifolds is independent of various choices involved in the construction (but depends on the gauge equivalence classes of b(0) , b(1) ), and of Hamiltonian perturbations of L(0) , L(1) . To be precise, there are two different cases, which we have to discuss in a different way. One is the case when we move two Lagrangian submanifolds L(0) and L(1) by the same symplectic diffeomorphism, the other is the case when we move L(0) and L(1) by different Hamiltonian isotopies. In the first case, Floer cohomology over Λ0,nov is preserved, while in the second case, only the Floer cohomology over Λnov (see (Conv.4) in Chapter 1) is preserved. We will discuss this point in Section 5.3 and again in Chapter 6 when we prove Theorem J. More precisely, we have the following two theorems. We first consider the following situation. Let (L(0) , L(1) ) and (L(0) , L(1) ) be relatively spin pairs of Lagrangian submanifolds of symplectic manifolds M and M  respectively. Let ψ : M → M  be a symplectic diffeomorphism such that ψ(L(0) ) = L(0) , ψ(L(1) ) = L(1) and preserving relative spin structure. We fix families of compatible almost complex structures Jt , Jt as well as other choices to define the filtered A∞ bimodules C(L(1) , L(0) ; Λ0,nov ), C(L(1) , L(0) ; Λ0,nov ) in Section 3.7.  (0) ), b(1) ∈ M(L  (1) ), b(0) ∈ M(L  (0) ), and Theorem 4.1.4. Let b(0) ∈ M(L (1) (0) (0) (1)  b ∈ M(L ). Suppose that ψ∗ ([b ]) = [b ] and ψ∗ ([b ]) = [b(1) ]. Then there exists an isomorphism (1)

ψ∗ : HF ((L(1) , b(1) ), (L(0) , b(0) ); Λ0,nov ) ∼ = HF ((L(1) , b(1) ), (L(0) , b(0) ); Λ0,nov ). Moreover, if {ψs }s (0 ≤ s ≤ 1) is an isotopy of symplectic diffeomorphisms such that ψs (L(0) ) = L(0) , ψs (L(1) ) = L(1) for each s, then ψ0∗ is equal to ψ1∗ . If we consider the case when M = M  and L(i) = L(i) (i = 0, 1) in Theorem 4.1.4, then it implies that Floer cohomology does not depend on the various choices involved in the definition. For example, let us consider the case when dimR M = 4, c1 (M ) = 0 and the Maslov indices of L(0) and L(1) vanish. Then for generic compatible almost complex structures on M , there is no pseudo-holomorphic disc bounding L(0) , L(1) . Hence our filtered A∞ structure is equal to the classical one. In particular, we can simply take b0 = 0, b1 = 0 as their bounding cochains. However the Floer cohomology HF ((L(1) , 0), (L(0) , 0); Λ0,nov ) depends on the choice of the almost complex structure. We can explain this phenomenon using Theorems 4.1.1, 4.1.4 as follows. If we change almost complex structures from J to J  , then we have a filtered A∞ homomorphism f : (C(L(i) ; Λ0,nov ), m) −→ (C(L(i) ; Λ0,nov ), m).

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(Note that since quantum effects are both zero, our filtered A∞ algebras do not  (i) ) → M(L  (i) ) is in general depend on J.) The induced homomorphism f∗ : M(L non trivial. Hence by Theorem 4.1.4 we have an isomorphism HF ((L(1) , 0), (L(0) , 0); Λ0,nov ) ∼ = HF ((L(1) , f∗ (0)), (L(0) , f∗ (0)); Λ0,nov ). This is an example of the wall crossing phenomenon we mentioned at the beginning of this section. Theorem 4.1.4 is deduced from Theorem 5.3.1 in Subsection 5.3.1 using an algebraic argument explained in Subsection 5.3.7. We next consider the following situation. Let (L(0) , L(1) ) and (L(0) , L(1) ) be relative spin pairs of Lagrangian (0) (1) submanifolds of a symplectic manifold M . Let {ψρ }ρ and {ψρ }ρ (0 ≤ ρ ≤ (0) (1) (0) 1) be Hamiltonian isotopies such that ψ0 = ψ0 = id, ψ1 (L(0) ) = L(0) and (1) ψ1 (L(1) ) = L(1) .  (0) ), b(1) ∈ M(L  (1) ), b(0) ∈ M(L  (0) ), and Theorem 4.1.5. Let b(0) ∈ M(L (0) (1)  (1) ). Suppose that ψ ([b(0) ]) = [b(0) ] and ψ ([b(1) ]) = [b(1) ]. We b(1) ∈ M(L 1∗ 1∗ consider the Floer cohomology over Λnov -coefficients instead of Λ0,nov . Then there exists an isomorphism ({ψρ(1) }ρ , {ψρ(0) }ρ )∗ :HF ((L(1) , b(1) ), (L(0) , b(0) ); Λnov ) ∼ = HF ((L(1) , b(1) ), (L(0) , b(0) ); Λnov ). (i)

Moreover, if {ψρ,s }ρ,s (i = 0, 1) be a two parameter family of hamiltonian (i) (i) isotopies such that ψ0,s = id and ψ1,s (L(i) ) = L(i) for each s, then we have (1)

(0)

(1)

(0)

({ψρ,0 }ρ , {ψρ,0 }ρ )∗ = ({ψρ,1 }ρ , {ψρ,1 }ρ )∗ . Here we restrict ourselves to Hamiltonian isotopies, because we need to show the energy estimate in Subsection 5.3.5. There are versions of Theorems 4.1.1-4.1.5 including unit (or homotopy unit) and also the versions which include the deformation by elements of H 2 (M ; Λ+ 0,nov ) we introduced in Section 3.8. We will state them later in Subsection 5.3.7 where we prove them. Theorem G (G.3) then will follow from it. The proof of Theorems A and F is completed in Section 5.4. The main part of Section 5.4 is devoted to more discussion on an algebraic side of the story. The main result there is Theorem W (= Theorem 5.4.2) and Theorem 5.4.18, which are proved in Subsection 5.4.4. Another purpose of Section 5.4 is to describe our M(L) as a zero set of a certain formal map (Kuranishi map). Namely we will prove Theorem M in Section 5.4. We also show that Floer cohomology can be regarded as an object of derived category of coherent sheaves on the product M(L(1) ) × M(L(0) ) regarded as formal scheme. Namely we prove Theorem N in Subsection 5.4.6. We also show that under certain assumption on Maslov class of L the Floer cohomology is defined on a ring a bit smaller than Λ0,nov . Especially it is defined on Λ0 defined in Subection 1.4 in case the Maslov class L is trivial. We use it in Subsection 5.4.6 to show Corollaries O and P. From now on we put ψ∗ = (ψ −1 )∗ where (ψ −1 )∗ : H ∗ (X) → H ∗ (Y ) is the induced by the diffeomorphism ψ : X → Y .

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197

4.2. Homotopy equivalence of A∞ algebras: the algebraic framework 4.2.1. Models of [0, 1] × C. In the next subsection, we define two (filtered) A∞ homomorphisms to be homotopic each other. There are various ways to define it and there are various references ([Kel01, MSS02, Smi00]) about it. To unify several approaches we proceed as follows. In algebraic topology, homotopy between two maps f, f  : X → Y is defined by a map H : [0, 1] × X → Y . So we start with defining an algebraic analog of the operation [0, 1]×, taking the product with [0, 1], for the context of (filtered) A∞ algebras. Let R be a commutative ring with unit. Let C be an A∞ algebra over R and let C be a filtered A∞ algebra over Λ0,nov (R). Here Λ0,nov (R) is the universal Novikov ring over R. See (Conv.4) in Chapter 1. We always assume that C and C are free and countably generated over R or Λ0,nov (R) and C is a R reduction of C (Definition 3.2.20). Hereafter we write Λ0,nov in place of Λ0,nov (R) in case no confusion can occur. Definition 4.2.1. An A∞ algebra C together with A∞ homomorphisms Incl : C → C,

Evals=0 : C → C,

Evals=1 : C → C

is said to be a model of [0, 1] × C if the following holds: (4.2.2.1) Inclk : Bk C[1] → C[1] is zero unless k = 1. The same holds for Evals=0 and Evals=1 . (4.2.2.2) Evals=0 ◦ Incl = Evals=1 ◦ Incl = identity. (4.2.2.3) Incl1 : (C,m) → (C, m) is a cochain homotopy equivalence and (Evals=0 )1 , (Evals=1 )1 : (C, m) → (C, m) are cochain homotopy equivalences. (4.2.2.4) The (cochain) homomorphism (Evals=0 )1 ⊕ (Evals=1 )1 : C → C ⊕ C is surjective. When C, C, Incl, Evals=0 , and Evals=1 are unital, we call C a unital model of [0, 1] × C. A filtered A∞ algebra C together with filtered A∞ homomorphisms Incl : C → C,

Evals=0 : C → C,

Evals=1 : C → C

is said to be a model of [0, 1] × C if the following holds. (4.2.3.1) Inclk : Bk C[1] → C is zero unless k = 1. The same holds for Evals=0 and Evals=1 . (4.2.3.2) Evals=0 ◦ Incl = Evals=1 ◦ Incl = identity. (4.2.3.3) Incl1 induces a cochain homotopy equivalence of the complex (C, m) → (C, m), and (Evals=0 )1 , (Evals=1 )1 induce cochain homotopy equivalences of the complex (C, m) → (C, m). (4.2.3.4) The homomorphism (Evals=0 )1 ⊕ (Evals=1 )1 : C → C ⊕ C is surjective. When C, C, Incl, Evals=0 , and Evals=1 are unital, we call C a unital model of [0, 1] × C. Remark 4.2.4. (1) We remark that in most of the cases (for example if R is a field or Z), (4.2.2.3) is equivalent to the condition that Incl1 , (Evals=0 )1 , (Evals=1 )1 induce isomorphisms on m1 cohomology. (Similar remark applies to (4.2.3.3).) Here

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we note that we assume C is free and countably generated R modules. Hence the map (Evals=0 )1 ⊕ (Evals=1 )1 in (4.2.2.4) turns out to be split surjective. (2) We remark that in (4.2.3.3) we only assume the cochain homotopy equivalences after reducing the coefficient ring Λ0,nov to R. This is because m1 ◦ m1 may be nonzero. (3) In the case of the models which we will construct below, we have: Incl1 = Incl1 , Eval1 = Eval1 . We do not take it as a part of the definition. Example 4.2.5. Let X be a smooth manifold. We take C = Λ(X) the de Rham complex of smooth forms on X and C = Λ(R × X) be the de Rham complex of smooth forms on R × X. Let Ii : X → R × X (i = 0, 1) be the inclusion Ii (x) = (i, x) and π : R × X → X be the projection. We take Incl1 = π ∗ and (Evals=i )1 = Ii∗ . Then clearly C satisfies Conditions (4.2.2.1) – (4.2.2.4). In the rest of this subsection, we will construct the models of [0, 1] × C for any filtered A∞ algebra C over Λ0,nov (R). Firstly, we will give an example of the model when R ⊇ Q. This model works for the most cases of our purpose. After that, we will construct the model of [0, 1] × C over general R, which leads us to be able to define the notion of homotopy equivalence over general R (for instance, over Z or Z/2Z). Now we begin to discuss the case that R ⊇ Q. The construction below is an A∞ analog to the model given in [GrMo81] Chapter X for the case of differential graded algebras. Let (C, m) be a filtered A∞ algebra. We consider ‘formal power series’   (4.2.6) x(s) = xi (s)T λi , y(s) = yi (s)T λi , i

i

where λi ≥ 0, limi→∞ λi = ∞. (We remark that λi are real numbers and is not necessary an integer. This is the reason we put ‘formal power series’ in the quote.) We assume that the coefficient functions xi : [0, 1] → C k [1] = C k+1 ,

yi : [0, 1] → C k−1 [1] = C k

satisfy one of the following conditions. (4.2.7.1) xi (s), yi (s) are polynomials of the variable s. (4.2.7.2) xi (s), yi (s) are smooth functions of s ∈ [0, 1]. (4.2.7.3) xi (s) is a continuous piecewise polynomial function of s ∈ [0, 1]. yi (s) is a piecewise polynomial function of s ∈ [0, 1] and is not necessarily continuous. (4.2.7.4) xi (s) is a continuous piecewise smooth function of s ∈ [0, 1]. yi (s) is a piecewise smooth function of s ∈ [0, 1] and is not necessarily continuous. For the cases of (4.2.7.2) and (4.2.7.4), we assume R = R or C. Remark 4.2.8. Note that we do not equip the vector space C with any particular topology other than the one induced by the energy filtration. By smoothness in the above definition we mean that each of the coefficients with respect to a basis is smooth. Definition 4.2.9. We define P oly([0, 1], C) to be the graded module such that P oly([0, 1], C)k+1 = P oly([0, 1], C)[1]k ∼ = P oly([0, 1], C[1])k

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199

is the set of all pairs (x, y) satisfying (4.2.7.1). We define the modules C ∞ ([0, 1], C), P P oly([0, 1], C) or P C ∞ ([0, 1], C) similarly with the condition (4.2.7.1) replaced by (4.2.7.2), (4.2.7.3) or (4.2.7.4) respectively. Let C be an A∞ algebra. We define P oly([0, 1], C) to be the graded module such that P oly([0, 1], C)k+1 = P oly([0, 1], C)[1]k ∼ = P oly([0, 1], C[1])k k+1

is the set of all pairs (x, y) such that x = x(s) is a C[1]k = C valued polynomial, k k−1 ∞ y = y(s) is a C[1] = C valued polynomial. C ([0, 1], C), P P oly([0, 1], C) or P C ∞ ([0, 1], C) are defined similarly. Remark 4.2.10. In the definition of P oly([0, 1], C)k we do not assume that the degree of xi (s) (as a polynomial of s) is bounded over i. So if we rewrite (4.2.6) as x(s) =

(4.2.11)

∞ 

x(k) (T )sk

k=0

where x(k) (T ) ∈ C, then the right hand side is not a finite sum. However it holds that x(k) (T ) ∈ F λk C and limk→∞ λk = ∞. In other words, x(s) is in the strictly convergent power series ring Λ0,nov s

(see [BGR84] 1.4.1 Definition 1). We next define a structure of filtered A∞ algebra on M ap([0, 1], C). From now on, M ap stands for one of P oly, C ∞ , P P oly or P C ∞ : M ap = P oly, C ∞ , P P oly or P C ∞ . We first define mk , k ≥ 1. Let xi = (x(i) , y (i) ) ∈ M ap([0, 1], C)di with (x(i) , y (i) ) as in Definition 4.2.9. In particular, di is the unshifted degree of x(i) (s) ∈ C. We consider the pair (˜ x, y˜) ∈ M ap([0, 1], C)di +1 defined by x ˜(s) = mk (x(1) (s), · · · , x(k) (s)), and (4.2.12.1)

y˜(s) =

k 

(−1)d1 +···+dj−1 +j mk (x(1) (s), · · · , y (j) (s), · · · , x(k) (s)),

j=1

for k = 1 and y˜(s) =

(4.2.12.2)

∂x(s) − m1 (y(s)), ∂s

for k = 1. We then define (4.2.12.3)

mk (x1 , · · · , xk ) = (˜ x, y˜)

for k ≥ 1.

Let x0 = m0 (1) ∈ C. (Here m0 is a part of the A∞ structure of C). We then define m0 on M ap([0, 1], C) by (4.2.12.4)

m0 (1) = (x0 , 0) ∈ M ap([0, 1], C).

In case C is an A∞ algebra we define mk on M ap([0, 1], C) by the same formulas as (4.2.12.1)-(4.2.12.3) and m0 = 0. Then we can show the following.

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Lemma 4.2.13. (M ap([0, 1], C), m) (resp. (M ap([0, 1], C), m)) is a filtered A∞ algebra (resp. an A∞ algebra). If C has a strict unit, (M ap([0, 1], C), m) also has a strict unit. The same holds for C and (M ap([0, 1], C), m). Proof. We prove the case of a filtered A∞ algebra only. We need to calculate  (1) , · · · , x(k) ). ( x, y) := (m ◦ d)(x  = 0. We now calculate It easily follows from the A∞ formula of (C, m) that x y = y(s). We put (j) x(s)

= x(1) (s) ⊗ · · · ⊗ x(j−1) (s),

x(j) (s) = x(j+1) (s) ⊗ · · · ⊗ x(k) (s), and Δm−1 x =



(m) x(1) a ⊗ · · · ⊗ xa ,

a

m−1

Δ

(j) x

=



(1) (j) xa

⊗ ··· ⊗

(m) (j) xa ,

a

m−1

Δ

x(j) =



(1)

(m)

x(j)a ⊗ · · · ⊗ x(j)a .

a

Then we have y(s) =

 a

+

(−1) 1 (a)+1 m



(1) (j) xa (s), m



 (2) x (s) , (j) a

j

 a

j

a

b

(3) (j) (s), x(j) (s) (j) xa (s), y

(−1) 2 (a)+1 m



(j) x(s), y

(j)



(1)

(s), x(j)a (s),

   (2) (3) m x(j)a (s) , x(j)a (s)    (2) + (−1) 3 (a) m (j) x(1) (s), m (j) xa (s), a

(4.2.14)

j

  (1) (2) y (j) (s), x(j)b (s) , x(j)b (s)    ∂x(j) (s) , x(j) (s) m (j) x(s), − ∂s j +

∂ mk (x(1) (s), · · · , x(k) (s)), ∂s

where

1 (a) = deg = deg

2 (a) = deg



3 (a) = deg = deg

(1) (j) xa (j) x (j) x

+ deg

(j)

+ deg

+ deg

(1) (j) xa

(1) (j) xa

+ deg y

(1) (j) xa (j) x

+ deg

+ deg

(2) (j) xa

+ 1 + deg

+ 1,

(s) + 1 + deg x(j)a + deg(j) x, (1)

(2) (j) xa

+ deg

(1) (j) xa .

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(1) (j) xa ,

(3) (j) xa

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201

Here deg is the shifted degree. The sum of the first three lines of (4.2.14) is zero by the A∞ formula for (C, m). The last two lines of (4.2.14) cancel each other. We have thus verified the A∞ formulae. If e is a unit of C, then (e, 0) is a unit of M ap([0, 1], C). The proof of Lemma 4.2.13 is complete.  Definition-Proposition 4.2.15. If R ⊇ Q, then (M ap([0, 1], C), m) is a model of [0, 1] × C and (M ap([0, 1], C), m) a model of [0, 1] × C. Proof. We take s0 ∈ [0, 1] and define an A∞ homomorphism Evals=s0 : M ap([0, 1], C) → C by (Evals=s0 )1 (x) = x(s0 ) ∈ C for x = (x, y) and (Evals0 )k = 0 for k = 1. We remark that (4.2.11) is not a polynomial but is a formal power series of s. But we can still put x(s0 ). This is because the right hand side of (4.2.11) converges in the adic topology of Λ0,nov as was remarked in Remark 4.2.10. It is easy to check that Evals=s0 is a filtered A∞ homomorphism. We next define Incl1 : C[1] → M ap([0, 1], C[1]), by regarding an element of C[1] as a constant map contained in M ap([0, 1], C[1]). It is easy to see that by putting Inclk = 0 for k = 1, we obtain a filtered A∞ homomorphism. We define Evals=s0 and Incl in the same way. It is easy to check (4.2.3.1), (4.2.3.2), (4.2.3.4), (4.2.2.1), (4.2.2.2), (4.2.2.4). Hence to prove Proposition 4.2.15, it suffices to show the following lemma.  Lemma 4.2.16. If R ⊇ Q, then Incl induces an isomorphism of m1 cohomology: H(C, m1 ) → H(M ap([0, 1], C), m1 ). Proof. The image of the map Incl1 is a sub-complex of (M ap([0, 1], C), m1 ) and the quotient complex M ap([0, 1], C)/ Im Incl1 is isomorphic to the sub-complex of M ap([0, 1], C) that consists of the elements x of the type x(s) = (sx(s), y(s)). We denote this sub-complex by (M ap ([0, 1], C), m1 ). Now we consider the case of (P oly  ([0, 1], C), m1 ). (The other case is similar.) We have an exact sequence 0 → (C, m1 ) ⊗ Q[s] → (P oly  ([0, 1], C), m1 ) → (C, m1 ) ⊗ Q[s] → 0. i

π

Diagram 4.2.1 Here Q[s] is the ring of polynomials. The maps i and π are defined by i(x(s)) = (0, −x(s)), π(sx(s), y(s)) = x(s). It follows from (4.2.12.2) that the connecting homomorphism of the long exact sequence induced by Diagram 4.2.1 is given by (4.2.17)

x(s) → −

∂ (sx(s)). ∂s

This map is an isomorphism on Q[s]. We remark that this is the place where we need to assume 'Q ⊆ R, because the inverse of (4.2.17) involves integration, i.e, is s given by g → 1s 0 g(t) dt. This proves the lemma for the case of P oly. The other cases can be proved in a similar way. The proof of Lemma 4.2.16 is complete.  We have thus constructed an example of the model of [0, 1] × C in the case R ⊇ Q. The model P oly([0, 1], C) works for the most cases of our purpose. However

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this model cannot be used if R has a torsion. So we will construct another model in general. From now on, we do not assume that R ⊇ Q in the rest of this subsection. We put C [0,1] = C ⊕ C[−1] ⊕ C. We define I0 , I1 : C → C [0,1] and J : C → C [0,1] by I0 (x) = (x, 0, 0),

I1 (x) = (0, 0, x),

J(x) = (0, x, 0).

Note that I0 , I1 preserve degree and J is of degree +1. We extend I0 ,I1 to BC[1] → BC [0,1] [1] and denote it by the same symbol. We are going to define an A∞ structure on C [0,1] , which will give another model of C. In the course of the construction of a filtered A∞ algebra on C [0,1] , we denote a filtered A∞ structure on C [0,1] by Mk and a filtered A∞ structure on C by mk . We define ⎧ deg x J(x), ⎪ ⎨ M1 (I0 (x)) = I0 (m1 x) + (−1)  deg x (4.2.18) M1 (I1 (x)) = I1 (m1 x) − (−1) J(x), ⎪ ⎩ M1 (J(x)) = J(m1 x), and  (4.2.19)

(4.2.20)

(Evals=0 )1 (x, y, z) = x, (Evals=1 )1 (x, y, z) = z, (Incl)1 (x) = I0 (x) + I1 (x) = (x, 0, x),

M0 (1) = (Incl)1 (m0 (1)).

Remark 4.2.21. (1) C [0,1] has the following geometric origin. We consider the case C = C(L). Then we consider the locally finite singular chain of (− , 1 + ) × L of the form (− , 1/2] × σ, [1/2, 1 + ) × σ, {1/2} × σ with σ ∈ C(L). The set of such chains defines a chain complex which is isomorphic to our C [0,1] . (We put m0 = 0 in this remark.) But if we consider product structure (cup product) in this geometric model, we are in trouble since the chains are not transversal to one another. (For example {1/2}×σ is not transversal to {1/2}×σ  .) To define a product structure we need to somehow break the symmetry as we will do in Chapter 7. The construction below might be regarded as its algebraic analog. (2) We remark although we write (− , 1 + ) × L, not L × (− , 1 + ) above, the sign we put in (4.2.18) etc. are ones in case we consider L × (− , 1 + ). Now we define Mk for k ≥ 2 on C [0,1] . It suffices to define the operators on the elements of the form I0 (x), I1 (x) and J(x). Let x ∈ Bk C, z ∈ B C, y ∈ C. (k,  = 0, 1, 2, · · · .) We define (4.2.22)



Mk++1 (I0 (x), J(y), I1 (z)) = (−1)deg z J(mk++1 (x, y, z)),

and (4.2.23)

Mk (I0 (x)) = I0 (mk (x)),

M (I1 (z)) = I1 (m (z)),

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203

for k,  ≥ 2. The operator Mk has values zero for all the elements except those given in (4.2.18), (4.2.20), (4.2.22), (4.2.23). This means the following M(· · · , J(y1 ), · · · , J(y2 ), · · · ) = 0, M(· · · , J(y1 ), · · · , I0 (x), · · · ) = 0, M(· · · , I1 (z), · · · , J(y), · · · ) = 0, M(· · · , I1 (z), · · · , I0 (x), · · · ) = 0, Mk+ (I0 (x), I1 (z)) = 0.

(4.2.24.1) (4.2.24.2) (4.2.24.3) (4.2.24.4) (4.2.24.5)

Then we can show the following. Lemma 4.2.25. (C [0,1] , Mk ) is a filtered A∞ algebra. Proof. We will check the A∞ relation M ◦ d = 0. We use (Conv.3) at the end of Chapter 1. Firstly we calculate  0 (x), J(y), I1 (z)) (M ◦ d)(I   (2;1)  (2;1) (2;2) (2;1) (2;2) (−1)deg xc1 +deg zc2 M(I0 (x(2;1) = c1 ), J(m(xc1 , y, zc2 )), I1 (zc2 )) c1 ,c2

+



(−1)deg



x(3;1) c



z(3;1) +deg x+deg y+1 c

M(I0 (x(3;1) ), M(I0 (x(3;2) )), I0 (x(3;3) ), J(y), I1 (z)) c c c

c

+



(−1)deg

c

M(I0 (x), J(y), I1 (z(3;1) ), M(I1 (z(3;2) )), I1 (z(3;3) )). c c c Here in the power of (−1) on the last line, +1 stands for deg J = +1. Thus we obtain  0 (x), J(y), I1 (z)) (M ◦ d)(I   (2;1)  (2;2) (2;1) (2;2) (−1)deg xc1 +deg z J ◦ m(x(2;1) = c1 , m(xc1 , y, zc2 ), zc2 ) c1 ,c2

+

  (3;1)  (−1)deg xc +deg z J ◦ m(x(3;1) , m(x(3;2) ), x(3;3) , y, z) c c c c

  (3;1)    + (−1)deg zc +deg x+deg y−1+deg z+1 c

J ◦ m(x, y, z(3;1) , m(z(3;2) ), z(3;3) ). c c c Here on the last line, deg z + 1 = deg(zc the A∞ relation on C.

(3;1)

(3;2)

, m(zc

(3;3)

), zc

(3;2)

). Then this is zero by

Remark 4.2.26. We remark that M(I0 (xc )) in the third line of the first (3;2) equality above contains the case when xc ∈ B1 C or ∈ B0 (C). In such case (3;2) (3;2) M(I0 (xc )) = I0 (m(xc )). However using (4.2.24) the term involving )) − I0 (m(x(3;2) )) M(I0 (x(3;2) c c vanishes.

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4.

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 0 (x)). We put x = x− ⊗ xlast where xlast ∈ B1 C[1]. We next calculate (M ◦ d)(I Then we have:  0 (x)) (M ◦ d)(I   (3;1) (−1)deg xc M(I0 (x(3;1) ), I0 (m(x(3;2) )), I0 (x(3;3) )) = c c c c 

(4.2.27)



+ (−1)deg x− +deg xlast M(I0 (x− ), J(xlast ))   (3;1) (−1)deg xc I0 (m(x(3;1) , m(x(3;2) ), x(3;3) )) = c c c c

+ (−1)deg + (−1)



x− +deg xlast

deg m(x)

J(m(x− , xlast ))

J(m(x)). (3;2)

Here the last line comes from the case when xc = x in the the first line. The last two lines cancel to each other. Hence (4.2.27) is zero.  1 (z)). We put z = zfirst ⊗ z− where zfirst ∈ B1 C[1]. We next calculate (M ◦ d)(I Then we have:  1 (z)) (M ◦ d)(I   (3;1) (−1)deg zc M(I1 (z(3;1) ), I1 (m(z(3;2) )), I1 (z(3;3) )) = c c c c 

(4.2.28)

− (−1)deg zfirst M(J(zfirst ), I1 (z− ))   (3;1) (−1)deg zc I1 (m(z(3;1) , m(z(3;2) ), z(3;3) )) = c c c c

− (−1)deg − (−1)



zfirst +deg z−

deg m(z)

J(m(zfirst , z− ))

J(m(z)). (3;2)

Here the last line comes from the case when xc = x in the the first line. The last two lines cancel to each other. Hence (4.2.28) is zero.  0 (x), I1 (z)) for x ∈ Bk C[1], z ∈ B C[1], k,  > 0. Finally we calculate (M ◦ d)(I We have:  0 (x), I1 (z)) (M ◦ d)(I = (−1)deg (4.2.29)

− (−1) = (−1)



x− +deg xlast

M(I0 (x− ), J(xlast ), I1 (z))

deg x+deg zfirst

M(I0 (x), J(zfirst )I1 (z− ))

deg x− +deg xlast +deg z

− (−1)deg





J(m(x− , xlast , z))



x+deg zfirst +deg z−

J(m(x, zfirst , z− ))

= 0. The proof of Lemma 4.2.25 is complete.



Definition-Lemma 4.2.30. (C [0,1] , Mk ) is a model of [0, 1] × C. If C has a unit then (C [0,1] , Mk ) also has a unit. Proof. We defined Evals=i and Incl already. It is easy to see that they induce cochain homotopy equivalences. If e is a unit of C, then (e, 0, e) is a unit of (C [0,1] , Mk ). 

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4.2. THE ALGEBRAIC FRAMEWORK

We can define (C thus proved

[0,1]

205

, Mk ) in the same way. It is a model of [0, 1] × C. We have

Theorem 4.2.31. The model C (resp. C) of [0, 1] × C (resp. [0, 1] × C) always exists. If C is gapped, then C can be taken as a gapped one. If C (resp. C) has a unit, then C (resp. C) can be taken as a unital model. Remark 4.2.32. We have defined C [0,1] by directly defining its operations. We can also prove Theorem 4.2.31 by proving the existence of the operations inductively via the obstruction theory. Remark 4.2.33. We remark that the A∞ algebra P oly([0, 1], C) is better than

[0,1]

C in various sense. For example, in case C is a (graded) commutative D.G.A., then the A∞ algebra P oly([0, 1], C) is also graded commutative D.G.A. On the other [0,1]

is always noncommutative. The authors do not know how to construct hand, C a model of [0, 1] × C over the torsion coefficient preserving commutativity. [0,1]

is somewhat similar to the model used by D. Sullivan in We remark that C the context of D.G.A. [Sul78]. However the model used by Sullivan is commutative and works only for the case of the minimal D.G.A. The following is the key result to establish basic properties of homotopy between (filtered) A∞ homomorphisms. Theorem 4.2.34. Let C1 ,C2 be gapped filtered A∞ algebras and C1 ,C2 be any of the models of [0, 1]×C1 , [0, 1]×C2 , which are gapped. Let f : C1 → C2 be a gapped filtered A∞ homomorphism. Then there exists a gapped filtered A∞ homomorphism F : C1 → C2 such that Evals=s0 ◦ F = f ◦ Evals=s0 for s0 = 0, 1 and Incl ◦ f = F ◦ Incl . The unfiltered version also holds. If f is unital, then so is F. If f is strict, then so is F. Incl

C1 −−−−→ ⏐ ⏐f  Incl

Eval

⊕Eval

Eval

⊕Eval

C1 −−−−s=0 −−−−−−s=1 −→ C1 ⊕ C1 ⏐ ⏐ ⏐F ⏐f⊕f  

C2 −−−−→ C2 −−−−s=0 −−−−−−s=1 −→ C2 ⊕ C2 Diagram 4.2.2 We say that F is compatible with Incl and Eval over f : C1 → C2 , if these two relations hold. We will prove Theorem 4.2.34 later in Section 4.4. 4.2.2. Homotopies between A∞ homomorphisms. From now on we assume that all filtered A∞ algebras and filtered A∞ homomorphisms are gapped. Definition 4.2.35. Let C1 ,C2 be filtered A∞ algebras and f, g : C1 → C2 filtered A∞ homomorphisms between them. Let C2 be a model of [0, 1] × C2 . We say f is homotopic to g via C2 and write f ∼C2 g, if there exists a filtered A∞ homomorphism F : C1 → C2 such that Evals=0 ◦F = f, Evals=1 ◦F = g. We call F the homotopy between f and g.

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Lemma 4.2.36. ∼C2 is independent of the choice of C2 . Proof. Let C2 be another model of [0, 1] × C2 . We assume f ∼C2 g and F is as in Definition 4.2.35. We apply Theorem 4.2.34 to the identity C2 → C2 and obtain ID : C2 → C2 which commutes with Eval. Now we put F = ID ◦ F. It is easy to see that Evals=0 ◦F = f, Evals=1 ◦F = g. Hence f ∼C2 g as required.  From now one we write ∼ in place of ∼C2 . Proposition 4.2.37. ∼ is an equivalence relation. Proof. f ∼ f is obvious. Let f(m) : C1 → C2 be filtered A∞ homomorphisms. We assume f(1) ∼ f(2) . Let C2 be a model of [0, 1] × C2 and F : C1 → C2 a homotopy from f(1) to f(2) . We consider another model C2 of [0, 1] × C2 as follows. C2 = C2 as an filtered A∞ algebra. We use the same Incl but we exchange Evals=0 with Evals=1 . Namely Evals=0 of C2 is Evals=1 of C2 and vice versa. Then F regarded an a filtered A∞ homomorphism from C1 to C2 is a homotopy from f(2) to f(1) . Namely f(2) ∼ f(1) . Next we assume that f(1) ∼ f(2) and f(2) ∼ f(3) . Let F(m) : C1 → C2 be homotopies from f(m) to f(m+1) . We are going to show f(1) ∼ f(3) . We define a filtered A∞ algebra C2 as follows. ⎧  C2 = {(x, y) ∈ C2 ⊕ C2 | deg x = deg y, Evals=1 (x) = Evals=0 (y)}. ⎪ ⎪ ⎪ ⎨ mk ((x1 , y1 ), · · · , (xk , yk )) = (mk (x1 , · · · , xk ), mk (y1 , · · · , yk )). ⎪ Evals=0 (x, y) = Evals=0 (x), Evals=1 (x, y) = Evals=1 (y), ⎪ ⎪ ⎩ Incl(x) = (Incl(x), Incl(x)). It is easy to see that C2 is a model of [0, 1] × C2 . We put F(1) #F(2) : C1 → C2 by (F(1) #F(2) )(z) = (F(1) (z), F(2) (z)). It is easy to see that F(1) #F(2) is a filtered A∞ homomorphism and is a homotopy from f(1) to f(3) . The proof of Proposition 4.2.37 is now completed.  Remark 4.2.38. The result similar to Proposition 4.2.37 is proved in the case of differential graded algebra in Corollary 10.7 [GrMo81]. However in the category of differential graded algebra and differential graded homomorphism, the result is correct only under additional assumption that C1 is minimal. Working in the category of (filtered) A∞ algebra thus makes the story simpler. Now we make a brief comparison of our definition with other definitions in the [0,1] [0,1] literature. Let us take C 2 as a model of C 2 . Let F : C 1 → C 2 be an A∞ (m)

homomorphism such that f

[0,1]

C2

= Evals=m ◦ F. We recall that

= I0 (C 2 ) ⊕ J(C 2 ) ⊕ I1 (C 2 ). (m)

Since we assume Evals=m ◦F = f of degree +1 such that

(0)



, it follows that there exists hk : Bk C1 [1] → C2 [1] 

(1)

Fk = (I0 ◦ fk ) ⊕ (J ◦ hk ) ⊕ (I1 ◦ fk ). We put





hk (x) = (−1)deg x hk (x).

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4.2. THE ALGEBRAIC FRAMEWORK

207

From the definition of A∞ structure of C, we derive that F is an A∞ homomorphism if and only if the following holds.  (0) (1)  (3;1) (−1)deg xc m(f (x(3;1) ), h(x(3;2) ), f (x(3;3) )) c c c (4.2.39)

c (0)

=f

(1)

(x) − f

(x) −



(−1)deg



x(3;1) c

h(x(3;1) , m(x(3;2) ), x(3;3) ). c c c

c

ˆ−d ˆ ◦ F)(x) ◦d  = 0.) ((4.2.39) is obtained by taking J(C 2 ) component of (F Hence, we have proved the following: (0)

: C 1 → C 2 is homotopic Proposition 4.2.40. An A∞ homomorphism f (1) to f : C 1 → C 2 if and only if there exists a sequences of homomorphisms hk : Bk C 1 [1] → C 2 [1] of degree −1 satisfying (4.2.39). The same holds for the filtered versions. (4.2.39) is indeed the definition of homotopy between A∞ homomorphisms which has been used in several literatures. (For example, in section 4.1 [Smi00]. It was also used in Section A5 of the preprint version [FOOO00] of this book.) In [Fuk02II], the first named author studied homotopy equivalence of A∞ categories. A∞ category with one object is an A∞ algebra. We also find easily that A∞ functor between two A∞ categories with one object is an A∞ homomorphism. If we apply the definition of two A∞ functors being homotopic given in Definition 8.5 [Fuk02II], it is easy to see that the definition is also equivalent to the existence of hk satisfying (4.2.39). (In fact two A∞ functors are defined to be homotopic if there is a natural transformation satisfying appropriate properties. The homomorphism hk above is such a natural transformation.) We next write down the condition for two filtered A∞ homomorphisms to be homotopic when we use the model P oly([0, 1], C2 ). An A∞ homomorphism h : C 1 → P oly([0, 1], C 2 ) can be written as h = (f(s), g(s)), where f(s) : BC 1 [1] → C 2 [1][s], g(s) : BC 1 [1] → C 2 [1][s] and f(s), g(s) preserve the degree. The condition that h is an A∞ homomorphism can be written as (4.2.41.1) (4.2.41.2)

f(s0 ) is an A∞ homomorphism : C 1 → C 2 for each s0 ∈ [0, 1].

∂f(s)  = m ◦ (f(s) ⊗  g(s) ⊗ f(s)) ◦ Δ2 − g(s) ◦ d. ∂s We do not check (4.2.41) here since we do not use it. We now turn to the study of basic properties of the homotopy. We know from Lemma 4.2.36 that the homotopy relation is independent of the choice of a model of [0, 1] × C. So, from now on, we use the script letter C for the model of [0, 1] × C without specifying the choice. (We will check this independence of the choice, only when it is not manifest.) Definition 4.2.42. An A∞ homomorphism f : C 1 → C 2 is called a homotopy equivalence if there exists an A∞ homomorphism g : C 2 → C 1 such that f ◦ g and g ◦ f are homotopic to identity. Two A∞ algebras are homotopy equivalent to each other if there exists a homotopy equivalence between them.

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The homotopy equivalences between (gapped) filtered A∞ algebras can be defined in a similar way. To show that the homotopy equivalence of A∞ algebras is an equivalence relation we need the following lemma. Lemma 4.2.43. Let f(m) , f(m) : Cm → Cm+1 be filtered A∞ homomorphisms. We assume f(m) ∼ f(m) . Then we have f(2) ◦ f(1) ∼ f(2) ◦ f(1) . The same holds for filtered A∞ homomorphisms. Proof. It is easy to see that f(2) ◦ f(1) ∼ f(2) ◦ f(1) . Hence we may assume f = f(2) . We apply Theorem 4.2.34 to f(2) and obtain a filtered A homomorphism F(2) : C2 → C3 such that Evals=s0 ◦ F(2) = f(2) ◦ Evals=s0 . Now let F : C1 → C2 be the homotopy between f(1) and f(1) . Then F(2) ◦ F is a homotopy from f(2) ◦ f(1) to f(2) ◦ f(1) .  (2)

The following corollary is immediate. Corollary 4.2.44. (1) A composition of homotopy equivalences is a homotopy equivalence. (2) The homotopy equivalence is an equivalence relation. We will prove the following theorem in Section 4.5. We recall that an A∞ homomorphism between unfiltered A∞ algebras is said to be a weak homotopy equivalence if it induces a chain homotopy equivalence and a filtered A∞ homomorphism f : C1 → C2 between filtered A∞ algebras (satisfying m0 = 0) is said to be a weak homotopy equivalence if f : C 1 → C 2 is a weak homotopy equivalence (Definitions 3.2.10 and 3.2.33). A filtered A∞ homomorphism f is called strict if f0 = 0 (Definition 3.2.29). Theorem 4.2.45. (Whitehead theorem for A∞ algebras) (1) A weak homotopy equivalence of A∞ algebra is a homotopy equivalence. (2) A gapped weak homotopy equivalence between gapped filtered A∞ algebras is a homotopy equivalence. The homotopy inverse of a strict weak homotopy equivalence can be taken to be strict. A similar result appears in the literature which leads that, for any weak homotopy equivalence f : C 1 → C 2 there exists an A∞ homomorphism g : C 2 → C 1 which induces inverse in the m1 cohomology. We remark that Theorem 4.2.45 (1) is slightly sharper than this statement, since A∞ homomorphism which induces identity on m1 cohomology may not be homotopic to identity. In a similar way we can define homotopy between (filtered) A∞ homomorphisms of (filtered) A∞ bimodules. We will discuss this later in Section 5.2 Chapter 5. 4.2.3. The unital or homotopy-unital cases. In this subsection, we specialize our study of homotopy to the cases of unital or homotopy-unital (filtered) A∞ homomorphisms. (See Subsection 3.3.2 for the definitions of unital or homotopyunital A∞ homomorphisms.) The main purpose of the subsection is to fix some notations and derive basic facts on them, which will be used in later sections. Although we explicitly discuss only the filtered version here, all the corresponding discussions equally apply to the unfiltered cases without change. We also assume that all filtered A∞ algebras and filtered A∞ homomorphisms are gapped. Let C1 , C2 be two unital filtered A∞ algebras. We take models Ci of [0, 1] × Ci . We may assume that they are unital.

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209

Definition 4.2.46. We say that unital filtered A∞ homomorphisms f, g : C1 → C2 are unitally homotopic to each other and write f ∼ g (or f ∼u g) if there exists a unital filtered A∞ homomorphism h : C1 → C2 such that Evals=0 ◦ h = f, Evals=1 ◦ h = g. We call h a unital homotopy between f and g. We define a strict unital homotopy between strict filtered A∞ algebras in the same way. We will use symbol ∼su for it. We sometimes say, for short, u-homotopic, su-homotopic, u-homotopy, suhomotopy in place of unitally homotopic, strictly-unitally homotopic, unital homotopy, strictly unital homotopy. Lemma 4.2.47. ∼u is independent of the choice of the model C2 and is an equivalence relation. If f ∼u f then f ◦ g ∼u f ◦ g and g ◦ f ∼u g ◦ f . The strict version is similar. The proof is the same as the proof of Lemma 4.2.36, Proposition 4.2.37 and Lemma 4.2.43. We can now define the notion of unital homotopy equivalence (uhomotopy equivalence) in an obvious way. Theorem 4.2.45a. The unital version of Theorem 4.2.45 holds. Moreover, if f : C1 → C2 is a unital filtered A∞ homomorphism and a homotopy equivalence, then it is a u-homotopy equivalence. The strict unital version also holds. We remark that the second half follows from the first half. The proof of the first half is given in Section 4.5. Let C1 , C2 be two homotopy-unital filtered A∞ algebras. We take Ci+ as in + Section 8.1. We take a model C+ 2 of [0, 1] × C2 , which is unital. Remark 4.2.48. It is possible to show that there exists a model of [0, 1] × C2 which is homotopy-unital and is unique in a sense similar to Theorem 4.2.34. Then we might use this model to define the homotopy-unital homotopy. However in the way we take here we do not need to prove it. Definition 4.2.49. We say that homotopically unital filtered A∞ homomorphisms f, g : C1 → C2 are homotopy-unitally homotopic and write f ∼ g (or f ∼hu g), if there exists a unital filtered A∞ homomorphism h : C1+ → C+ 2 such that Evals=0 ◦ h = f+ , Evals=1 ◦ h = g+ . Here f+ , g+ : C1+ → C2+ are as in Definition 3.3.13. We call h a homotopy-unital homotopy between f and g. We sometimes call hu-homotopic or hu-homotopy in place of homotopy-unitally homotopic or homotopy-unital homotopy. The strict version can be defined in a similar way and abbreviated by shu-homotopic or shu-homotopy. Lemma 4.2.50. ∼hu is independent of the choice of the model C2 and is an equivalence relation. If f ∼hu f then f ◦ g ∼hu f ◦ g and g ◦ f ∼hu g ◦ f . The same holds for ∼shu The proof is the same as that of Lemma 4.2.47. We can now define the notion of homotopy-unital homotopy equivalence (hu-homotopy equivalence) in an obvious way. Theorem 4.2.45b. The homotopy-unital version of Theorem 4.2.45 holds. Moreover, if f : C1 → C2 is a hu-filtered A∞ homomorphism and a homotopy equivalence, then it is a hu-homotopy equivalence. The strict and/or homotopyunital version also holds.

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The proof will be given in Section 4.5. Before going further, let us remind one technical but delicate issue which is likely to be confused. Remark 4.2.51. We have defined a homotopy-unital filtered A∞ homomorphism f : C1 → C2 as a unital filtered A∞ homomorphism f+ : C1+ → C2+ . This means that there are several choices of f+ to a given f : C1 → C2 . Moreover the different f+ (with the same f) are regarded as a different homotopy unital filtered A∞ homomorphisms. The authors do not know whether different choice f+ : C1+ → C2+ is always unitally homotopic to f+ or not. (It is easy to show that f+ is homotopic to f+ .) In case we regard a unital filtered A∞ homomorphism f as a homotopy-unital filtered A∞ homomorphism, we take a canonical choice of f+ such that f+ (· · · ⊗ f ⊗ · · · ) = 0. By the choice specified in Remark 4.2.51, we can define a functor Iunit : [unital filtered A∞ algebra] → [homotopy-unital filtered A∞ algebra]. On the other hand C → C + defines a functor I+ : [homotopy-unital filtered A∞ algebra] → [unital filtered A∞ algebra]. Proposition 4.2.52. Iunit and I+ induce equivalences of categories [unital filtered A∞ algebra; u-homotopy class of unital filtered A∞ homomorphisms] ∼ = [homotopy-unital filtered A∞ algebra; hu-homotopy class of homotopy-unital filtered A∞ homomorphisms]. The same holds for the strict version. Proof. We first prove the following lemma. Lemma 4.2.53. Let f, g : C → C  be two unital filtered A∞ homomorphisms between unital filtered A∞ algebras. They are homotopy-unitally homotopic, if and only if they are unitally homotopic. For the proof we use the following sublemma. Sublemma 4.2.54. If C is a unital filtered A∞ algebra, then there exists a strict unital and filtered A∞ homomorphism ret : C + → C such that the composition ret ◦ i : C → C is identity, where i : C → C + is the canonical embedding. Proof of Sublemma 4.2.54. We put ret1 (e+ ) = e, ret1 (f) = 0, retk = 0 for k = 1. We also put ret1 = id on C ⊆ C + . It is easy to see that ret has the required property.  We remark that i is not unital. Proof of Lemma 4.2.53. Applying Theorem 4.2.45a we choose a unital homotopy inverse ret−1 : C → C + (resp. ret−1 : C  → C + ) of ret (resp. ret ). Now suppose f ∼hu g. Then f+ ∼u g+ . Hence f ∼u f ◦ ret ◦ ret−1 = ret ◦ f+ ◦ ret−1 ∼u ret ◦ g+ ◦ ret−1 = g ◦ ret ◦ ret−1 ∼u g.

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4.3. GAUGE EQUIVALENCE OF BOUNDING COCHAINS

211

Here the equality f ◦ ret = ret ◦ f+ follows from the choice of f+ we specified in Remark 4.2.51 and by the definition of ret above. We next suppose f ∼u g. Then we have f+ ∼u ret−1 ◦ ret ◦ f+ = ret−1 ◦ f ◦ ret ∼u ret−1 ◦ g ◦ ret = ret−1 ◦ ret ◦ g+ ∼u g+ . 

Hence f ∼hu g.

Lemma 4.2.53 implies that Iunit induces an injective functor on the homotopy category. On the other hand, it is easy to see that I+ induces a functor between homotopy categories. Hence to complete the proof of Proposition 4.2.52 it suffices to show the following lemma. (In fact Lemma 4.2.55 implies that Iunit ◦ I+ is equivalent to the identity functor.) Lemma 4.2.55. If C be a homotopy-unital filtered A∞ algebra, then C is huhomotopy equivalent to C + . C

+

Proof. We extend i : C → C + to a unital filtered A∞ homomorphism i+ : → C ++ as follows. We write C + = C ⊕ Λ0,nov e+ ⊕ Λ0,nov f, C ++ = C + ⊕ Λ0,nov e++ ⊕ Λ0,nov f+ ,

+ + ++ + , i1 (f) = f+ + f. We also set i+ and put i+ 1 (x) = x for x ∈ C, i1 (e ) = e k = 0 for + k = 1. It is easy to check that i is a unital filtered A∞ homomorphism. (Note that the unit of C + is e+ and the unit of C ++ is e++ .) Therefore i is homotopy unital. It then follows from Theorem 4.2.45b that i is a hu-filtered homotopy equivalence.

The proof of Proposition 4.2.52 is now complete.



We can reduce most of algebraic discussions of homotopy-unital filtered A∞ algebras to unital ones, by using Proposition 4.2.52. So we discuss homotopy-unital filtered A∞ algebras only when we absolutely need to. 4.3. Gauge equivalence of bounding cochains In this section we define an equivalence relation ∼ on the set of bounding   cochains M(C) and show that the quotient space M(C) = M(C)/ ∼ is an invariant of the homotopy type of filtered A∞ algebras C. 4.3.1. Basic properties and the category HA∞ . Let C be a filtered A∞ algebra and C a model of [0, 1] × C.  Definition 4.3.1. Let b0 , b1 ∈ M(C). b1 and write b0 ∼C b1 if there exists ˜b ∈ Evals=1∗ (˜b) = b1 . We say such ˜b, a homotopy

We say b0 is C-gauge equivalent to  M(C) such that Evals=0∗ (˜b) = b0 , from b0 to b1 .

 Remark 4.3.2. We define the set M(C) and an equivalence relation ∼ on it only for the filtered A∞ algebra. In fact, for the unfiltered A∞ algebra the set of  b ) = 0 does not make sense, since in solutions b of the Maurer-Cartan equation d(e b  ) is an infinite sum. Therefore the set of all bounding cochains M(C)  general d(e  does not make sense. However we can still define M(C) as a functor from the category of Artin rings ([Schl68]). See for example [Fuk03II] Section 2.1.

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Lemma 4.3.3. ∼C is independent of the choice of the model C of [0, 1] × C. Proof. The proof is the same as the proof of Lemma 4.2.36.



From now on, we write ∼ in place of ∼C and just say ‘gauge equivalent’ in place of ‘C-gauge equivalent’. Lemma 4.3.4. The gauge equivalence ∼ on the set of bounding cochains is an equivalence relation. Proof. The proof is the same as the proof of Proposition 4.2.37.



Let us describe the gauge equivalence more explicitly. Let us first take C = P oly([0, 1], C). Then we have   Proposition 4.3.5. b0 ∈ M(C) is gauge equivalent to b1 ∈ M(C) if and only if there exists ˜b = (b(s), c(s)) ∈ P oly([0, 1], C)1 such that ∂b(s) = δb(s) c(s) ∂s b(0) = b0 , b(1) = b1 .

(4.3.6.1) (4.3.6.2)

Proof. Let ˜b be a homotopy from b0 to b1 . We put ˜b = (b(s), c(s)). Then the  oly([0, 1], C)) is equivalent to (4.3.6.1) and Evals=0∗ (˜b) = b0 , condition ˜b ∈ M(P ˜  Evals=1∗ (b) = b1 is equivalent to (4.3.6.2). For the case of a differential graded algebras (C; d, ◦) where d is a derivation and ◦ is an associative product, we can prove that if b is gauge equivalent to b there exists g such that b = g −1 ◦ d(g) + g −1 ◦ b ◦ g. In other words, b is gauge equivalent to b in the usual sense. We do not prove this since we do not need it. (See Lemma 2.2.2 [Fuk03II].) We also remark the following: Lemma 4.3.7. If (b(s), c(s)) satisfies (4.3.6.1) and if b(0) is a bounding cochain, then b(s) is also a bounding cochain for each s. Proof. Using (4.3.6.1), we calculate   ∂ m(eb(s) ) = m eb(s) , δb(s) (c(s)), eb(s) = δb(s) (δb(s) (c(s))) = 0. ∂s Lemma 4.3.7 follows.



Remark 4.3.8. The conclusion of Proposition 4.3.5 is an analog of the corresponding definition in the literature concerning D.G.A. (See [GoMi88, Sche98, Kon03]). In [Kon03] a similar definition is proposed for the case of L∞ algebras. However we did not find a proof of Lemmas 4.3.3 and 4.3.4 in the literature. Next we take C = C [0,1] .

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Proposition 4.3.9. Two bounding cochains b0 and b1 are gauge equivalent to each other if and only if there exists c ∈ C[1]−1 such that (4.3.10)

b1 − b0 = m(eb0 ceb1 ).

 [0,1] ) be a homotopy from b0 to b1 , then we can write Proof. If ˜b ∈ M(C it as ˜b = (b0 , c, b1 ). The equation (4.3.10) is equivalent to the condition that ˜b ∈ M(C  [0,1] ).  Definition 4.3.11. Let (C, m) be a filtered A∞ algebra. We define M(C, m)  as the set of all ∼ equivalence classes of M(C, m). We sometimes write them as  M(C), M(C) for simplicity. Definition 4.3.12. For a relatively spin Lagrangian submanifold of a symplectic manifold M , we associated a filtered A∞ algebra (C(L; Λ0,nov ), m) in Section 3.5. We denote by M(L) the set M(C(L; Λ0,nov ), m). One of the main results of this chapter is that M(C(L; Λ0,nov ), m) is invariant of the various choices involved (Theorem 4.1.3). The next theorem is an algebraic part of its proof. Let HA∞ = [filtered A∞ algebra; homotopy class of filtered A∞ homomorphism] be a category whose object is a filtered A∞ algebra and whose morphism is a homotopy class of a filtered A∞ homomorphism. Theorem 4.3.13. The assignment (C, m) → M(C, m) defines a covariant functor HA∞ −→ [set]. Corollary 4.3.14 below is a filtered A∞ algebra analog of a result by GoldmanMillson [GoMi88,90]. Corollary 4.3.14. If f : (C1 , m1 ) → (C2 , m2 ) is a homotopy equivalence, then it induces a bijection f∗ : M(C1 , m1 ) → M(C2 , m2 ). To prove Theorem 4.3.13 it suffices to show the following Lemma 4.3.15. For a filtered A∞ homomorphism f : (C1 , m1 ) → (C2 , m2 ) we defined a map f∗ :  1 , m1 ) → M(C  2 , m2 ) by f∗ (b) = f(eb ). See Lemma 3.6.36. M(C  1 , m1 ), then f∗ (b0 ) ∼ f∗ (b1 ). Lemma 4.3.15. (1) If b0 ∼ b1 in M(C   1 , m1 ). (2) If f is homotopic to f , then f∗ (b) ∼ f∗ (b) for any b ∈ M(C Proof. Let Ci (i = 1, 2) be a model of [0, 1] × Ci . (1) We apply Theorem 4.2.34 to f and obtain a filtered A∞ homomorphism  1 ) be a homotopy F : C1 → C2 , such that Evals=s0 ◦ F = f ◦ Evals=s0 . Let ˜b ∈ M(C from b0 to b1 . Then F∗ (˜b) is a homotopy from f∗ (b0 ) to f∗ (b1 ). Thus f∗ (b0 ) ∼ f∗ (b1 ). (2) Let H : C1 → C2 be a homotopy from f to f . Then H∗ b is a bounding cochain such that Evals=0 H∗ b = f∗ (b) and Evals=1 H∗ b = f∗ (b). Hence f∗ (b) ∼ f∗ (b).  In the proof of the next proposition, we will use Theorem 4.2.45 which will be proved in Section 4.5.

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 Proposition 4.3.16. If b0 ∼ b1 in M(C, m), there exists a strict homotopy b0 b1 bi is the equivalence f : (C, m ) → (C, m ) such that f ≡ id mod Λ+ 0,nov . Here m deformed A∞ structure defined by Definition 3.6.9. Proof. Let C be a model of [0, 1] × C and ˜b as in Definition 4.3.1. Then, Evals=0 and Evals=1 define strict filtered A∞ homomorphisms ˜

Evals=0 : (C, mb ) → (C, mb0 ),

˜

Evals=1 : (C, mb ) → (C, mb1 ).

Using Theorem 4.2.45, we can prove that they are strict homotopy equivalences. Hence we obtain f. We will check the property f ≡ id mod Λ+ 0,nov later at the end of Section 4.5.   Remark 4.3.17. For any b ∈ M(C, m), the filtered A∞ algebra (C, mb ) is homotopy equivalent to (C, m). In fact, we define ib : (C, mb ) → (C, m) by ib0 (1) = b, ib1 = id, ibk = 0 for k ≥ 2. It is a homotopy equivalence by Theorem 4.2.45. However, note that ib is not strict in the sense of Definition 3.2.29. The following proposition is a special case when L = L(0) = L(1) in Corollary 5.2.40. b0

 Proposition 4.3.18. Let b0 , b1 , b0 , b1 ∈ M(C, m). We assume b0 ∼ b1 and  ∼ b1 . Then there exists a canonical isomorphism H ∗ (C; δb0 ,b0 ) ∼ = H ∗ (C; δb1 ,b1 ).

Proof. Let ˜b, ˜b ∈ C [0,1] be bounding cochains such that Evals=i˜b = bi and Evals=i˜b = bi for i = 0, 1. We consider the following exact sequence of graded modules: 0 → C[−1] ⊕ C → C [0,1] → C [0,1] /(C[−1] ⊕ C) → 0, where the second homomorphism is the inclusion. Note that C[−1] ⊕ C is closed [0,1]

, m1 ).) It is also under δ˜b,˜b as well as m1,0 . (C[−1] ⊕ C is a subcomplex of (C easy to see that (C[−1] ⊕ C, m1 ) is an acyclic complex, hence (C[−1] ⊕ C, m1,0 ) is acyclic. We show that (C[−1] ⊕ C, δ˜b,˜b ) is acyclic. ∞ ν j λj Let c = ∈ C[−1] ⊕ C be a δ˜b,˜b -cocycle. For simplicity, we j=0 cj e T assume that λj is strictly monotone increasing: 0 = λ0 < λ1 < λ2 < · · · . Since δ˜b,˜b ≡ m1 modΛ+ 0,nov , c0 ∈ C[−1] ⊕ C is a m1 -cocycle. Recall that C[−1] ⊕ C is m1 -acyclic, there is a0 ∈ C[−1]⊕C such that c0 = m1 a0 . Consider c = c−δ˜b,˜b (a0 ), which is a δ˜b,˜b -cocycle. (Recall that μ0 = 0 by our assumption on Λ0,nov .) Note that the energy level of c is strictly higher than c, i.e., c ∈ F λ C for some λ > 0. Because of the gapped condition, we can repeat the above procedure to write c as a δ˜b,˜b -coboundary. Thus C[−1] ⊕ C is δ˜b,˜b -acyclic. Since the quotient complex C [0,1] /(C[−1] ⊕ C) with the coboundary operator induced from δ˜b,˜b is naturally isomorphic to the complex (C, δb0 ,b0 ), we obtain H ∗ (C [0,1] ; δ˜b,˜b ) ∼ = H ∗ (C; δb0 ,b0 ). Similarly, we have H ∗ (C [0,1] ; δ˜b,˜b ) ∼ = H ∗ (C; δb1 ,b1 ). Hence we obtain Proposition 4.3.18.  Proposition 4.3.18, in particular, implies that the Floer cohomology group HF ((L, b), (L, b ); Λ0,nov ) depends only on gauge equivalence classes of b, b . We study M(C, m) more in Sections 5.3 and 5.4.

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4.3.2. Mweak (C) and its homotopy invariance. Let C be a unital filtered weak (C) in Subsection 3.6.3. Namely b ∈ C 1 = A∞ algebra. We defined the set M weak (C) if and only if m(eb ) = cee for some c ∈ Λ+(0) . We called C[1]0 is in M 0,nov such b a weak bounding cochain. See Definition 3.6.29. Further we recall that a unital filtered A∞ homomorphism f : C1 → C2 induces weak (C1 ) → M weak (C2 ) f∗ : M so that f∗ (b) = f(eb ). See Lemma 3.6.36. Let C be a unital model of [0, 1] × C. weak (C). We say that b0 is gauge equivalent Definition 4.3.19. Let b0 , b1 ∈ M weak (C) such that Evals=0∗ ˜b = b0 , to b1 and write b0 ∼ b1 , if there exists ˜b ∈ M Evals=1∗ ˜b = b1 . Lemma 4.3.20. ∼ is independent of the choice of the model of [0, 1] × C and it is an equivalence relation. The proof is the same as the proofs of Lemmas 4.3.3 and 4.3.4. Definition 4.3.21. Let (C, m) be a unital filtered A∞ algebra. We define weak (C, m). Mweak (C, m) as the set of all ∼ equivalence classes of M The next theorem can be proved in the same way as Theorem 4.3.13. Theorem 4.3.22. The assignment (C, m) → Mweak (C, m) defines a covariant functor HA∞u := [unital filtered A∞ algebra; u-homotopy class of unital filtered A∞ homomorphisms] −→ [set]. If f : (C1 , m1 ) → (C2 , m2 ) is a unital homotopy equivalence, then it induces a bijection f∗ : Mweak (C1 , m1 ) → Mweak (C2 , m2 ). weak (C) → In Subsection 3.6.3 we introduced the potential function PO : M by m(eb ) = PO(b)ee. (Definition 3.6.33.) Then we can show the following.

+(0) Λ0,nov

Lemma 4.3.23. PO(b) depends only on the gauge equivalence class of b ∈ +(0)  Mweak (C). Namely PO induces PO : Mweak (C) → Λ0,nov . Moreover if Ci is a unital filtered A∞ algebra and f : C1 → C2 is a unital filtered A∞ homomorphism, we have PO2 ◦ f∗ (b) = PO1 (b) for b ∈ Mweak (C1 ), where POi is the potential function for Ci . weak (C) and ˜b a homotopy from b0 to b1 as in Proof. Let b0 ∼ b1 in M Definition 4.3.1. Then by Lemma 3.6.36 (3) we have PO(b0 ) = PO(Evals=0∗˜b) = PO(˜b) = PO(Evals=1∗˜b) = PO(b1 ), which proves the first assertion. The second assertion also follows from Lemma 3.6.36 (3). 

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When C is a homotopy-unital filtered A∞ algebra, we define (4.3.24)

Mweak (C) = Mweak (C + ).

It is independent of the hu-homotopy type of C. We can define the potential function in the same way for this case. The potential function is defined as a function on Mweak (C) = Mweak (C + ). Also we can define a category HA∞hu in a similar way. Now let us consider the case of filtered A∞ bimodules. Let C0 , C1 be unital filtered A∞ algebras and D a left C1 and right C0 unital filtered A∞ bimodule. weak (C0 ) and b1 ∈ M weak (C1 ), we defined δb ,b : (Definition 3.7.5). For b0 ∈ M 1 0 D → D in Definition-Lemma 3.7.13. Namely δb1 ,b0 (x) = n(eb1 xeb0 ). We denote by POi , (i = 0, 1) the potential function of Ci . Proposition 3.7.17 shows that δb1 ,b0 ◦ δb1 ,b0 (x) = (−PO1 (b1 ) + PO0 (b0 ))ex. Thus if PO0 (b0 ) = PO1 (b1 ), we can define the cohomology of (D, δb1 ,b0 ). weak (C0 ) and Proposition 4.3.25. Under the situation above, let b0 , b0 ∈ M   ∈ Mweak (C1 ). Assume that PO(b0 ) = PO(b1 ). If b0 ∼ b0 and b1 ∼ b1 , then

b1 , b1

H(D, δb1 ,b0 ) ∼ = H(D, δb1 ,b0 ). The isomorphism is canonical. The homotopy-unital version also holds. The proof will be given in Subsection 5.2.4 (Corollary 5.2.40). 4.3.3. Mweak,def (L) and its homotopy invariance. In Section 3.8 we introduced the infinitesimal deformations (bulk deformations) by using cocycles b in the ambient symplectic manifold M . We recall that weak,def (L) = {(b, b) ∈ C 2 (M ; Λ+ ) × C 1 (L; Λ+ ) M 0,nov 0,nov = cee+ }. | δM b = 0, m+b,b 0 See Definition 3.8.40. + + 2 We fix H2 (M ; Λ+ 0,nov ) ⊂ C(M ; Λ0,nov ) representing H (M ; Λ0,nov ). weak,def (L), b ∈ H2 (M ; Λ+ ). Definition 4.3.26. Let (b, b0 ), (b, b1 ) ∈ M 0,nov We say that (b, b0 ) is gauge equivalent to (b, b1 ) and write (b, b0 ) ∼ (b, b1 ), if b0 is b gauge equivalent to b1 in (C(M ; Λ+ 0,nov ), m ) in the sense of Definition 4.3.19. Then ∼ obviously defines an equivalence relation and we define Mweak,def (L) as the set of all gauge equivalence classes of such (b, b). We will prove in Subsection 4.6.4 that Mweak,def (L) is independent of the choice of the space H2 (M ; Λ+ 0,nov ). Theorem 4.6.47 proven there also implies that the b filtered A∞ algebra (C(M ; Λ+ 0,nov ), m ) depends only on the homology class of b up to strict homotopy equivalence. Then, in Definition 4.6.50, we will generalize the Definition 4.3.26 to the pair (b0 , b0 ), (b1 , b1 ) where b0 is homologous to but may not be equal to b1 . By definition we have the well-defined map (4.3.27)

πamb : Mweak,def (L)

(b, b) −→ [b] ∈ H 2 (M ; Λ+ 0,nov )

which we mentioned in Theorem B in Chapter 1. Now we have:

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Lemma 4.3.28. The potential function PO(b, b) defined in Definition 3.8.40 depends only on the gauge equivalence class of the bounding cochain weak,def (C(M ; Λ+ ), mb ). b∈M 0,nov Namely PO induces +(0)

PO : Mweak,def (L) → Λ0,nov . Proof. The lemma follows from Lemma 4.3.23 applied to the filtered A∞ b algebra (C(M ; Λ+  0,nov ), m ). We will prove PO is independent of the choice of H2 (M ; Λ+ 0,nov ) later. (See Subsection 4.6.4 Lemma 4.6.51.) We refer Corollary 4.6.53 for the relevant property of the potential function related to a symplectic diffeomorphism ψ : M → M  . Now we consider the case of filtered A∞ bimodules as in Subsection 3.8.7. Here we use the same notations as in Subsection 3.8.7. The following is an analog of Proposition 4.3.18 and Proposition 4.3.25. The proof will be also given in Subsection 5.2.4 (Corollary 5.2.40). Proposition 4.3.29. Under the situation in Subsection 3.8.7, let weak,def (L(0) ) (b, b0 ), (b, b0 ) ∈ M

and

weak,def (L(1) ). (b, b1 ), (b, b1 ) ∈ M

Assume the equality PO((b, b0 )) = PO((b, b1 )). (b, b0 )

and (b, b1 ) ∼ (b, b1 ), then we have an isomorphism between the If (b, b0 ) ∼ Floer cohomologies HF ((L(1) , (b, b1 )), (L(0) , (b, b0 )); Λ0,nov ) ∼ = HF ((L(1) , (b, b )), (L(0) , (b, b )); Λ0,nov ). 1

0

The isomorphism is canonical. This proposition implies that the Floer cohomology is well defined and is parametrized by the pair of the gauge equivalence classes of (b, b0 ), (b, b1 ) ∈ Mweak,def (L) such that πamb (b, b0 ) = πamb (b, b1 )

and PO(b, b0 ) = PO(b, b1 ).

This is nothing but Theorem B (B.1) in Chapter 1. From now on, we will simply denote by the same symbol (b, b) ∈ Mweak,def (L) weak,def (L), if no confusion can occur. the gauge equivalence class of (b, b) ∈ M 4.4. Uniqueness of the model of [0, 1] × C This section is devoted to the proof of Theorem 4.2.34 which plays the fundamental role to study the homotopy equivalence of filtered A∞ algebras.

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4.4.1. Induction on the number filtration I. In Subsections 4.4.1-4.4.4, we first prove the case of unfiltered A∞ algebras. Namely we consider the following situation. Let C 1 , C 2 be A∞ algebras and C1 , C2 models of [0, 1] × C 1 and of [0, 1] × C 2 respectively. Let f : C 1 → C 2 be an A∞ homomorphism. We will construct an A∞ homomorphism F from C1 to C2 such that (4.4.1.1) (4.4.1.2)

  / / Eval s=s0 ◦ F = f ◦ Evals=s0 ,  ◦ Incl. / / ◦ f = F Incl

where s0 = 0, 1,

Here  stands for the induced coalgebra homomorphism on the bar complex. See (3.2.6). We say that F is compatible with Incl and Eval over f, if (4.4.1) is satisfied. Recall that an A∞ homomorphism F consists of a sequence Fi : Bi C[1] → C2 [1],

for i = 1, 2, 3, · · ·

satisfying the condition given in Definition 3.2.7. If Fi i = 1, · · · , k are defined and (4.4.1.1),(4.4.1.2) hold on ⊕ki=1 Bi C1 [1], we say that it is compatible with Incl and Eval. The construction of F is by induction on the number filtration. Namely we will construct Fk : Bk C1 [1] → C2 [1] inductively on k. We start the induction with the following: Lemma 4.4.2. There exists a cochain homomorphism F1 : C1 → C2 compatible with Incl and Eval. 

Proof. We first remark that there exists a cochain map F1 which satisfies (4.4.1.2) (on B1 C 1 = C 1 ) but not necessarily (4.4.1.1). (For example we may put  F1 = Incl1 ◦ f1 ◦ (Evals=1 )1 .) Now we put 

Err1s0 = (Evals=s0 )1 ◦ F1 − f1 ◦ (Evals=s0 )1 and Err1 = (Err10 , Err 11 ) ∈ Hom(C1 , C 2 ) ⊕ Hom(C1 , C 2 ). Then Err1 is a cochain map. In other words δ1 Err1 = 0, where δ1 is a coboundary ˆ 1 and m1 on C1 and C 2 . We operator of Hom(C1 , C 2 ) ⊕ Hom(C1 , C 2 ) induced by m derive 

Err1 ◦ Incl = (Evals=s0 )1 ◦ F1 ◦ Incl − f1 ◦ (Evals=s0 )1 ◦ Incl = 0, 

since F1 satisfies (4.4.1.2). The following lemma will be used several times in this section. Lemma 4.4.3. Let (Dj , d), j = 1, 2, 3 be cochain complexes over R, and i : D1 → D2 be a cochain homomorphism. Suppose that i is a cochain homotopy equivalence that is split injective as an R module homomorphism. Let A ∈ HomR (D2 , D3 ) such that dA = 0, A ◦ i = 0. Then there exists B ∈ HomR (D2 , D3 ) such that dB = A and B ◦ i = 0.

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Proof. By assumption i∗ : H ∗ (HomR (D2 , D3 )) → H ∗ (HomR (D1 , D3 )) is an isomorphism. Hence there exists B  ∈ HomR (D2 , D3 ) such that dB  = A. Then, by assumption d(B  ◦ i) = 0. Therefore, there exists α ∈ HomR (D2 , D3 ), β ∈ HomR (D1 , D3 ) such that B  ◦ i = α ◦ i + dβ and dα = 0. Since i is split injective, there exists β˜ ∈ HomR (D2 , D3 ) such that β˜ ◦ i = β. It is easy to see that  B = B  − α − dβ˜ has required properties. We return to the proof of Lemma 4.4.2. We apply Lemma 4.4.3 to A = Err1    and obtain Corr such that Err1 = δCorr and Corr ◦ Incl1 = 0. Since the homomorphism (Evals=0 )1 ⊕ (Evals=1 )1 : C2 −→ C 2 ⊕ C 2 is split surjective by (4.2.2.4) and Remark 4.2.4 and since Incl1 : C 1 → C1 is split injective, it follows that we have Corr ∈ Hom(C1 , C2 ) such that ((Evals=0 )1 , (Evals=1 )1 ) ◦ Corr = Corr

(4.4.4.1)



and Corr ◦ Incl1 = 0.

(4.4.4.2) Now we put



F1 = F1 − δCorr. It is easy to see that it has required properties.



Hence we have finished the first step of the induction. To proceed further we introduce the notion of AK structures. 4.4.2. AK structures and homomorphisms. To define AK structure, we need some more notations. Let C be a graded R module and m1 < m2 be natural numbers. We put

m2 Bk C[1] (4.4.5) Bm1 ···m2 C[1] ∼ . = mk=1 1 −1 k=1 Bk C[1] We consider a series of degree 1 homomorphisms mk : Bk C[1] → C[1], for  k : B1···K C[1] → B1···K C[1] by (3.2.1). k = 1, · · · , K. It induces a coderivation m K   . = m We put d 1···K

k=1

k

Definition 4.4.6. m = {m1 , · · · , mK } defines a structure of AK algebra over d1···K ◦  d1···K = 0. R if  Remark 4.4.7. We do not try to generalize the notion of AK algebras to one for the filtered A∞ algebra in this section. Because m0 does not preserve B1···K C[1], the definition thereof is more involved. We will introduce the filtered An,K algebras in Subsection 7.2.6, where we use them. Let (C i , m), i = 1, 2 be AK algebras. We consider a sequence of R module homomorphisms fk : Bk C 1 [1] → C 2 [1] of degree 0, for k = 1, 2, 3, · · · , K. It induces coalgebra homomorphisms fk : B1···K C 1 [1] → B1···K C 2 [1], by (3.2.6). We put K  f 1···K = k=1 fk .

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Definition 4.4.8. We call f = {f1 , · · · , fK } an AK homomorphism from C 1 to C 2 if f1···K ◦  d1···K =  d1···K ◦ f1···K . The composition of two AK homomorphisms are defined in the same way as in Definition 3.2.8. Further, the notions of the AK homotopy between AK homomorphisms and the AK homotopy equivalence are defined in a similar way in Subsection 4.2.2. We remark that an AK structure induces an AK  structure for K  ≤ K ≤ ∞. We state the following two lemmas which will be used in the next subsection. Lemma 4.4.9. If (C, m) is an AK algebra and m2 − m1 < K, then m indm1 ···m2 : Bm1 ···m2 C[1] → Bm1 ···m2 C[1] such that  dm1 ···m2 ◦ duces a homomorphism   d = 0. m1 ···m2

Lemma 4.4.10. If f : C 1 → C 2 is an AK homomorphism and if m2 − m1 < K, then f induces a homomorphism fm1 ···m2 : Bm1 ···m2 C[1] → Bm1 ···m2 C[1] such that f ◦ d = d ◦ f . m1 ···m2

m1 ···m2

m1 ···m2

m1 ···m2

The proofs are easy and omitted. 4.4.3. Induction on the number filtration II. Now we prove the following proposition by induction on K. We consider the situation of Theorem 4.2.34. (K−1)

Proposition 4.4.11. For any given AK−1 homomorphism F compatible (K) from C1 to C2 with Incl and Eval over f, there exists an AK homomorphism F (K) compatible with Incl and Eval over f. Moreover if we regard F as an AK−1 homomorphism, it coincides with F

(K−1)

.

Proof. The proof is by induction on K. The case of K = 1 is already taken care of in Lemma 4.4.2. Assume that Proposition 4.4.11 is proved for K − 1. Then (K−1) F induces a coalgebra homomorphism (K−1) : BC [1] → BC [1]. F 1 2 We consider the restriction of (4.4.12)

(K−1) − F (K−1) ◦   d◦F d

to B1···K C1 [1] and denote it by ErrK . Lemma 4.4.13. The restriction of ErrK to B1···K−1 C1 [1] vanishes. The image of ErrK is in B1 C2 [1] = C2 [1]. (K−1) to B Proof. It is easy to see that the restriction of F 1···K−1 C1 [1] is

 F 1···K−1 . The first statement then follows from Lemma 4.4.10.  (K−1) induces F F 2···K on Hom(B2···K C1 [1], B2···K C2 [1]). Then the second half also follows from Lemma 4.4.10.  By Lemma 4.4.13, we may regard ErrK ∈ Hom(BK C1 [1], C2 [1])).

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The coboundary operator m1 induces δ1 : Hom(BK C1 [1], C2 [1])) → Hom(BK C1 [1], C2 [1])) by

 1. δ1 (A) = m1 ◦ A − (−1)deg A A ◦ m Lemma 4.4.14. δ1 ErrK = 0. d induces a coboundary operator on Hom(BC1 [1], BC2 [1])) Proof. Note that 

by

φ →  d(φ) =  d ◦ φ − (−1)deg φ φ ◦  d. (K−1)  ) to BK C1 [1]. Hence, by Lemma By definition, ErrK is a restriction of d(F (K−1) d( d(F )) = 0 to B C [1] is δ Err . The lemma 4.4.13, the restriction of  1

K

1

K

follows.



Lemma 4.4.15. ErrK ∈ Im δ1 in Hom(BK C1 [1], C2 [1]). (K−1) Proof. Let f : BC 1 [1] → BC 2 [1] be the coalgebra homomorphism in  duced by f1 , · · · , fK−1 . Then since f is an A∞ homomorphism, it follows from the A∞ formula that (K−1) (K−1)  d ◦ f − f ◦ d + δ (f ) = 0 1

K

on B1···K C 1 . Here δ1 : Hom(BK C 1 [1], C 2 [1])) → Hom(BK C 1 [1], C 2 [1])) is induced by m1 . (K−1) Now, we calculate, using the compatibility of F with Incl, to obtain the following formula in Hom(Bk C 1 [1], C2 [1]): / = (m ◦ F (K−1) − F(K−1) ◦  / ErrK ◦ Incl d) ◦ Incl (K−1) (K−1)  = Incl1 ◦ (m ◦ f −f ◦ d) = −δ (Incl ◦ f ).

(4.4.16)

1

1

K

The lemma then follows from the fact that Incl1 is a cochain homotopy equivalence.  

Lemma 4.4.15 implies there exists FK ∈ Hom(BK C1 [1], C2 [1]) such that 

δ1 (FK ) + ErrK = 0. 



Lemma 4.4.17. We may choose FK such that FK ◦Incl1 = Incl1 ◦fK in addition. 



Proof. Let us first take any FK such that δ1 (FK ) + ErrK = 0. (4.4.16) then implies    / − Incl ◦ f δ1 FK ◦ Incl 1 K = 0. Here there exist α ∈ Hom(BK C1 [1], C2 [1]) and β ∈ Hom(BK C 1 [1], C2 [1]) such that δ1 α = 0 and  / − Incl ◦ f = α ◦ Incl / + δ β. F ◦ Incl K

1

K

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1

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4.

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/ = β. We put We have β such that β ◦ Incl    FK = FK − α − δ1 β. 



Then it is easy to check that FK has required properties. By definition, F ible with Incl.

(K−1)



together with FK defines an AK homomorphism compat

(K)

Lemma 4.4.18. We can modify FK to FK so that the AK homomorphism F (K−1)

obtained from F

and FK is compatible with Incl and Eval over f. 

(K)

Proof. Let us use FK to obtain an AK homomorphism F (K)

(s0 )

= Evals=s0 ◦ F

ErrK

(4.4.19)

/ − f ◦ Eval s=s0

. We put

for s0 = 0, 1.

(K−1)

is compatible with Eval by induction hypothesis, it follows that the Since F (s0 ) restriction of ErrK to B1···K−1 C1 [1] is zero. Moreover the image of its restriction to B1···K C1 [1] is in B1 C 2 [1] = C 2 [1]. Namely we have (0)

(1)

(ErrK , ErrK ) ∈ Hom(BK C1 [1], C 2 [1] ⊕ C 2 [1]). (K)

Since F

and f both are AK homomorphisms, it follows that (0)

(1)

δ1 (ErrK , ErrK ) = 0. Moreover using the compatibility with Incl we have (s0 ) / = 0. ErrK ◦ Incl (0)

(1)

Now we apply Lemma 4.4.3 to A = (ErrK , ErrK ). Then, there exists (0)

(1)

(Cor1K , Cor1K ) ∈ Hom(BK C1 [1], C 2 [1] ⊕ C 2 [1]) such that (0)

(1)

(0)

(1)

δ1 (Cor1K , Cor1K ) = (ErrK , ErrK ), (s0 ) / = 0. Cor1 ◦ Incl

(4.4.20.1) (4.4.20.2)

K

Using the property (4.2.2.4) we find Cor2K ∈ Hom(BK C1 [1], C2 [1]) such that (s0 )

(Evals=s0 )1 ◦ (Cor2K ) = Cor1K , / = 0. Cor2 ◦ Incl

(4.4.21.1) (4.4.21.2)

K



We put FK = FK − Cor2K . Then (4.4.20.1), (4.4.21.1) and (4.4.21.2) imply that FK satisfies the required properties.  Therefore we have now finished the proof of Proposition 4.4.11.



Proposition 4.4.11 immediately implies the unfiltered version of Theorem 4.2.34.

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223

4.4.4. Unital case I: the unfiltered version. Before proceeding to the filtered version, we explain how we can modify the argument above to treat the unital case. Suppose f is unital. We will construct F that is compatible with Incl and Eval over f and also is unital. We consider an A∞ algebra C with unit e. Definition 4.4.22. We define the partially reduced bar complex denoted by red  B C[1] as follows. Putting C red = C/R[e], we define  C[1] ⊗ Bk−2 C red [1] ⊗ C[1], if k ≥ 2, red (4.4.23) Bk C[1] = C[1], if k = 1,

red  red C[1] =  and B k≥1 Bk C[1]. red d : BC[1] → BC[1] induces a differential  d on Lemma-Definition 4.4.24.  red  B C[1].

Proof. Let us consider an element x1 ⊗ e ⊗ x2 ∈ BC[1] with xi ∈ Bki C[1], ki ≥ 1. We write x1 = x1− ⊗ xlast , x2 = xfirst ⊗ x2− . Then we have   d(x1 ⊗ e ⊗ x2 ) = (−1)deg x1− x1− ⊗ m2 (xlast , e)⊗x2 

+(−1)deg x x ⊗ m2 (e, xfirst )⊗x2− = 0  red C[1] by Condition 3.2.18. Hence the lemma follows. in B



 red [1] → BC  red [1] is not well defined. We remark that  d : BC red Next we define operators sleft , sright : Bkred C[1] → Bk+1 C[1] by

sleft (x) = e ⊗ x,

sright (x) = (−1)deg x x ⊗ e.

red red red red Lemma 4.4.25.  d ◦ sleft + sleft ◦  d = d ◦ sright + sright ◦  d = identity.

The proof is an easy calculation using Condition 3.2.18. Let C 1 , C 2 be unital A∞ algebras. Suppose we have a sequence of homomorphisms fk,red : Bkred C 1 [1] → C 2 [1] of degree 0, with f1,red (e) = e. It induces f : B  red C 1 [1] −→ BC  2 [1] red

(4.4.26)

as follows. (We remark that right hand side is the usual bar complex and is not the reduced one.) We choose a splitting C = C red ⊕ R[e].

(4.4.27)

 red C 1 [1]. The See Definition 3.2.20 (3). We use (4.4.27) to regard Bk C 1red [1] ⊂ B k  1red . Finally we put we define f by the same formula as (3.2.28) on BC f (e ⊗ x) = e ⊗ f (x), f (x ⊗ e) = f (x) ⊗ e, red red red red f (e ⊗ x ⊗ e) = e ⊗ f (x) ⊗ e, f (e) = e, red

red

red

 1red . We also define f : BC  1 [1] → BC  2 [1] by putting for x ∈ BC (4.4.28)

f(x ⊗ e ⊗ · · · ⊗ e ⊗ x ) = f (x ) ⊗ e ⊗ · · · ⊗ e ⊗ f (x ) 1  red 1 red 

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4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

 red C 1 [1] and put for xi ∈ Bki C 1red [1]. We use (4.4.27) to regard Bk C 1red [1] ⊂ B k f (1) = 1 for 1 ∈ B C red  [1] which is not contained in B C [1]. 0 1red 1 red Remark 4.4.29. We remark that C red is not an A∞ subalgebra in general. Namely the image of the restriction of mk to Bk C red may be ce. Moreover the splitting (4.4.27) is not necessarily canonical in the algebraic situation. (In our geometric situation there is a canonical splitting but we do not use it.) Lemma 4.4.30. Let fk,red be a sequence of homomorphisms as in (4.4.26). We assume red (4.4.31.1) fred ◦  d = d ◦ fred .

Then the homomorphism (4.4.28) is a unital A∞ homomorphism. Moreover every unital A∞ homomorphism is obtained from a sequence fk,red satisfying (4.4.31.1) and (4.4.31.2)

fk,red ◦ sleft = fk,red ◦ sright = 0.

Proof. Let us assume (4.4.31). We calculate, for xi ∈ Bki C 1red [1], (4.4.32)

m(f(x1 ⊗ e ⊗ · · · ⊗ e ⊗ x )) = m(f (x ) ⊗ e ⊗ · · · ⊗ e ⊗ f 1

red

red (x )).

Case 1: If k1 , k > 0, then (4.4.32) is zero. Also we have (4.4.33)

ˆ 1 ⊗ e ⊗ · · · ⊗ e ⊗ x )) = 0. (f ◦ d)(x

Case 2: If  > 2, then (4.4.32) is zero. (4.4.33) also holds. Case 3: If  = 1, then red d ◦ f)(x1 ) = ( d ◦ fred )(x1 ) = (fred ◦  d )(x1 ). (

By taking its component we have d)(x1 ), (m ◦ f)(x1 ) = (f ◦  since B1 C2 [1] component of fred is f. Case 4: If  = 2, k1 = 0, we have m(f(e ⊗ x)) = m(e ⊗ fred (x)) = m2 (e ⊗ f(x)) = f(x). On the other hand, since  d(e ⊗ x) = x − e ⊗  d(x), we have f( d(e ⊗ x)) = f(x). Case 5:  = 2, k2 = 0. Similar to Case 4. Thus we proved that f is a cochain homomorphism. On the other hand, (4.4.28) implies that f is a coalgebra homomorphism. We proved that f is a filtered A ∞

homomorphism. It is easy to see that it is unital. The converse is easy.



Now we are ready to give the proof of the unfiltered unital version of Theorem 4.2.34. Let fk,red be as in Lemma 4.4.30, and let C1 , C2 be models of [0, 1] × C 1 and

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4.4. UNIQUENESS OF THE MODEL OF [0, 1] × C

225

[0, 1] × C 2 respectively. We take a splitting (4.4.27) which is compatible with Incl and Eval. Proposition 4.4.34. For k = 1, · · · , K, there exists Fk,red : Bk C1,red [1] → (K)

C2 [1] satisfying (4.4.31) so that the induced unital AK homomorphism F C2 is compatible with Incl and Eval over f.

: C1 →

Proof. The proof goes in a similar way as that of Proposition 4.4.11. So we discuss only the point which needs modification. The first step of induction is the same as the proof of Lemma 4.4.2. The only point to check is that F1,red satisfies F1,red (e) = e. This follows automatically from the compatibility with Incl. We suppose that Proposition 4.4.34 is proved for K − 1. Namely we have Fk,red (K−1)

for k = 1, · · · , K − 1. We then obtain F

which induces

(K−1) : B red C [1] → B F 1···K C2 [1]. 1···K 1 red We consider red (K−1) − F (K−1) ◦  red ErrK =  d◦F d ∈ Hom(B1···K C1 [1], B1···K C2 [1]). red red

In the same way as the proof of Proposition 4.4.11 we can prove red ErrK ∈ Hom(BK C1 [1], C2 [1]), red (K−1) − F(K−1) ◦  red ErrK = m ◦ F d ∈ Hom(B1···K C1 [1], B1···K C2 [1]) red red

and δ1 (ErrK ) = 0,

ErrK ∈ Im δ1 .

(K−1) Fred

red is a homomorphism B1···K C1 [1] → C2 [1] which is Fred,k on Bkred C1 [1] Here, red (k = 1, · · · , K) and is 0 on BK C1 [1]. The new point to check is the following.

Lemma 4.4.35. ErrK ◦ sleft = ErrK ◦ sright = 0. Proof. We first calculate (K−1)

(4.4.36)

 (F red

(K−1) (x) ⊗ F ◦ sright )(x) = (−1)deg x F 1,red (e) red (K−1) )(x) = (sright ◦ F red

by (4.4.31.2). Hence using Lemma 4.4.25 and (4.4.36), we have ErrK ◦ sright (4.4.37)

(K−1) red (K−1) ◦ s =m◦F ◦ d ◦ sright right − Fred red

(K−1) + F(K−1) ◦ s red − F(K−1) . = m ◦ sright ◦ F right ◦ d red red red (K−1)

From (4.4.31.2) we have Fred ◦ sright = 0. (Note there is no ‘hat’ in this formula.) On the other hand, by the definition of sright we have (4.4.38)

(K−1) )(x) = (−1)deg x m(F (K−1) (x) ⊗ e). (m ◦ sright ◦ F red red

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226

4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

In the right hand side, only the term in B1 C2 [1] ⊗ e is nonzero. Hence (4.4.38) (K−1)

is equal to Fred similar.

(x). Therefore (4.4.37) is zero. The proof of ErrK ◦ sleft = 0 is 

Now the rest of the proof is similar to the proof of Proposition 4.4.11. Using Lemma 4.4.35, we may regard ErrK ∈ Hom(BK C1red [1], C2 [1]). We can define a coboundary operator δ1 on Hom(BK C1red [1], C2 [1]) using m1 . (We can not use m2 etc. to define operators on BCred [1].) So we can apply the argument of the proof of Proposition 4.4.11 to show   that ErrK is in the image of δ1 . Hence we have ErrK = δ1 FK,red with FK,red ∈ red Hom(BK C1red [1], C2 [1]). It can be identified with a homomorphism : BK C1 [1] → C1 [1] satisfying (4.4.31.2). Using it we obtain a unital AK homomorphism : C1 → C2 . We can modify it so that it is compatible with Incl and Eval in exactly the same way as in the proof of Proposition 4.4.11. The proof of Proposition 4.4.34 is now complete.  Proposition 4.4.34 immediately implies that the unital unfiltered version of Theorem 4.2.34 also holds. 4.4.5. Coderivation and Hochschild cohomology. Now we proceed to the proof of Theorem 4.2.34 for filtered A∞ algebras. The proof will be done by induction on the energy filtration. We start with introducing some notations. Let C i [1], i = 1, 2 be free graded ∞ R modules. The free graded R module BC i [1] = k=0 Bk C i [1] has a coalgebra  structure of Δ by (3.2.14). Let f : BC [1] → BC [1] be a coalgebra homomorphism 1

2

of degree 0 and put ϕ = (ϕ0 , ϕ1 , ϕ2 , · · · ) ∈

∞ 0

Hom(Bk C 1 [1], C 2 [1]),

k=0

where deg ϕk = deg ϕ is independent of k. We define ϕ  : BC 1 [1] → BC 2 [1] as follows. We use (Conv.3) (1.35) below. We define ϕ  : BC 1 [1] → BC 2 [1] by   (3;1)  ϕ(x(3;2)  f(x(3;3) (−1)deg ϕ deg xa f(x(3;1) )⊗ )⊗ ) (4.4.39) ϕ(x)  = a a a a

where ϕ : BC 1 [1] → C2 [1] is a homomorphism which restricts to ϕk on Bk C 1 [1]. We can easily check that ϕ  is a coderivation, i.e., satisfies (4.4.40)

ˆ f) + (f ⊗  ϕ)) ((ϕ ⊗  ◦ Δ = Δ ◦ ϕ. 

 is a graded tensor product defined by Here ⊗   ϕ)(x (f ⊗  ⊗ y) = (−1)deg ϕ deg xf(x) ⊗ ϕ(y). 

We remark that the image of ϕ  is in B1···∞ C 2 [1]. Namely its B0 C 2 [1] = R component is zero.

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4.4. UNIQUENESS OF THE MODEL OF [0, 1] × C

We put B1···∞ C i [1] =

∞

227

Bk C i [1].

k=1

Definition 4.4.41. We denote by Der(BC 1 [1], BC 2 [1]; f) the set of all graded coderivations ϕ  whose images lie in B1···∞ C 2 [1]. The assignment ϕ → ϕ  determines a homomorphism ∞ 0

(4.4.42)

Hom(Bk C 1 [1], C 2 [1]) → Der(BC 1 [1], BC 2 [1]; f).

k=0

Then we can show the following. Lemma 4.4.43. (4.4.42) is an isomorphism. Proof. Let ϕ  ∈ Der(BC 1 [1], BC 2 [1]; f) and ϕk ∈ Hom(Bk C 1 [1], B1 C 2 [1]; f) be its component. Thus we obtain a homomorphism Der(BC 1 [1], BC 2 [1]; f) →

(4.4.44)

∞ 0

Hom(Bk C 1 [1], C 2 [1]).

k=0

It is easy to see the composition ∞of (4.4.42) followed by (4.4.44) is the identity ∞ Hom(B C [1], C [1]) → k 1 2 k=0 k=0 Hom(Bk C 1 [1], C 2 [1]). Hence it is enough to show that (4.4.44) is an injection. Now we suppose that ϕ  ∈ Der(BC [1], BC [1]; f) is in the kernel of (4.4.44). 1

2

=0 Let π1···n : BC 2 [1] → B1···n C 2 [1] be the projection. We will prove that π1···n ◦ ϕ for all n by induction. (Note π0 ◦ ϕ  = 0 by Definition 4.4.41.) The case n = 1 is immediate from the assumption that ϕ  is in the kernel of (4.4.44). Assuming π1···n ◦ ϕ  = 0, we will prove π1···n+1 ◦ ϕ  = 0. Let πk : BC i [1] → Bk C i [1] be the projection. By the induction hypothesis, we have (π1···∞ ⊗ π1···∞ ) ◦ Δ ◦ π1···n+1 ◦ ϕ =

(4.4.45)

n+1 

(πk ⊗ πn+1−k ) ◦ Δ ◦ ϕ. 

k=0

On the other hand, we have n 

(πk ⊗ πn+1−k ) ◦ Δ ◦ ϕ =

k=1

n 

ˆ f) + (f ⊗  ϕ)) (πk ⊗ πn+1−k ) ◦ ((ϕ ⊗  ◦ Δ.

k=1

The right hand side vanishes by the induction hypothesis. Moreover, since π0 ◦ ϕˆ = 0, it follows that  = Δ ◦ (π0 ⊗ πn+1 + πn+1 ⊗ π0 ) ◦ ϕ  = 0. (π0 ⊗ πn+1 + πn+1 ⊗ π0 ) ◦ Δ ◦ ϕ The proof of Lemma 4.4.43 is now complete.



d be Now we assume that C i has the structure of an A∞ algebra and let   the coboundary operator on BC[1]. We also assume that f is induced by an A∞ homomorphism f. Hence f is a  d cochain map.

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 ∈ Der(BC 1 [1], BC 2 [1]; f), then Lemma & Definition 4.4.46. If ϕ   − (−1)deg ϕ ϕ  ◦ d◦ϕ d ∈ Der(BC 1 [1], BC 2 [1]; f). We define δ : Der(BC 1 [1], BC 2 [1]; f) → Der(BC 1 [1], BC 2 [1]; f)   =d◦ϕ  − (−1)deg ϕ ϕ  ◦ by δ(ϕ) d. The proof is standard and omitted. It is easy to see that δ ◦ δ = 0. Definition 4.4.47. We define HH(C 1 , C 2 ; f) = Ker δ/ Im δ and call it the Hochschild cohomology of an A∞ homomorphism f. We remark that the left hand side of (4.4.42) is independent of f. However, the right hand side does depend on f. So the Hochschild cohomology depends on f in general.   Let g : C 1 → C 1 and g : C 2 → C 2 be A∞ homomorphisms. In an obvious way they induce a cochain map denoted by (g, g )∗ 



(g, g )∗ : Der(BC 1 [1], BC 2 [1]; f) → Der(BC 1 [1], BC 2 [1]; g ◦ f ◦ g). We denote the map induced by (g, g )∗ on the Hochschild cohomology by the same symbol. Proposition 4.4.48. If g and g are homotopy equivalences, then the homo  morphism (g, g )∗ : HH(C 1 , C 2 ; f) → HH(C 1 , C 2 ; g ◦ f ◦ g) is an isomorphism. Proof. Let C1 , C2 be models of [0, 1] × C 1 , [0, 1] × C 2 , respectively. Incl induces homomorphisms (Incl, id)∗ : HH(C1 , C 2 ) → HH(C 1 , C 2 ) and (id, Incl)∗ : HH(C 1 , C 2 ) → HH(C 1 , C2 ). Lemma 4.4.49. (Incl, id)∗ and (id, Incl)∗ are isomorphisms. Proof. We first prove the case of (Incl, id)∗ . Since (Incl, id)∗ ◦ (Evals=0 , id)∗ = id, it follows that (Incl, id)∗ is surjective. We will prove its injectivity. Let ϕ = (ϕ0 , ϕ1 , · · · ) ∈ Hom(Bk C1 [1], C 2 [1]) k

/ = δψ. Then such that δϕ = 0 and ϕ ◦ Incl   / δ ϕ − δ(ψ ◦ Eval s=0 ) = 0 and

  / / ϕ − δ(ψ ◦ Eval s=0 ) ◦ Incl = 0.

Therefore it suffices to consider ϕ such that (4.4.50)

δϕ = 0,

/ = 0, ϕ ◦ Incl

and prove the following sublemma.

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4.4. UNIQUENESS OF THE MODEL OF [0, 1] × C

of

229

4.4.51. If ϕ = (ϕ0 , ϕ1 , · · · ) satisfies (4.4.50) and if ϕ is an element

Sublemma ∞ Hom(B  C1 [1], C 2 [1]), then there exists an element =k ψ∈

∞

Hom(B C1 [1], C 2 [1])

=k

such that

∞

ϕ − δψ ∈

Hom(B C1 [1], C 2 [1])

=k+1

and that ϕ − δψ satisfies (4.4.50). Proof. We remark that / : Hom(B C [1], C [1]) → Hom(B C [1], C [1]) Incl k 1 2 k 1 2 satisfies

/ = Incl / ◦δ , δ1 ◦ Incl 1

and induces an isomorphism on δ1 cohomology. (Note δ1 (A) = m1 ◦A−(−1)deg A A◦ ˆ 1 .) m Since ϕˆ = 0 and ∞ ϕ∈ Hom(B C1 [1], C 2 [1]), =k

/ = 0. Therefore, there it follows that δ1 (ϕk ) = 0. Moreover, by assumption ϕk ◦ Incl   exists ψk ∈ Hom(Bk C1 [1], C 2 [1]) such that ϕk = δ1 (ψk ). We put 

ψ = (0, · · ·

, 0, ψk , 0, · · · )

∈

∞

Hom(B C1 [1], C 2 [1])

=k

and

/ ◦ Eval / ψ = ψ  − ψ  ◦ Incl s=0 .

We can check that ψ has the required property by an easy calculation.



Lemma 4.4.49 follows immediately from Sublemma 4.4.51 for the case of (Incl, id)∗ . We next prove the case of (id, Incl)∗ . Since (id, Evals=0 )∗ ◦ (id, Incl)∗ = id, it follows that (id, Incl)∗ is injective. Thus it suffices to show its surjectivity. Let ψ∈ Hom(Bk C 1 [1], C2 [1]) k

ˆ = 0. Note deg ψ be a δ-cocycle, i.e., δψ = 0. More ψ◦d

concretely, m1 ◦ ψ − (−1) that Evals=0 ◦ ψ is a δ-cocycle in k Hom(Bk C 1 [1], C 2 [1]). Since Incl ◦ Evals=0 is chain homotopic to the identity, i.e., id − Incl ◦ Evals=0 = m1 ◦ h + h ◦ m1 ,

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4.

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where h is a chain homotopy, we find that ψ − Incl ◦ Evals=0 ◦ ψ = (m1 ◦ h + h ◦ m1 )ψ ˆ = m1 (h ◦ ψ) + (−1)deg ψ (h ◦ ψ) ◦ d ˆ = m1 (h ◦ ψ) − (−1)deg h◦ψ (h ◦ ψ) ◦ d. Thus we have ψ = Incl(Evals=0 ◦ ψ) + δ(h ◦ ψ). Hence (id, Incl)∗ : HH(C 1 , C 2 ) → HH(C 1 , C2 ) is surjective. The proof of Lemma 4.4.49 is now complete.  Using Lemma 4.4.49, we can prove easily that (g, g )∗ : HH(C 1 , C 2 ; f) →   HH(C 1 , C 2 ; g ◦ f ◦ g) depends only on homotopy classes of g and g . This implies Proposition 4.4.48.  4.4.6. Induction on the energy filtration. In this subsection and the next, we complete the proof of Theorem 4.2.34. We use the notations of Theorem 4.2.34. We take G ⊆ R≥0 × 2Z satisfying Condition 3.1.6 and assume that all filtered A∞ algebras and filtered A∞ homomorphisms we consider are G-gapped. We can enumerate the image of the projection of G to R≥0 denoted by Spec(G) = {λ0 , λ1 , · · · } so that λi < λi+1 , limi→∞ λi = ∞, because Spec(G) is discrete by Condition 3.1.6. We will construct Fk,i : Bk C1 [1] → C2 [1],

k = 0, 1, 2 · · · , i = 0, 1, 2, · · · ,

and define (n) Fk

=

n 

Fk,i T λi : Bk C1 [1] → C2 [1],

k = 0, 1, 2 · · · .

i=0

 (n) (n) : (We assume F0,0 = 0.) We extend k Fk to a coalgebra homomorphism F   BC1 [1] → BC2 [1]. Now we show the following by induction on n. Proposition 4.4.52. For any n = 0, 1, 2, · · · , there exists Fk,i for i ≤ n such that (4.4.53.1)

(n) ≡ F (n) ◦ d mod T λn+1 , d ◦ F

(4.4.53.2)

/ ◦ f ≡ F / (n) ◦ Incl Incl

(4.4.53.3)

mod T λn+1 ,

/ s=s ◦ F / s=s (n) ≡ f ◦ Eval Eval 0 0

mod T λn+1 for s0 = 0, 1.

Moreover we have (4.4.53.4)

(n−1) ≡ 0. (n) − F F

mod T λn .

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4.4. UNIQUENESS OF THE MODEL OF [0, 1] × C

231

Proof. We apply the unfiltered version of Theorem 4.2.34 to f and get Fk . We put Fk,0 = Fk . Thus, we proved the case of n = 0. We assume Proposition 4.4.52 for n and will prove it for n + 1. By induction hypothesis (4.4.53), we have Err1n ∈ Der(BC1 [1], BC2 [1]), Err2n ∈ Der(BC 1 [1], BC2 [1]), Err3n ∈ Der(BC1 [1], BC 2 [1]), such that (4.4.54.1) (4.4.54.2) (4.4.54.3)

(n) − F (n) ◦ d − T λn+1 Err1n ≡ 0 mod T λn+2 , d ◦ F / ◦ f − F (n) ◦ Incl / − T λn+1 Err2n ≡ 0 mod T λn+2 , Incl (n) / / s=s − T λn+1 Err3n ≡ 0 mod T λn+2 .  Evals=s0 ◦ F − f ◦ Eval 0

We note that d mod T λ1 . d ≡ 

(4.4.55)

Since Incl is split injective, we can take Err2∼ n ∈ Der(BC1 [1], BC2 [1]) such that ∼ / (n) (n)   Err2 ◦Incl = Err2 . We replace F by F − T λn+1 Err2∼ . Then after changing n

n

n

the choices of Err1n and Err3n , if necessary, we may assume that (4.4.54.1) and (4.4.54.3) hold and / ◦ f − F / ≡0 (n) ◦ Incl Incl

(4.4.54.2+)

mod T λn+2 .

We use (4.4.54.2+) and (4.4.54.3) to obtain / = 0 ∈ Der(BC [1], BC [1]). Err3n ◦ Incl 1 2

(4.4.56)

Using the fact that ((Evals=0 )1 , (Evals=1 )1 ) is surjective and Incl1 is split injective together with (4.4.56) and Lemma 4.4.43, we find a coderivation Err3∼ n ∈ Der(BC1 [1], BC2 [1]) such that (4.4.57.1) (4.4.57.2)

/ / ∼ ∼ (Eval s=0 ◦ Err3n , Evals=1 ◦ Err3n ) = Err3n , / = 0. Err3∼ ◦ Incl n

(n) by F (n) − T λn+1 Err3∼ . Then, (4.4.54.1), (4.4.54.2+) and We replace F n (4.4.54.3+)

/ s=s ≡ 0 mod T λn+2 / s=s ◦ F (n) − f ◦ Eval Eval 0 0

hold. Now we have Lemma 4.4.58. δ(Err1n ) = 0. Here δ is as in Definition 4.4.46. Proof. By definition  d ◦ Err1n + Err1n ◦ d = 0. Then the lemma follows from (4.4.55). By (4.4.54.2+), (4.4.54.3+) we have (4.4.59.1)

Err1n ◦ Incl1 = 0,

(4.4.59.2)

Evals=s0 ◦ Err1n = 0.

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We put D3 = Ker Eval0∗ ∩ Ker Eval1∗ ⊆ C2 [1]. We can now apply Lemma 4.4.3 and Lemma 4.4.49 to A = Err1n to obtain Fk,n+1 ∈ Hom(Bk C1 [1], D3 ) ⊆ Der(BC1 [1], BC2 [1]) k = 0, 1, 2, · · · such that (4.4.60.1) (4.4.60.2)

/ = 0, F∗,n+1 ◦ Incl 1 δ(F∗,n+1 ) + Err1n = 0.

It is easy to see that Fk,n+1 has the required properties. The proof of Proposition 4.4.52 is now complete.  To prove Theorem 4.2.34 from Proposition 4.4.52, we only need additional arguments in the case when f is strict and/or unital. In the case when f is strict, we can take Fk,i such that F0,i is zero, inductively as follows. Suppose F0,i is zero for i ≤ n. Then since f is strict, Err1n , Err2n , (n) so that (4.4.54.2+) Err3n is zero on B0 C 1 or B0 C1 . Hence when we modify F and (4.4.54.3+) will be satisfied, we may still assume F0,i = 0. Hence Err1n is zero on B0 C1 . Therefore we may take F0,n+1 = 0 by construction. The case where f is unital is treated in the next subsection. 4.4.7. Unital case II: the filtered version. In this subsection we prove the unital filtered version of Proposition 4.4.52. Let (C, m) be a unital filtered A∞ algebra.  red C[1] as folDefinition 4.4.61. We define partially reduced bar complex B lows. Putting Cred = C/Λ0,nov [e], we define ⎧ if k ≥ 2, ⎪ ⎨ C[1] ⊗ Bk−2 Cred [1] ⊗ C[1], red if k = 1, (4.4.62) Bk C[1] = C[1], ⎪ ⎩ if k = 0, Λ0,nov ,

 red C[1] =  B red C[1]. and B k k  red C[1]. (We can prove it in d : BC[1] → BC[1] induces a differential dred on B the same way as Lemma 4.4.24.) red We define operators sleft , sright : Bkred C[1] → Bk+1 C[1] by sleft (x) = e ⊗ x,

sright (x) = (−1)deg x x ⊗ e,

for k ≥ 1 and sleft = sright = 0 on B0red C[1].



Lemma 4.4.63. dred ◦ sleft + sleft ◦ dred = dred ◦ sright + sright ◦ dred = id, on red k≥1 Bk . The proof is an easy calculation so omitted. We choose a splitting

(4.4.64)

C = Cred ⊕ Λ0,nov [e].

Let fk,red : Bkred C1 [1] → C2 [1], k = 0, 1, 2, · · · be a series of homomorphisms. We define ˆfred : BC1 [1] → BC2 [1] in a way similar to (4.4.28) by ˆfred (x1 ⊗ e ⊗ · · · ⊗ e ⊗ xk ) = fk,red (x1 ) ⊗ e ⊗ · · · ⊗ e ⊗ fk,red (xk ),

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4.5. WHITEHEAD THEOREM IN A∞ ALGEBRAS

233

where xi ∈ Bki C1,red [1] ⊂ Bki C1 [1]. Lemma 4.4.65. A series of operators fk,red : Bkred C1 [1] → C2 [1], k = 0, 1, · · · satisfying the following conditions (4.4.66) which corresponds one to one to the unital filtered A∞ homomorphism C1 → C2 . (4.4.66.1) (4.4.66.2) (4.4.66.3) (4.4.66.4)

fred ◦ dred = d ◦ fred . fk,red ◦ sleft = fk,red ◦ sright = 0. f1,red (e) = e. f0,red ≡ 0 mod Λ+ 0,nov .

The proof is straightforward and is analogous to the proof of Lemma 4.4.30. Now we prove a unital version of Proposition 4.4.52. Namely suppose we have Fk,i : Bkred C1 [1] → C2 [1], which induces (n)

Fk

=

n 

k = 0, 1, 2 · · · , i = 0, 1, 2, · · · , n

Fk,i T λi : Bkred C1 [1] → C2 [1],

i=0

satisfying (4.4.66.2), (4.4.66.3), (4.4.66.4). We extend it to a coalgebra homomorphism (n)red : B  red C1 [1] → BC  2 [1], F which induces a unital homomorphism (n) : BC  1 [1] → BC  2 [1]. F We assume that it satisfies (4.4.53.1), (4.4.53.2), (4.4.53.3). We then want to find Fk,n+1 . (We remark that the case n = 0 is already proved in Subsection 4.4.4.) We define Err1n ∈ Der(BC1 [1], BC2 [1]), Err2n ∈ Der(BC 1 [1], BC2 [1]), Err3n ∈ Der(BC1 [1], BC 2 [1]), by formulas (4.4.54.1), (4.4.54.2), (4.4.54.3). Lemma 4.4.67. For ∗ = 1, 2, 3, the operators Err ∗n are induced by those on B red C 1 ⊕ B0 C 1 or B red C1 ⊕ B0 C1 . We also have: Err ∗n ◦ sleft = Err ∗n ◦ sright = 0. Proof. For Err2n and Err3n , the lemma is immediate from definition. For Err3n , the proof is similar to that of Lemma 4.4.35.  Using Lemma 4.4.67, the rest of the proof goes in the same way as that of Proposition 4.4.52. Hence we have proved the unital filtered version of Theorem 4.2.34. Therefore the proof of Theorem 4.2.34 is now complete.  4.5. Whitehead theorem in A∞ algebras In this section we prove Theorem 4.2.45. The strategy of the proof is similar to that of Theorem 4.2.34. Namely we will construct the required homotopy inverse by an induction on the number and the energy filtrations. To make the exposition transparent we develop a kind of obstruction theory. Then the proof looks quite similar to the proof of the well-known Whitehead theorem in algebraic topology.

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234

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HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

4.5.1. Extending AK homomorphisms to AK+1 homomorphisms. The main result of this subsection is the following Theorem 4.5.1. Let C i be AK+1 algebras and f : C 1 → C 2 an AK homomorphism. We consider an R module Hom(BK+1 C 1 [1], C 2 [1]). A coboundary operator δ1 on Hom(BK+1 C 1 [1], C 2 [1]) is defined by  1, δ1 (ϕ) = m1 ◦ ϕ + (−1)deg ϕ+1 ϕ ◦ m  1 : BC 1 [1] → BC 1 [1] is a coderivation induced by m1 on C 1 . (See (3.2.1).) where m We denote by H(Hom(BK+1 C 1 [1], C 2 [1]), δ1 ) the set of δ1 -cohomology classes. Theorem 4.5.1. There exists oK+1 (f) ∈ Hom(BK+1 C 1 [1], C 2 [1]) of degree 1, such that the following holds. (4.5.2.1) δ1 (oK+1 (f)) = 0. (4.5.2.2) [oK+1 (f)] = 0 as a δ1 -cohomology class if and only if there exists an AK+1 homomorphism extending f.  (4.5.2.3) If f is AK homotopic to f, then 

[oK+1 (f)] = [oK+1 (f )] ∈ H(Hom(BK+1 C 1 [1], C 2 [1]), δ1 ). (4.5.2.4)





If g : C 1 → C 1 , g : C 2 → C 2 are AK+1 homomorphisms, then [oK+1 (g ◦ f ◦ g)] = (g1 )∗ ◦ [oK+1 (f)] ◦ (g1 ⊗ · · · ⊗ g1 )∗ 



in H(Hom(BK+1 C 1 [1], C 2 [1]), δ1 ). Here 

g1 ⊗ · · · ⊗ g1 : BK+1 C 1 [1] → BK+1 C 1 [1] 

is induced by g1 : C 1 [1] → C 1 [1]. It induces homomorphism (g1 ⊗ · · · ⊗ g1 )∗ on δ1 cohomology. Remark 4.5.3. In fact, we already used a similar results in the course of the proof of Theorem 4.2.34. Since the notion of homotopy appears in the statement of Theorem 4.5.1, we repeat the arguments here in order to make clear that our argument is not circular. 

Proof of Theorem 4.5.1. We first take fK+1 = 0 and consider f 1···K+1 ∈ Hom(B1···K+1 C 1 [1], B1···K+1 C 2 [1]). It induces cochain homomorphisms f2···K+1 and f1···K by Lemmas 4.4.9 and 4.4.10. Therefore  d1···K+1 ◦f1···K+1 −f1···K+1 ◦  d1···K+1 is zero on B1···K C 1 [1] and its image is in B1 C 2 [1] = C 2 [1]. We define d1···K+1 ◦ f1···K+1 − f1···K+1 ◦  d1···K+1 ∈ Hom(BK+1 C 1 [1], C 2 [1]). oK+1 (f) =  Then oK+1 (f) satisfies  1 = 0. δ1 (oK+1 (f)) = m1 ◦ oK+1 (f) − oK+1 (f) ◦ m This gives (4.5.2.1).

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4.5. WHITEHEAD THEOREM IN A∞ ALGEBRAS

235

To prove (4.5.2.2), we only need to remark that fi , i = 1, · · · , K together with fK+1 define an AK+1 homomorphism if and only if δ1 (fK+1 ) = −oK+1 (f). Let us prove (4.5.2.4). We consider the coalgebra homomorphisms  ˆg : BC1···K+1 → BC1···K+1

and

 ˆ : BC2···K+1 g → BC2···K+1

which are induced by gˆi i = 1, · · · , K. Then the process to take hat commutes with composition. Now we consider the commutator ˆg ˆ ˆg } ◦ ˆf ◦ gˆ + g ˆ ˆf} ◦ ˆg + g ˆ ˆg}. ˆ ◦ ˆf ◦ g} ˆ = {d, ˆ ◦ {d, ˆ ◦ ˆf ◦ {d, {d,    ˆ The Hom(BK+1 C 1 [1], C 2 [1]) component of the left hand side is [oK+1 (ˆg ◦ ˆf ◦ g)]. ⊗K+1  ⊗K+1 The first term of the right hand side is −δ1 gK+1 ◦ f1 ◦ g1 and is a δ1 boundary. The second term of the right hand side is g1 ◦ oK+1 (f) ◦ (g1 ⊗ · · · ⊗ g1 ). The third term of the right hand side is −g1 ◦ f1 ◦ δ1 gK+1 and is a δ1 boundary. (4.5.2.4) follows. To prove (4.5.2.3) we need the following lemma.

Lemma 4.5.4. If f : C 1 → C 2 is A1 homotopic to g : C 1 → C 2 , then f∗ = g∗ : H ∗ (C 1 ; m1 ) → H ∗ (C 2 ; m1 ). Proof. Let h : C 1 → C2 be an A1 homotopy between f and g. By (4.2.2.3), Incl induces an isomorphism on m1 -cohomology. On the other hand, by (4.2.2.2), Evals=s0 ◦ Incl is identity for any s0 . Hence Evals=0∗ = Evals=1∗ : H ∗ (C2 , m1 ) → H ∗ (C 2 , m1 ). Therefore f∗ = Evals=0∗ ◦ h∗ = Evals=1∗ ◦ h∗ = g∗ , as required.



Now we prove (4.5.2.3). Let h : C 1 → C2 be an AK homotopy between f and f . By (4.5.2.4) and Lemma 4.5.4, we have 

[oK+1 (f)] = [oK+1 (Evals=0 ◦ h)] = Evals=0∗ [oK+1 (h)] 

= Evals=1∗ [oK+1 (h)] = [oK+1 (Evals=1 ◦ h)] = [oK+1 (f )]. 

The proof of Theorem 4.5.1 is now complete. We will use the following corollary in the next subsection.

Corollary 4.5.5. Let f : C 1 → C 2 be an AK+1 homomorphism, g : C 1 → C 2 an AK homomorphism, and let h : C 1 → C2 be an AK homotopy from f to g. Then g is extended to an AK+1 homomorphism g and h is extended to an AK+1 homotopy from f to g . Proof. Since (Evals=0 )1∗ [oK+1 (h)] = [oK+1 (f)] = 0 by (4.5.2.2) and (4.5.2.4) and (Evals=0 )1∗ induces an isomorphism on cohomology, it follows that h is extended to an AK+1 homomorphism. To prove the corollary we need to extend h with some additional properties. So we proceed as follows.  We consider hK+1 = Incl ◦ fK+1 . Then we find 

(Evals=0 )1 (oK+1 (h) + δ1 (hK+1 )) = 0

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236

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and



(Evals=0 )1 ◦ hK+1 = fK+1 . Since the kernel of Evals=0,∗ is δ1 -acyclic, it follows that we have ΔhK+1 such that 

oK+1 (h) + δ1 (hK+1 ) = δ1 (ΔhK+1 ) and (Evals=0 )1 (Δhk+1 ) = 0.

 hK+1

− ΔhK+1 . Then hk , k = 1, · · · , K + 1 define an AK+1 homomorphism denoted by  h. We define g = (Evals=1 )1 ◦  h. It is easy to see that   g is an AK+1 homomorphism which extends g. Moreover since (Evals=0 )1 ◦ hK+1 = f , it follows that  h is an A homotopy between f and g . 

We put hK+1 =

K+1

K+1

4.5.2. Proof of Theorem 4.2.45 I: the number filtration. Let f : C 1 → C 2 be a weak homotopy equivalence. (K)

: Proposition 4.5.6. Let g(K) : C 2 → C 1 be an AK homomorphism and h C 1 → C1 an AK homotopy from identity to g(K) ◦ f. Then g(K) can be extended (K) to an AK+1 homomorphism g(K+1) and h is extended to an AK+1 homotopy h

(K+1)

from identity to g(K+1) ◦ f. Eval

s=0 C 1 ←−−− −− 2 2 2

C1

g(K)

Eval

C1 −−−−s=1 −→ C 1 ←−−−− 3 ⏐ ⏐h

C2 2 2 2

f

C1 C 1 −−−−→ C 2 Diagram 4.5.1 



Proof. By Corollary 4.5.5, we have hK+1 such that hk , k = 1, 2, · · · , K and (K+1)

(K+1)

hK+1 define an AK+1 homomorphism h from C 1 to C1 and Evals=0 ◦h =  (K) id. Moreover Evals=1 ◦ hK+1 extends g ◦ f to an AK+1 homomorphism. Namely we have 

oK+1 (g(K) ◦ f) = −δ1 (Evals=1 ◦ hK+1 ).

(4.5.7)

Since f : C 1 → C 2 is a weak homotopy equivalence, it follows from (4.5.2.4) and (4.5.7) that [oK+1 (g(K) )] = 0. Hence we have gK+1 such that oK+1 (g(K) ) = −δ1 (gK+1 ).

(4.5.8) We define



Ξ = gK+1 ◦ (f1 )⊗K+1 − Eval1 ◦ hK+1 ∈ Hom(BK+1 C 1 [1], C 1 [1]). By (4.5.7) we have δ1 (Ξ) = 0. Since f is a weak homotopy equivalence, there exists a ⊗K+1 δ1 -cocycle ΔgK+1 ∈ Hom(BK+1 C 2 [1], C 1 [1]) such that [Ξ + (ΔgK+1 ◦ f1 )] = 0. Therefore, there exists Δ1 hK+1 ∈ Hom(BK+1 C 1 [1], C 1 [1]) such that (4.5.9)

⊗K+1

δ1 (Δ1 hK+1 ) = (gK+1 + ΔgK+1 ) ◦ f1

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− (Evals=1 )1 ◦ hK+1 .

4.5. WHITEHEAD THEOREM IN A∞ ALGEBRAS

237

Using (4.2.2.4), we find ΔhK+1 ∈ Hom(BK+1 C 1 [1], C1 [1]) such that (4.5.10)

(Evals=0 )1 ◦ ΔhK+1 = 0,

(Evals=1 )1 ◦ ΔhK+1 = Δ1 hK+1 . 

We now put gK+1 = gK+1 + ΔgK+1 , hK+1 = hK+1 + δ1 (ΔhK+1 ). Let us check that they have the required properties. Since ΔgK+1 is a δ1 -cocycle, it follows from (4.5.8) that oK+1 (g(K) ) = −δ1 (gK+1 ). Namely gK+1 gives an extension of g(K) to an AK+1 homomorphism g(K+1) . Sim(K)

to an AK+1 homomorphism h ilarly, hK+1 extends h By (4.5.10), we have (Evals=0 )1 ◦ h

(K+1)

= (Evals=0 )1 ◦ h

(K+1)

(K+1)

.

= id.

Finally by (4.5.9) and (4.5.10), we obtain 

⊗K+1

(Evals=1 )1 ◦ hK+1 = (Evals=1 )1 ◦ hK+1 + δ1 (Δ1 hK+1 ) = gK+1 ◦ f1 Thus h

(K+1)

is an AK+1 homotopy from id to g(K+1) ◦ f.

. 

Now we are ready to complete the proof of Theorem 4.2.45 (1). Let f : C 1 → C 2 be a weak homotopy equivalence. Since f1 : (C 1 , m1 ) → (C 2 , m1 ) is a cochain homotopy equivalence, there exists a cochain map g1 : (C 2 , m1 ) → (C 1 , m1 ) such that g1 ◦ f1 is cochain homotopic to identity. g1 defines an A1 homomorphism: a g(1) : C 2 → C 1 . Let h1 : C 1 → C 1 be a cochain homotopy from identity to g1 ◦ f1 . b We use (4.2.2.4) to obtain h1 : C 1 → C1 such that b

a

(Evals=1 )1 ◦ h1 = h1 ,

b

(Evals=0 )1 ◦ h1 = 0.

We put b

b

h1 = (Incl)1 + m1 ◦ h1 + h1 ◦ m1 : C 1 → C1 . It is easy to check that h1 defines an A1 homotopy from id to g(1) ◦ f. We now apply Proposition 4.5.6 inductively and obtain an A∞ homomorphism g : C 2 → C 1 and an A∞ homotopy h from id to g ◦ f. We finally prove that f ◦ g is homotopic to identity. We remark that g is a weak homotopy equivalence because g1 is chain homotopic to a chain homotopy inverse  of f1 . So we apply the above argument to obtain an A∞ homomorphism f such   that f ◦ g is homotopic to identity. Therefore f ◦ g ◦ f is homotopic to f by Lemma 4.2.43. On the other hand, since g ◦ f is homotopic to identity, Lemma 4.2.43 again    implies that f ◦ g ◦ f is homotopic to f . Thus we proved that f is homotopic to f.  Then, since f ◦ g is homotopic to identity, f ◦ g is homotopic to identity. The proof of Theorem 4.2.45 (1) is now complete.  4.5.3. Unital case: the unfiltered version. We prove the unfiltered versions of Theorem 4.2.45a and 4.2.45b in this subsection. As for unfiltered versions of Theorem 4.2.45a the proof is parallel to the proof of Theorem 4.2.45 (1). The main difference is that we replace Theorem 4.5.1 by the following Theorem 4.5.1a. Let f : C 1 → C 2 be a unital AK homomorphism.

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Let us recall Bk C 1,red [1] = C 1 [1]/R[e] ⊗ · · · ⊗ C 1 [1]/R[e] . 

 k times

ˆ 1 induces an operator on Bk C 1,red [1]. We denote Since m1 e = 0, it follows that m ˆ 1 and m1 : C 2 [1] → C 2 [1] induce a differential δ1 on ˆ 1 . Hence m it also by m Hom(Bk C 1,red [1], C 2 [1]). Theorem 4.5.1a. There exists ored K+1 (f) ∈ Hom(BK+1 C 1,red [1], C 2 [1]) of degree 1, such that the following holds. (4.5.2a.1) δ1 (ored K+1 (f)) = 0. (4.5.2a.2) [ored K+1 (f)] = 0 as a δ1 -cohomology class if and only if there exists an AK+1 homomorphism extending f.   red (4.5.2a.3) If f is AK homotopic to f, then [ored K+1 (f)] = [oK+1 (f )] as δ1-cohomology class.   (4.5.2a.4) If g : C 1 → C 1 and g : C 2 → C 2 are unital AK+1 homomorphisms, then   red [ored K+1 (g ◦ f ◦ g)] = (g1 )∗ ◦ [oK+1 (f)] ◦ (g1 ⊗ · · · ⊗ g1 )∗ in H(Hom(BK+1 C 1,red [1], C 2 [1]), δ1 ). Here red

red

g1 ⊗ · · · ⊗ g1 : BK+1 C 1 [1] → BK+1 C 1 [1] red

red

is induced by g1 : C 1 [1] → C 1 [1]. Proof. We use Lemma 4.4.30 (or its AK analogue) to associate an R module

homomorphism fk,red : Bkred C 1 [1] → C 2 [1] for k = 1, · · · , K. They induce fred 1···K+1 ∈ Hom(B1···K+1,red C 1 [1], B1···K+1 C 2 [1]) 

by putting fK+1,red = 0. We then consider     red ored,0 K+1 (f) = d1···K+1 ◦ f1···K+1 − f1···K+1 ◦ d1···K+1 ∈ Hom(BK+1 C 1 [1], C 2 [1]). red,0 We can show that ored,0 K+1 ◦ sleft = oK+1 ◦ sright = 0 in a way similar to the proof of Lemma 4.4.35. Hence we obtain ored K+1 (f) ∈ Hom(BK+1 C 1,red [1], C 2 [1]). Then it is a straightforward analog to the proof of Theorem 4.5.1 to check that it has required properties. 

We use Theorem 4.5.1a in place of Theorem 4.5.1 and the rest of the proof goes in the same way. We thus proved the unfiltered version of Theorem 4.2.45a.  We next turn to the unfiltered version of Theorem 4.2.45b. We start with a homotopy-unital A∞ homomorphism C 1 → C 2 which is, by definition, a unital A∞ + + + homomorphism f : C 1 → C 2 such that (4.5.11)

+

fk (Bk (C 1 [1])) ⊆ C 2 [1]. +

+

We then need to find a homotopy inverse g+ : C 2 [1] → C 1 [1] such that (4.5.12)

g+ k (Bk (C 2 [1])) ⊆ C 1 [1].

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4.5. WHITEHEAD THEOREM IN A∞ ALGEBRAS

239 +

So we need some additional argument to apply Theorem 4.2.45a to f . (+) + Let C1 be a model of [0, 1] × C 1 which is unital. We put (4.5.13)

−1 C1 = (Evals=0 )−1 1 (C1 ) ∩ (Evals=1 )1 (C1 ).

Since (Evals=s0 )k = 0 for k = 1, it follows that C1 is an A∞ subalgebra. (We remark that C1 is an A∞ subalgebra of C1+ .) It is easy to see that C1 is a model of [0, 1] × C 1 . Now we prove the following by induction on K. +

+

+

Proposition 4.5.6b. Let f : C 1 → C 2 be a unital weak homotopy equiva+ + lence satisfying (4.5.11) and g+(K) : C 2 → C 1 a unital AK homomorphism satis+(K)

fying (4.5.12). Let h + g+(K) ◦ f such that (4.5.14)

(+)

+

: C 1 → C1

be a unital AK homotopy from identity to

+

hk (Bk (C 1 [1])) ⊆ C1 .

Then g+(K) can be extended to a unital AK+1 homomorphism g+(K+1) sat+(K) +(K+1) is extended to a unital AK+1 homotopy h from isfying (4.5.12) and h + identity to g(K+1) ◦ f satisfying (4.5.14). Proof. The proof is parallel to that of Theorem 4.5.1a after we remark the following two points. + + +red First, the A∞ algebra C i has a canonical splitting C i = C i ⊕ Re, where +red Ci = C i ⊕ Rf. Second, to keep Conditions (4.5.12) and (4.5.14), we modify the obstruction +(K) theory Theorem 4.5.1a a bit. Namely, for example, the obstruction ored ) for K+1 (g +(K) g to be extended to a unital AK+1 homomorphism satisfying (4.5.12) lies in the +red +red + group Hom(BK+1 C 1 [1],C 2 [1]) in place of the group Hom(BK+1 C 1 [1],C 2 [1]). The proof now goes in the same way. The proof of Proposition 4.5.6b and the unfiltered version of Theorem 4.2.45b is now complete.  4.5.4. Extending filtered A∞ homomorphism modulo T λi to modulo T . We next proceed to the proof of Theorem 4.2.45 (2). We will use the induction on the energy filtration for this purpose. Let G ⊂ R≥0 × 2Z satisfy Condition 3.1.6 and we assume all filtered A∞ algebras as well as filtered A∞ homomorphisms we consider are G-gapped. As in Subsection 4.4.6 we can enumerate the image of the projection of G to R≥0 denoted by Spec(G) = {λ0 , λ1 , · · · } so that λi < λi+1 , limi→∞ λi = ∞. Let λi+1

fk : Bk C1 [1] → C2 [1],

k = 0, 1, 2, · · ·

be a sequence of degree 0 homomorphisms which satisfy (3.2.27). They induce a  2 [1] as in (3.2.28).  1 [1] → BC coalgebra homomorphism f : BC Definition 4.5.15. We say f is a filtered A∞ homomorphism modulo T λi if d ◦ f − f ◦ d ≡ 0

mod T λi .

Here i = 1, 2, · · ·

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HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

Let f, g be filtered A∞ homomorphisms modulo T λi from C1 to C2 and C2 a model of [0, 1] × C2 . We say that they are homotopic modulo T λi if there is a filtered A∞ homomorphism h modulo T λi from C1 to C2 and such that Evals=0 ◦ h ≡ f mod T λi ,

Evals=1 ◦h ≡ g

mod T λi .

We can define the composition of filtered A∞ homomorphisms modulo T λi in an obvious way. We remark that a filtered A∞ homomorphism f : C1 → C2 modulo T λi induces an A∞ homomorphism f : C 1 → C 2 . Now we are ready to start the proof of Theorem 4.2.45 (2). Definition-Proposition 4.5.16. Assume that f(i) is a filtered A∞ homomorphism modulo T λi in the sense of Definition 4.5.15. Then there exists a Hochschild (i) cohomology class [oλi (f(i) )] ∈ HH(C 1 , C 2 ; f ) with the following properties. (4.5.17.1) [oλi (f(i) )] = 0 if and only if there exists f(i) such that: (4.5.17.1.1) f(i) is a filtered A∞ homomorphism modulo T λi+1 . (4.5.17.1.2) f(i) ≡ f(i) mod T λi . (4.5.17.2) If f(i) is a filtered A∞ homomorphism modulo T λi homotopic to f(i) modulo T λi , then we have [oλi (f(i) )] = [oλi (f(i) )]. (4.5.17.3) If g : C1 → C1 , g : C2 → C2 be filtered A∞ homomorphisms modulo T λi+1 , then [oλi (g ◦ f(i) ◦ g)] = (g, g )∗ [oλi (f(i) )]. Remark 4.5.18. We need some explanation on the meaning of (4.5.17.2). (i)

(i)

Since we assume that f , f : C 1 → C 2 are homotopic by a homotopy h : C 1 → C2 , we have a canonical isomorphism HH(C 1 , C 2 ; f ) ∼ = HH(C 1 , C 2 ; f (i)

(i)

).

Therefore the equality (4.5.17.2) makes sense under the identification via this canonical isomorphism. The proof of this isomorphism follows from the fact that the homomorphisms (Evals=0 )1 and Incl1 induce an isomorphism HH(C 1 , C2 ; h) ∼ = HH(C 1 , C 2 ; f) and (Evals=1 )1 and Incl1 induce an isomorphism  HH(C 1 , C2 ; h) ∼ = HH(C 1 , C 2 ; f ).

This follows from Proposition 4.4.48. Proof of Proposition 4.5.16. By assumption, there exists oλi (f(i) ) ∈ Hom(BC 1 [1], BC 2 [1]) such that

d ◦ f(i) − f(i) ◦ d ≡ T λi oλi (f(i) ) mod T λi+1 .

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4.5. WHITEHEAD THEOREM IN A∞ ALGEBRAS

241

Since d is a coderivation and f(i) is a coalgebra homomorphism, it easily follows that oλi (f(i) ) is a coderivation. Moreover, since  d ◦ f(i) − f(i) ◦ d)  + (d ◦ f(i) − d ◦ f(i) )d = 0, d( (i)

we have δ(oλi (f(i) )) = 0. Therefore it defines [oλi (f(i) )] ∈ HH(C 1 , C 2 ; f ). We can check the properties (4.5.17.1), (4.5.17.2), (4.5.17.3) in a way similar to the proof of Theorem 4.5.1.  4.5.5. Proof of Theorem 4.2.45 II: the energy filtration. Now we go back to the proof of Theorem 4.2.45 (2). We take G and λi as in the last subsection. Let Ci be G-gapped filtered A∞ algebras, and f : C1 → C2 be a G-gapped filtered A∞ homomorphism. We assume that f is a weak homotopy equivalence, i.e., the unfiltered A∞ homomorphism f : C 1 → C 2 induced by f defines an isomorphism in m1 -cohomology. Then f is a homotopy equivalence by Theorem 4.2.45 (1), which we already proved. We prove the following proposition by induction on n. Proposition 4.5.19. Let g(n) : C2 → C1 be a filtered A∞ homomorphism modulo T λn and let h(n) : C1 → C1 be a filtered A∞ homotopy modulo T λn between id and g(n) ◦ f. Then there exists g(n+1) : C2 → C1 a filtered A∞ homomorphism modulo T λn+1 and h(n+1) : C1 → C1 a filtered A∞ homotopy modulo T λn+1 between id and g(n+1) ◦ f. Moreover we have g(n+1) ≡ g(n) mod T λn and h(n+1) ≡ h(n) mod T λn . Eval

s=0 C1 ←−−− −− 2 2 2

C1

Eval

g(n)

C1 −−−−s=1 −→ C1 ←−−−− 3 ⏐h ⏐

C2 2 2 2

f

C1 C1 −−−−→ C2 Diagram 4.5.2

Proof. We consider oλn (hn ). Since (Eval0 , id)∗ [oλn (hn )] = oλn (id) = 0, we have [oλn (hn )] = 0 by Proposition 4.4.48 and (4.5.2.4). Indeed, we can prove more. Namely, we remark that oλn (hn ) lies in: , + (4.5.20) K := Ker (Evals=0 , id)∗ : Der(BC 1 [1], BC2 [1]) → Der(BC 1 [1], BC 2 [1]) . We also remark that K is an acyclic complex by Proposition 4.4.48 and (4.2.3.3). Thus there exists hn+1 ∈ K such that −δ(hn+1 ) = oλn (hn ) and (Evals=0 , id)∗ (hn+1 ) = 0. It is easy to see that h(n+1) = hn + T λn+1 hn+1 is an A∞ homomorphism modulo T λn+1 . Since −δ((Evals=1 , id)∗ (hn+1 )) = (Evals=1 , id)∗ oλn (h(n) ) = oλn (g(n) ◦ f) and f induces an isomorphism on the Hochschild cohomology (Proposition 4.4.48), there exists gn+1 ∈ Der(BC 2 [1], BC 1 [1]) such that δ(gn+1 ) = oλn (g(n) ). Then we have δ((Evals=1 , id)∗ (hn+1 ) − (id, f)∗ (gn+1 )) = 0.

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HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

Moreover since f induces an isomorphism on the Hochschild cohomology (Proposition 4.4.48), there exist Δgn+1 ∈ Der(BC 2 [1], BC 1 [1])

and Δ1 hn+1 ∈ Der(BC 2 [1], BC 1 [1])

such that (id, f)∗ (gK+1 ) + (id, f)∗ (Δgn+1 ) − (Eval1 , id)∗ (hn+1 ) = δ(Δ1 hn+1 ). By (4.2.3.3) and Lemma 4.4.43 we have Δhn+1 ∈ Der(BC 2 [1], BC1 [1]) such that / Eval s=1 ◦ Δhn+1 = Δ1 hn+1 ,

/ Eval s=0 ◦ Δhn+1 = 0.

We put gn+1 = gn+1 + Δgn+1 , hn+1 = hn+1 + δ(Δhn+1 ). Then it is easy to check that they satisfy the required properties.  Now we are ready to complete the proof of Theorem 4.2.45 (2). We remark that Theorem 4.2.45 (1) which we proved in Subsection 4.5.2 implies that the assumption of Proposition 4.5.19 holds for n = 1. Therefore suppose we have g(n) and h(n) by induction. Since g(n+1) ≡ g(n) mod T λn and h(n+1) ≡ h(n) mod T λn , it follows that lim g(n) , lim h(n) exist. Hence we have g such that g ◦ f is homotopic to the identity. The proof that f ◦ g ∼ id goes as in the last step of the proof of Theorem 4.2.45 (1). The proof of Theorem 4.2.45 is now complete.  For the unital or the homotopy-unital version Theorem 4.2.45a, 4.2.45b, the argument is a modification of the above. Since the required modification of the proof is similar to those of Subsection 4.5.3 and Subsection 4.4.7, we omit the details.  Proof of Proposition 4.3.16. We prove the property f ≡ id mod Λ+ 0,nov . We use the notation in the proof at Section 4.3. We remark that if we reduce the ˜ coefficient of (C, mb ) to R then we have a model of [0, 1] × C. Hence we can use Incl to start the induction of the construction of the homotopy inverse of Evals=0 : ˜ (C, mb ) → (C, mb0 ). Therefore the homotopy equivalence f : (C, mb0 ) → (C, mb1 )  has a property f = Evals=1 ◦ Incl = id as required. 4.6. Homotopy equivalence of A∞ algebras: the geometric realization 4.6.1. Construction of A∞ homomorphisms. The aim of this section is to prove that the filtered A∞ algebra constructed in Theorem 3.5.11 is independent of various choices involved. More precisely we prove the following Theorem 4.6.1 (and Theorem 4.1.1). We recall the notations from Section 4.1. Let (M, ω), (M  , ω  ) be symplectic manifolds and L ⊂ M and L ⊂ M  be n dimensional Lagrangian submanifolds. Let ψ : M −→ M  be a symplectic diffeomorphism such that ψ(L) = L . Let J be a compatible almost complex structure on M . We use J to construct the moduli spaces Mmain k+1 (β) as in Section 3.5. To specify the choices of L and J ((L, J), β). Let J  be a compatible almost structure on M  . We we write Mmain k+1 main  then define Mk+1 ((L , J  ), β) in a similar way. (We remark that we do not assume any genericity condition on J, J  . The transversality we need will be achieved by abstract perturbations, not by perturbation of J.)

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4.6. GEOMETRIC REALIZATION

243

In Section 3.5 we took a countable set X1 (L) of smooth singular simplices and considered a subcomplex of S(L; Q) spanned by it. We denote this subcomplex by C1 (L; Q). For Pi ∈ X1 (L) we considered also the fiber product main  Mmain k+1 ((L, J), β; P ) = Mk+1 ((L, J), β)ev × (P1 × · · · × Pk ).

It is a space with Kuranishi structure. Proposition 3.5.2 (whose proof will be completed in Section 7.2) asserts that we can choose X1 (L) and a system of multisections  ) such that s on Mk+1 ((L, J), β; P −1  s (0), ev0 ) Mmain k+1 ((L, J), β; P ) = (s

is contained in C1 (L; Q) for any Pi ∈ X1 (L), by taking appropriate triangulation of s−1 (0). Proposition 3.5.2 also asserts that the multisections s are compatible in the  compactification of Mmain k+1 ((L, J), β; P ). We then defined  s mk,β (P1 , · · · , Pk ) = Mmain k+1 ((L, J), β; P ) . To specify the choices of J and s we write mJ,s k,β . These maps define a structure of filtered A∞ algebra on C1 (L; Λ0,nov ). We make similar choices for L , J  , and obtain a countably set of smooth singular simplices X2 (L ) which generates a subcomplex C2 (L ; Q) and a moduli       s space Mmain k+1 ((L , J ), β; P ) for Pi ∈ X2 (L ). It induces a structure of filtered A∞ 



algebra on C2 (L ; Λ0,nov ). We denote it by mJk,β,s . We recall that we chose C1 (L; Q) and C2 (L ; Q) so that their cohomology groups are isomorphic to the cohomology groups of L and L respectively. Theorem 4.6.1. There exists a countable set of smooth singular simplices X3 (L ) which generates a subcomplex C3 (L ; Q) of S(L ; Q), a structure of filtered   A∞ algebra mJk,β,s on C3 (L ; Λ0,nov ), and a gapped filtered A∞ homomorphism f : C1 (L; Λ0,nov ) → C3 (L ; Λ0,nov ) with the following properties:    (4.6.2.1) s is a system of multisections on Mmain k+1 ((L , J ), β; P ) which is com patible at their boundaries. Here Pi ∈ C3 (L ; Q). (4.6.2.2) X3 (L ) ⊃ X2 (L ). Moreover, if Pi ∈ X2 (L ), then s = s on the moduli    space Mmain k+1 ((L , J ), β; P ). (4.6.2.3) The cohomology of C3 (L ; Q) (with respect to the usual coboundary operator) is isomorphic to the cohomology of L . (4.6.2.4) The homomorphism

(f1 )∗ = ψ −1∗ : H ∗ (C1 (L; Q), m1 ) ∼ = H ∗ (L; Q) → H ∗ (C3 (L ; Q), m1 ) ∼ = H ∗ (L ; Q) induced by f coincides with the isomorphism ψ −1∗ induced by the diffeomorphism ψ −1 . Corollary 4.6.3. The filtered A∞ algebras (C1 (L; Λ0,nov ), mJ,s k ) is homotopy   equivalent to (C2 (L ; Λ0,nov ), mJk ,s ). Corollary 4.6.3 is a restatement of Theorem 4.1.1.

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Proof of Theorem 4.6.1 ⇒ Corollary 4.6.3. By (4.6.2.4), f is a weak homotopy equivalence. Hence by Theorem 4.2.45 it is a homotopy equivalence. On the other hand, by (4.6.2.2) the natural inclusion C2 (L ; Q) ⊂ C3 (L ; Q) induces a filtered A∞ homomorphism. It is a homotopy equivalence by (4.6.2.3) and Theorem 4.2.45. Corollary 4.6.3 now follows from Corollary 4.2.44.  The proof of Theorem 4.6.1 occupies the rest of this subsection. Strictly speaking, we presume Proposition 4.6.14 holds in this subsection, which is related to how to choose the countable set X3 (L ) and will be proved in Section 7.2. So presuming Proposition 4.6.14, we will complete the proof of Theorem 4.6.1 in this subsection. We start with preparing more notations. Let J ψ = ψ∗ J. Since the set of compatible almost complex structures are contractible, we can find a family of compatible almost complex structures {Jρ }ρ = {Jρ | ρ ∈ [0, 1]} such that J0 = J ψ (= ψ∗ J),

J1 = J  .

We remark that the diffeomorphism ψ : L → L induces an isomorphism between the vector spaces of smooth singular chains on L and ones on L . We denote by X1 (L ) and by C1 (L ; Q) the images of X1 (L) and C1 (L; Q) by this isomor ψ  phism respectively. Using J ψ we obtain a moduli space Mmain k+1 ((L , J ), β; P ). If  Pi = ψ(Pi ), then obviously we have  ψ   ∼ main  Mmain k+1 ((L , J ), β; P ) = Mk+1 ((L, J), β; P )

(4.6.4)

as spaces with Kuranishi structures. Therefore a system of multisections s on main  ψ   Mmain k+1 ((L, J), β; P ) induces a system of multisections on Mk+1 ((L ,J ),β;P ). We ψ denote it by s . We thus obtain a structure of filtered A∞ algebra on C1 (L ; Λ0,nov ). We deψ

ψ

note it by mJk ,s . It follows from (4.6.4) that ψ induces (not only a homotopy equivalence but also) a canonical isomorphism J  ψ∗ : (C1 (L; Λ0,nov ), mJ,s k ) −→ (C1 (L ; Λ0,nov ), mk

ψ

,sψ

)

of filtered A∞ algebras. Remark 4.6.5. By the discussion above we may replace L, J, s by L , J ψ , sψ and may assume ψ = id in the proof of Theorem 4.6.1. However we do not do so, since in the next subsection we show that the homotopy equivalence constructed in Theorem 4.6.1 is independent of the symplectic isotopy of ψ, where we need to consider a non-constant family of isomorphisms (4.6.4). Now to construct f in Theorem 4.6.1 we use another moduli space of bordered pseudo-holomorphic curves of genus zero. Here we need to consider the case where components are pseudo-holomorphic with respect to the compatible almost complex structure which may vary on the components. Defining such a moduli space is our next task. Let (Σ, z) be an element of Mmain k+1 . Let {Σα | α ∈ A} be the set of the components of Σ. (When Σ has sphere bubbles, we include each of the tree of sphere bubbles to the disc component where it is rooted. In particular the boundary ∂Σα of each component Σα is S 1 .) We define a partial order on A as follows: Let

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4.6. GEOMETRIC REALIZATION

245

α1 , α2 ∈ A, and let Σα1 , Σα2 be the corresponding components of Σ. Let Σα0 be the component containing the 0-th marked point. Definition 4.6.6. We say α1 ≤ α2 if every path joining Σα1 to Σα0 intersects with Σα2 . It is easy to see that Definition 4.6.6 defines a partial order on A. (α0 is maximum among α’s.) Now we consider the system ((Σ, z), (uα ), (ρα )) with the following properties: (4.6.7.1) (4.6.7.2) (4.6.7.3) (4.6.7.4) (4.6.7.5)

uα : (Σα , ∂Σα ) → (M  , L ) is Jρα holomorphic. ρα ∈ [0, 1]. If α1 ≤ α2 , then ρα1 ≤ ρα2 . ((Σ, z), (uα )) is stable in the sense of Definition 2.1.24.  The homology class of uα is β(α) and α∈A β(α) = β. If z ∈ Σα ∩ Σα , then uα (z) = uα (z).

ρ ≥

uα5

ρ4 ≤ ρ2 ρ ρ

≤

≤ ≤

ρ0

uα4

uα2

z2 z1

1

z0

uα0 zk

uα1 uα3

uαL is Jρ holomorphic. L

Figure 4.6.1 We identify two such systems if there exists an isomorphism between them in an obvious sense. We remark that Definition 4.6.6 is similar to the “time ordered product” which appears in quantum field theory.   Definition 4.6.8. We define Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) to be the set of all equivalence classes ((Σ, z), (uα ), (ρα )) satisfying (4.6.7).   Here and hereafter top(ρ) in the symbol Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) means that we are taking time ordered product with respect to the ρ-parameter in the sense mentioned above. We are considering family of almost complex structures parametrized by several variables and consider moduli spaces using it. In the later argument, we take either time ordered product as above or take fiber product at each time (time-wise product). For the latter case we use the notation twp(s) to specify which kind of product we take there. These two moduli spaces are related to two methods to prove independence of Floer cohomologies of various choices, respectively. See the discussion right after (2.3.13) and also Subsection 7.2.14.   As usual, our moduli space Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) has a Kuran(i)

(i)

ishi structure with boundary. A sequence of elements ((Σ(i) , z(i) ), (uα ), (ρα )) of

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  Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) goes to the boundary in the sense of Kuranishi structure if one of the following occurs.

(4.6.9.1) (4.6.9.2) (4.6.9.3) (4.6.9.4)

(i)

One of the components Σα splits into two components. (i) (i) (i) (i) limi→∞ ρα = limi→∞ ρα , where Σα ∩ Σα = ∅. (i) limi→∞ ρα = 0. (i) limi→∞ ρα = 1.

(See Figure 4.6.2-4.6.5.) Each of the boundary components corresponding to (4.6.9.1)–(4.6.9.4) is described as a fiber product of moduli spaces. (See Section A1.2 for the definition of the fiber product of spaces with Kuranishi structure.)

ev0

L

ev0

Figure 4.6.2 (4.6.9.1)

(i)

ρα

(i)

ρα

ev0

(i) −

ρα

(i)

ρα

0

Figure 4.6.3 (4.6.9.2)

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4.6. GEOMETRIC REALIZATION

247

P3 P4 ρ = 0 ρ P5

ρ3

P2

ρ

P2

P3

ρ2

ρ2

P1

P1

P6

ev0

ρ

0

ρ1

ρ1

P8

ev0

ρ0 = 1

P4

P6

P5

P7

Figure 4.6.4 (4.6.9.3)

Figure 4.6.5 (4.6.9.4)

We observe that (4.6.9.1) cancels with (4.6.9.2). Therefore we only need to consider (4.6.9.3) and (4.6.9.4). We remark that the moduli space describing (4.6.9.3)     is a fiber product of the moduli spaces Mmain +1 (M , L , {Jρ }ρ : β ; top(ρ)) for β ≤ β main  ψ  ψ and of M +1 ((L , J ), β ), since J0 = J . Similarly the moduli space describing    (4.6.9.4) is a fiber product of the moduli spaces Mmain +1 (M , L , {Jρ }ρ : β ; top(ρ))    for β  ≤ β and of Mmain  +1 ((L , J ), β ). (We go back to this point, which is the main idea of the proof of Theorem 4.6.1, at the end of the proof.) We define   k  ev = (ev1 , · · · , evk , ev0 ) : Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) → L × L

by



ev0 (((Σ, z), (uα ), (ρα ))) = u(z0 ), evi (((Σ, z), (uα ), (ρα ))) = ψ −1 (u(zi )),

(i = 1, · · · , k).

Definition 4.6.10. For Pi ∈ X1 (L), we define    Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P )

=

  Mmain k+1 (M , L , {Jρ }ρ

: β; top(ρ))(ev1 ,··· ,evk ) ×



k 0

 Pi

,

i=1

where the right hand side is the fiber product over Lk .   Let stop(ρ) be a system of multisections of the space Mmain k+1 (M , L , {Jρ }ρ :  ). (We will describe the properties which stop(ρ) satisfies in Proposition β; top(ρ); P 4.6.14 below.) We put

(4.6.11)

   stop(ρ) = ((stop(ρ) )−1 (0), ev0 ). Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P )

Here the right hand side is regarded as a singular chain by taking an appropriate triangulation. See Definition A1.28. (As for the orientation of this space, see Section 8.9.) We now define:

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Definition 4.6.12. (4.6.13.1) (4.6.13.2) = (1, β0 ).

f1,β0 (P ) = ψ(P ).    stop(ρ) , for (k, β) fk,β (P1 , · · · , Pk ) = Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P )

(4.6.13.3)

fk (P1 , · · · , Pk ) =



β fk,β (P1 , · · ·

, Pk ) ⊗ T E(β) eμL (β)/2 .

We remark that the set of β for which   Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) = ∅

and E(β) < E0 is finite for any E0 by the Gromov compactness. Therefore fk (P1 , · · · , Pk ) is well defined and f is gapped. In order to prove Theorem 4.6.1 we need to clarify where the right hand side of (4.6.13) belongs. This is related to the choice of the countable set X3 (L ). We will prove this point in detail in Subsection 7.2.9 (see Proposition 7.2.100). Now we can state the properties we need for stop(ρ) . Proposition 4.6.14. We can find X3 (L ), s and stop(ρ) with the following properties: (4.6.15.1) (4.6.2.1), (4.6.2.2), (4.6.2.3) are satisfied. (4.6.15.2) For any Pi ∈ X1 (L), the Q chain fk,β (P1 , · · · , Pk ) defined by (4.6.13) is in C3 (L , Q). In other words, fk (P1 , · · · , Pk ) ∈ C3 (L , Λ0,nov ). In particular, ψ(P ) ∈ C3 (L , Λ0,nov ) for each P ∈ X1 (L).   (4.6.15.3) stop(ρ) is compatible with sψ at the boundary of Mmain k+1 (M , L , {Jρ }ρ :  ) corresponding to (4.6.9.3). β; top(ρ); P   (4.6.15.4) stop(ρ) is compatible with s at the boundary of Mmain k+1 (M , L , {Jρ }ρ :  ) corresponding to (4.6.9.4). β; top(ρ); P We will prove Proposition 4.6.14 in Subsection 7.2.9. Precisely speaking, in Subsection 7.2.9 we fix E0 and prove Proposition 4.6.14 only for β with ω[β] ≤ E0 . It defines filtered An,K homomorphism (in the sense defined in Subsection 7.2.6) for any n, K. Then using homological algebra we will construct filtered A∞ homomorphism in Subsection 7.2.11. Compare this also with Remark 3.5.13. Here we complete the proof of Theorem 4.6.1 assuming Proposition 4.6.14. For this purpose we only need to check that f is a filtered A∞ homomorphism. Let Pj ∈ X1 (L). We put Pj = (|Pj |, fj ), where |Pj | is a simplex and fj : |Pj | → L is a smooth map. We consider the boundary of the chains fk,β (P1 , · · · , Pk ). Let (i) (i) (i) ((Σ(i) , z(i) ), (uα ), (ρα )) be a sequence and xj ∈ |Pj | such that (i)

(i) evj ((Σ(i) , z(i) ), (u(i) α ), (ρα )) = fj (xj )

for j = 1, · · · , k. We thus have a sequence (i)

(i)

(i) (((Σ(i) , z(i) ), (u(i) α ), (ρα )); x1 , · · · , xk )    of elements of the moduli space Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P ). Now this sequence will go to the boundary if one of the following happens.

(4.6.16.1) (4.6.16.2) (4.6.16.3)

(4.6.9.1) occurs. (4.6.9.2) occurs. (4.6.9.3) occurs.

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4.6. GEOMETRIC REALIZATION

(4.6.16.4) (4.6.16.5)

249

(4.6.9.4) occurs. (i) One of xj goes to the boundary of Pj .

(4.6.16.1) cancels with (4.6.16.2) as we remarked before. (4.6.16.3) corresponds to the sum   fk+−m,β1 (P1 , · · · , P−1 , ,m β ,β =  0;β=β +β 1 2 1 2 (4.6.17) mJ,s m−+1,β2 (P , · · · , Pm ), Pm+1 , · · · , Pk ). For example, Figure 4.6.4 corresponds f7 (P1 , P2 , m2 (P3 , P4 ), P5 , P6 , P7 , P8 ). (4.6.16.4) corresponds to the sum 



m,1=1 ≤2 ···≤m =k+1 β1 ,β2,i =0;β=β1 +

(4.6.18)

m−1

β2,i

 f2 −1 ,β2,1 (P1 , · · · , P2 −1 ), · · · fi+1 −i ,β2,i (Pi , · · ·  · · · , Pi+1 −1 ), fm −m−1 ,β2,m−1 (Pm−1 , · · · , Pm −1 ) . i=1

  ,s mJm−1,β 1

For example Figure 4.6.5 corresponds to m5 (ψ(P1 ), f2 (P2 , P3 ), ψ(P4 ), f1 (P5 ), ψ(P6 )). (We remark that ψ(P ) = f1,β0 (P ).) (4.6.16.5) corresponds to the sum  (4.6.19) fk,β (P1 , · · · , m1 (Pj ), · · · , Pk ). j

Thus we find that m1 (fk,β (P1 , · · · , Pk )) is sum of (4.6.17), (4.6.18) and (4.6.19). (We do not discuss sign here. The sign is discussed in Section 8.9.) This implies that f is a filtered A∞ homomorphism.  Remark 4.6.20. In the discussion above, we need to perturb the moduli space

   Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P ) by our multisection and use the perturbed modmain    )s . However the compatibility of the uli space Mk+1 (M , L , {Jρ }ρ : β; top(ρ); P

multisections asserted in Proposition 4.6.14 is designed so that the argument above goes through including the perturbations induced by stop(ρ) and etc. 4.6.2. Homotopies between A∞ homomorphisms. The purpose of this subsection is to prove that the homotopy class of the filtered A∞ homomorphism g obtained in Corollary 4.6.3 is independent of various choices we made and of a symplectic isotopy of ψ. Namely we will prove Theorem 4.1.2. We continue to use the notations in the last subsection. To specify the dependence of f on ψ, {Jρ }ρ top(ρ) and stop(ρ) we write fψ,{Jρ }ρ ,s in place of f. We have constructed the filtered A∞ homomorphism (4.6.21)

top(ρ)

fψ,{Jρ }ρ ,s

J  : (C1 (L; Λ0,nov ), mJ,s k ) → (C3 (L ; Λ0,nov ), mk

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,s

)

250

4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

in the previous subsection. We compose this with the homotopy inverse of the filtered A∞ homomorphism (C2 (L ; Λ0,nov ), mJk



,s

) → (C3 (L ; Λ0,nov ), mJk 



,s

)



induced by the natural inclusion C2 (L ; Q) ⊂ C3 (L ; Q) (see (4.6.2.2)) and obtain top(ρ)

gψ,{Jρ }ρ ,s

(4.6.22)

J  : (C1 (L; Λ0,nov ), mJ,s k ) → (C2 (L ; Λ0,nov ), mk



,s

).

top(ρ)

gives a homotopy equivalence. Hereafter in this subsection we write gψ,{Jρ }ρ ,s top(ρ)  ss=0 , ss=0 in place of s , stop(ρ) . We also write C3s=0 (L ; Λ0,nov ), χs=0 3 (L ). We top(ρ)

ψ,{J s=0 }ρ ,s

top(ρ)

ψ,{J s=0 }ρ ,s

s=0 s=0 , fs=0 ρ , also. write gs=0 ρ  Let ψs : M → M , s ∈ [0, 1] be a symplectic isotopy with ψ0 = ψ, ψ1 = ψ  . (In order to clarify which we are considering on the source or the target space of ψs , we denote the target space by M  instead of M .) We write ψ = {ψs }s . We assume

ψs (L) = L

(4.6.23)

for each s. We apply the construction of the last subsection to ψ1 = ψ  in place of ψ0 = ψ and obtain the following: (4.6.24.1) {Jρs=1 }ρ , a homotopy of compatible almost complex structures on M  from J ψ1 to J  . (4.6.24.2) X3s=1 (L ), a countable set of smooth singular chains which generates a subcomplex C3s=1 (L , Q) of S(L ). top(ρ)

top(ρ)

(4.6.24.3) ss=1 , ss=1 . Here ss=1 is a system of multisections of the moduli   s=1  ) and s is a system of multisections }ρ : β; top(ρ); P spaces Mmain s=1 k+1 (M , L , {Jρ main     s=1   of Mk+1 ((L , J ), β; P ) for Pi ∈ X3 (L ). They satisfy (4.6.15) and hence induce filtered A∞ homomorphisms. We denote them by top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

fs=1 and

top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

gs=1

J  ,s s=1

s=1 : (C1 (L; Λ0,nov ), mJ,s (L ; Λ0,nov ), mk k ) → (C3 J  : (C1 (L; Λ0,nov ), mJ,s k ) → (C2 (L ; Λ0,nov ), mk



,s

)

).

Now our main result of this subsection is: top(ρ)

ψ0 ,{Jρs=0 }ρ ,ss=0

Theorem 4.6.25. gs=0

top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

is homotopic to gs=1

.

Proof. We use a homotopy between two homotopies {Jρs=1 }ρ , {Jρs=0 }ρ of compatible almost complex structures. Namely we consider a two-parameter family {Jρ,s }ρ,s of compatible almost complex structures such that (4.6.26.1) (4.6.26.2) (4.6.26.3) (4.6.26.4)

{Jρ,0 }ρ coincides with the family {Jρs=0 }ρ . {Jρ,1 }ρ coincides with the family {Jρs=1 }ρ . J0,s = (ψs )∗ J for any s ∈ [0.1]. J1,s = J  for any s ∈ [0, 1].

The contractibility of the set of compatible almost complex structures implies that there exists such a homotopy {Jρ,s }ρ,s . We also denote this two parameter family by {Jρ,s }ρ,s .

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4.6. GEOMETRIC REALIZATION

251

For each fixed s, we have a family of compatible almost complex structures ρ → Jρ,s , which we denote by {Jρ,s }ρ . Using {Jρ,s }ρ we obtain moduli spaces   Mmain k+1 (M , L , {Jρ,s }ρ : β; top(ρ)). We put   Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s))    {s} × Mmain = k+1 (M , L , {Jρ,s }ρ : β; top(ρ)).

(4.6.27)

s∈[0,1]

We define evaluation maps    ev0 : Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s)) → L   evi : Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s)) → L,

i = 1, · · · , k

and   evs : Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s)) → [0, 1]

as follows.   Let ((Σ, z), (uα ), (ρα )) ∈ Mmain k+1 (M , L , {Jρ,s }ρ : β; top(ρ)). We put (4.6.28.1) (4.6.28.2) (4.6.28.3)

ev0 (((Σ, z), (uα ), (ρα )), s) = uα0 (z0 ), evi (((Σ, z), (uα ), (ρα )), s) = ψs−1 (uαi (zi )), evs (((Σ, z), (uα ), (ρα )), s) = s.

For Pi ∈ X1 (L) (i = 1, · · · , k) we use evi to define    Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P )

(4.6.29)

=

  Mmain k+1 (M , L , {Jρ,s }ρ,s

: β; top(ρ), twp(s))(ev1 ,··· ,evk ) ×



k 0

 Pi

.

i=1

The idea of the proof of Theorem 4.6.25 is to use these moduli spaces to construct a filtered A∞ homomorphism from (C1 (L ; Λ0,nov ), mJ,s k ) to a filtered A∞ algebra defined on the module of smooth singular chains on [0, 1]×L . To define this filtered A∞ algebra we need more notations. Note we will use homomorphisms Evals=s0 : C([0, 1] × L ) → C(L ) for appropriate subcomplexs of singular chain complexes on [0, 1] × L and on L . The homomorphism Evals=s0 induces a homomorphism H ∗ ([0, 1] × L ) → H ∗ (L ) which coincides with the one induced by the inclusion L → {s0 } × L ⊂ [0, 1] × L . We remark that we are working with cohomology groups rather than homology groups. We use singular chains to represent elements of cohomology group. In other words, we use Poincar´e duality. The Poincar´e dual to the homomorphism H ∗ ([0, 1] × L ) → H ∗ (L ) is the composition H∗ ([0, 1] × L ; {0, 1} × L ) → H∗−1 ({0, 1} × L ) → H∗−1 ({s0 } × L ) of the connecting homomorphism H∗ ([0, 1]×L ; {0, 1}×L ) → H∗−1 ({0, 1}×L ) and the projection to the H∗−1 ({s0 }×L ) factor. We are going to define chain complexes C([0, 1] × L ), C(L ) so that the connecting homomorphism can be defined in the chain level. From now on, we use the script letter P for singular simplices on [0, 1] × L and roman letter P ,P  for singular simplices on L,L .

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252

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HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

Definition 4.6.30. Let P = (|P|, f ) (f : |P| → [0, 1] × L ) be a smooth singular simplex. We say P is adapted if the following condition are satisfied for s0 = 0, 1. (4.6.31.1) f −1 ({s0 } × L ) ∩ |P| is either |P|, empty or consists of a single face (of arbitrary codimension). (4.6.31.2) If f −1 ({0} × L ) ∩ |P| is nonempty then |P| ⊂ [0, 1/3) × L . If f −1 ({1} × L ) ∩ |P| is nonempty then |P| ⊂ (2/3, 1] × L . m be another face containing Δm (4.6.31.3) If Δm a is a face of |P| and Δb a . We m −1  m assume x ∈ Δa ⊂ f ({s0 } × L ) and Δb is not contained in f −1 ({s0 } × L ).  m Then, for any vector N ∈ Tx Δm b \ Tx (∂Δb ), the [0, 1] component of f∗ (N ) is nonzero. A singular simplex of dimension m on [0, 1] × L is regarded as a cochain of d degree d = 1 + dim L − m. We write S+ ([0, 1] × L ) the abelian group of adapted cochains of degree d. It is easy to see that it is a cochain complex. The smooth singular chain complex S({0, 1} × L ) is a subcomplex of it. We define S([0, 1] × L ) =

S+ ([0, 1] × L ) . S({0, 1} × L )

S([0, 1] × L ) is free on the basis of singular simplices which satisfy Condition 4.6.30 and which are not contained in {0, 1} × L . We define Evals=s0 : S([0, 1] × L ) → S(L ) as follows. (Here S(L ) is the smooth singular chain complex of L .) If f −1 ({s0 } × L ) ∩ |P| is not codimension one in |P|, then (4.6.32.1)

Evals=s0 (P) = 0.

If f −1 ({s0 } × L ) ∩ |P| is codimension one in |P|, then (4.6.32.2)

Evals=s0 (P) = ±(f −1 ({s0 } × L ) ∩ |P|, f ).

See Subsection 7.2.10 for the sign ±. In Lemma 7.2.136 Section 7.2, we will prove that Evals=s0 : S([0, 1] × L ) → S(L ) is a cochain map and is a cochain homotopy equivalence. We also consider the following condition for P ∈ S([0, 1] × L ). Condition 4.6.33. Evals=s0 (P) ∈ C3s=s0 (L , Q). We next define Incl1,β0 : S(L ) → S([0, 1] × L ) by the following formula. Incl1,β0 (|P |, f ) = ([0, 1] × |P |, id × f ), where we take an appropriate prism type simplicial decomposition of [0, 1] × |P | to regard the right hand side as a singular chain. (See Subsection 7.2.10 for detail.) We will construct a structure of filtered A∞ algebra on a cochain complex which is generated by countably many adapted smooth singular simplices. For this purpose we use another parametrized version of moduli space of pseudo-holomorphic discs. For the family {Jρ,s }ρ,s satisfying (4.6.26), we put ρ = 1. Then we have a family {J1,s }s of compatible almost complex structures J1,s , s ∈ [0, 1]. From

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4.6. GEOMETRIC REALIZATION

253

(4.6.26.4) this is a constant family for ρ = 1. Namely J1,s = J  for each s. Using  {J1,s }s , we obtain the moduli space Mmain k+1 ((L , J1,s ), β). We now put   Mmain k+1 (M , L , {J1,s }s : β; twp(s))   main   {s} × Mmain = k+1 ((L , J1,s ), β) = [0, 1] × Mk+1 ((L , J ), β).

(4.6.34)

s∈[0,1]   We remark that we have defined Mmain k+1 (M , L , {J1,s }s : β; twp(s)) using the modmain  uli spaces Mk+1 ((L , J1,s ), β) “time-wise”. So this is different from the mod  uli space Mmain k+1 (M , L , {J1,s }s : β; top(s)) for the family s → J1,s . Namely, main   Mk+1 (M , L , {J1,s }s : β; top(s)) is a moduli space of pseudo-holomorphic maps from semi stable genus zero bordered Riemann surface with respect to the almost complex structure depending on the component. But an element of the moduli space   Mmain k+1 (M , L , {J1,s }s : β; twp(s)) is a pseudo-holomorphic map from a semi stable bordered Riemann surface with respect to some J1,s where s is independent of the component. We can define evaluation maps    evi : Mmain k+1 (M , L , {J1,s }s : β; twp(s)) → L ,

evs :

  Mmain k+1 (M , L , {J1,s }s

for i = 0, · · · , k,

: β; twp(s)) → [0, 1],

by (4.6.35.1) (4.6.35.2)

evi ((Σ, z), (uα )), s) = uαi (zi ), evs ((Σ, z), (uα )), s) = s.

(Here we note that evi for i = 0, · · · , k maps to L .) We put evi+ = (evs , evi ). For Pi ∈ S([0, 1] × L ), (i = 1, · · · , k), we define    Mmain k+1 (M , L , {J1,s }s : β; twp(s); P)



  = Mmain k+1 (M , L , {J1,s }s : β; twp(s))(ev + ,··· ,ev + ) × 1

k

k 0

 Pi

.

i=1

  Here the fiber product is taken on ([0, 1] × L )k . The space Mmain k+1 (M , L , {J1,s }s : twp(s)  β; twp(s); P) has a Kuranishi structure. If we take a multisection s transversal to 0, we obtain   twp(s) −1  s Mmain ) (0), ev0+ ). k+1 (M , L , {J1,s }s : β; twp(s); P) = ((s

We can take a simplicial decomposition of (stwp(s) )−1 (0) so that ((stwp(s) )−1 (0), ev0+ ) can be regarded as a singular chain in [0, 1] × L . Moreover we may choose our simplicial decomposition so that each singular simplex of (stwp(s) )−1 (0) is adapted. (See Subsection 7.2.10.) We can easily prove the following:     s (4.6.36.1) deg Mmain deg Pi − μ(β), k+1 (M , L , {J1,s }s ; β; twp(s); P) = 2 − k +

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254

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HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

where deg is the codimension (of the chain), which is equal to the cohomology degree. We also remark that the boundary of the moduli space    s Mmain k+1 (M , L , {J1,s }s ; β; twp(s); P)

can be described as a fiber product of similar moduli spaces. On the other hand, if we take a multisection stop(ρ),twp(s) on the space    Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P )

transversal 0, we obtain   top(ρ),twp(s) −1  s ) (0), ev0+ ). Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P ) = ((s

We can take a simplicial decomposition of (stop(ρ),twp(s) )−1 (0) so that the pair ((stop(ρ),twp(s) )−1 (0), ev0+ ) can be regarded as a singular chain in [0, 1] × L . Moreover we may choose our simplicial decomposition such that each singular simplex of (stwp(s) )−1 (0) satisfies Condition 4.6.30. (See Subsection 7.2.10.) We can easily calculate its degree as: (4.6.36.2)

   s deg Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P )  =3−k+ deg Pi − μ(β).

Now the main result we need to construct a filtered A∞ structure on a countably generated subcomplex of S([0, 1] × L ; ∂) is the following. The proof will be given in Subsection 7.2.10. Strictly speaking we prove the An,K version of Proposition 4.6.37 (that is Propositions 7.2.160 and 7.2.162) in Subsection 7.2.10 and use it to prove the A∞ version in Subsections 7.2.11-7.2.13. Proposition 4.6.37. There exist a countable set X ([0, 1] × L ) of singular simplices which generates a subcomplex C([0, 1] × L ) of S([0, 1] × L ) and a system of multisections stwp(s) , stop(ρ),twp(s) transversal to 0 with the following properties: (4.6.38.1) Every element of X ([0, 1]×L ) is adapted and satisfies Condition 4.6.33. Elements of X ([0, 1] × L ) are not contained in S({0, 1} × L ). (4.6.38.2) If Pi ∈ X ([0, 1] × L ), then     s Mmain k+1 (M , L , {J1,s }s : β; twp(s); P) ∈ C([0, 1] × L ).

(4.6.38.3) The system of multisections stwp(s) , stop(ρ),twp(s) are compatible at the   main    boundaries of Mmain k+1 (M , L , {J1,s }s : β; twp(s); P) and Mk+1 (M , L , {Jρ,s }ρ,s : s ) . β; top(ρ), twp(s); P (4.6.38.4) If P ∈ X2 (L ), then Incl1,β0 (P ) ∈ C([0, 1] × L ). (4.6.38.5) The inclusion C([0, 1] × L ) → S([0, 1] × L ) induces an isomorphism on the cohomology groups.  = (P1 , · · · , Pk ). We put Pi |s=0 = (4.6.38.6) Let Pi ∈ C([0, 1] × L ) and denote P  Evals=0 (Pi ) and P|s=0 = (P1 |s=0 , · · · , Pk |s=0 ). We then have   main    s  s  Evals=0 (Mmain k+1 (M , L , {J1,s }s : β; twp(s); P) ) = Mk+1 ((M , L ), J : β; P|s=0 ) .

 s=1 = (P1 |s=1 , · · · , Pk |s=1 ), then we have If we put Pi |s=1 = Evals=1 (Pi ) and P| 

  main    s  s  Evals=1 (Mmain k+1 (M , L , {J1,s }s : β; twp(s); P) ) = Mk+1 ((M , L ), J : β; P|s=1 ) .

(In the above equalities, we recall that J1,0 = J1,1 = J  from (4.6.26.4)).

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4.6. GEOMETRIC REALIZATION

(4.6.38.7)

255

For Pi ∈ C(L), we have     s Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P ) ∈ C([0, 1] × L ).

(4.6.38.8)

We can define Inclk,β : Bk (C(L )[1]) → C([0, 1] × L )

such that Incl1,β0 and Inclk,β define a filtered A∞ homomorphism Incl : C(L )[1] → C([0, 1] × L ) satisfying Evals=s0 ◦ Incl = id. We will use Proposition 4.6.37 to construct the following Diagram 4.6.1. C3s=1 (L; Λ0 ψ1

fs=1

Jρs=1 ρ

) ⊇ C2 (L; Λ0

)

top(ρ) ss=1

Evals= ψ

C( L; Λ0

)

fk sβ

stop(ρ)

J

twp(s)

C((− 1 + ) × L) ⊗ Λ0 Evals=

ψ0 J s=0 fs=0 ρ

ρ

top(ρ)

ss=0

C3s=0 (L; Λ0

) ⊇ C2 (L; Λ0

)

Diagram 4.6.1 Proposition 4.6.37 will be proved in Subsection 7.2.10. We remark that Incl in Proposition 4.6.37 does not satisfy (4.2.3.1). (See Remark 7.2.172.) By using this proposition, we define a structure of filtered A∞ algebra on C([0, 1] × L , Λ0,nov ). Here C([0, 1] × L , Λ0,nov ) is a completion of C([0, 1] × L ) ⊗ Λ0,nov . Let Pi ∈ C([0, 1] × L ). We define     s mk,β (P1 , · · · , Pk ) = Mmain k+1 (M , L , {J1,s }s : β; twp(s); P) ∈ C([0, 1] × L )

for (k, β) = (1, β0 )(= (1, 0)) and m1,β0 (P) = (−1)n+1 ∂P, where n = dim L. (Note P is a singular simplex of the (n + 1)-dimensional space [0, 1] × L .) We then put:  μ(β) mk = mk,β ⊗ T ω(β) e 2 . β 

Lemma 4.6.39. (C([0, 1] × L , Λ0,nov ), m) is a filtered A∞ algebra. The proof is similar to the proof of Theorem 3.5.11, using Proposition 4.6.37. We omit it. We next use (4.6.38.4), (4.6.38.6) and (4.6.38.7) to construct several A∞ homomorphisms. We first define J  ,s s=0

Evals=0 : (C([0, 1] × L , Λ0,nov ), m) → (C3s=0 (L ; Λ0,nov ), mk 

Evals=1 : (C([0, 1] × L , Λ0,nov ), m) →

),

J  ,s (C3s=1 (L ; Λ0,nov ), mk s=1 )

as follows: (Evals=s0 )1 is defined by (4.6.32). We put (Evals=s0 )k = 0 for k = 1. Then (4.6.38.6) implies that Evals=s0 is a filtered A∞ homomorphism.

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We define a filtered A∞ homomorphism Incl : (C2 (L ; Λ0,nov ), mJk



,s

) → (C([0, 1] × L , Λ0,nov ), m)

by (4.6.38.8). We remark that we can take X (([0, 1] × L ) so that (Evals=0 )1 ⊕ (Evals=1 )1 given above is surjective. (This fact will be used in Subsection 7.2.13.) Then (C([0, 1]×L , Λ0,nov ), m) is a model of [0, 1]×(C(L ; Λ0,nov ), mk ) in some generalized sense. (See Definition 7.2.174.) Furthermore, we define top(ρ),twp(s)

fψ,{Jρ,s }ρ,s ,s

 : (C1 (L; Λ0,nov ), mJ,s k ) → (C([0, 1] × L ) ⊗ Λ0,nov , m)

by ψ,{Jρ,s }ρ,s ,stop(ρ),twp(s)

fk

=



ψ,{Jρ,s }ρ,s ,stop(ρ),twp(s)

fk,β

T ω(β) eμL (β)/2 ,

β

where

ψ,{Jρ,s }ρ,s ,stop(ρ),twp(s)

fk,β

(P1 , · · · , Pk )

   s = Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P ) .

(See Diagram 4.6.1.) Here we recall that the right hand side is defined by using (4.6.29) and the multisection stop(ρ),twp(s) . We remark that the right hand side is contained in C([0, 1] × L ) by (4.6.38.7). top(ρ),twp(s) We can prove that fψ,{Jρ,s }ρ,s ,s is a filtered A∞ homomorphism in a way similar to the proof of Theorem 4.6.1. The following lemma is immediate from definitions. Lemma 4.6.40. We have (4.6.41.1) (4.6.41.2)

top(ρ),twp(s)

Evals=0 ◦ fψ,{Jρ,s }ρ,s ,s

top(ρ),twp(s)

Evals=1 ◦ fψ,{Jρ,s }ρ,s ,s

top(ρ)

ψ0 ,{Jρs=0 }ρ ,ss=0

= fs=0 =

,

top(ρ)

ψ1 ,{J s=1 }ρ ,ss=1 fs=1 ρ

.

Now let I0 : (C2 (L ; Λ0,nov ), mJk



,s

I1 : (C2 (L ; Λ0,nov ), mJk





,s

J  ,s s=0

) → (C3s=0 (L ; Λ0,nov ), mk ) → (C3s=1 (L ; Λ0,nov ), mk

J



,s s=1

) )

be inclusions and let K0 , K1 be their homotopy inverses respectively. Then the following lemma also follows from (4.6.38.8). Lemma 4.6.42. Evals=0 ◦ Incl = I0 . Evals=1 ◦ Incl = I1 . Finally we show Lemma 4.6.43. K0 ◦ Evals=0 is homotopic to K1 ◦ Evals=1 . Proof. We have K0 ◦ Evals=0 ◦ Incl = K0 ◦ I0 ∼ id ∼ K1 ◦ I1 = K1 ◦ Evals=1 ◦ Incl . By (4.6.38) and Theorem 4.2.45, the filtered A∞ homomorphism Incl is a homotopy equivalence. The lemma follows. 

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4.6. GEOMETRIC REALIZATION

s=0 } ,stop(ρ) ψ0 ,{Jρ ρ s=0

f

s=0 −− −−−−−−−−−−→

C1 

257

I

0 C3s=0 ←−− −− C2 3 ⏐Eval  ⏐ s=0

top(ρ),twp(s)

fψ,{Jρ,s }ρ,s ,s

Incl

C1 −−−−−−−−−−−−−−−−→ C([0, 1] × L ) ←−−−− C2 ⏐ ⏐Eval    s=1 s=1 } ,stop(ρ) ψ1 ,{Jρ ρ s=1

fs=1

−−−−−−−−−−−−→ C3s=1 Diagram 4.6.2

C1

I

1 ←−− −− C2



Then Theorem 4.6.25 now follows from Lemmas 4.6.40 and 4.6.43.

We have thus completed the proof of Theorem 4.1.2. We can prove the homotopy-unital versions of Theorems 4.1.1 and 4.1.2 as well. top(ρ) Namely the homotopy equivalence gψ,{Jρ }ρ ,s is unital and the “homotopic” in the statement of Theorem 4.6.25 can be replaced by “u-homotopic”. This version can be proved by a combination of the arguments of this section and Sections 7.2,7.3. 4.6.3. Compositions. Let L ⊂ M be a relatively spin Lagrangian submanifold and ψ : M → M  a symplectic diffeomorphism. In Subsection 4.6.1 we constructed a filtered A∞ homomorphism associated to ψ from the filtered A∞ algebra associated to L to one associated to L = ψ(L). For another symplectic diffeomorphism ψ  : M  → M  , we defined a filtered A∞ homomorphism from the filtered A∞ algebra associated to L to one associated to L = (ψ  ◦ ψ)(L). In this subsection, we show the composition of these A∞ homomorphisms coincides up to homotopy to the A∞ homomorphism associated to ψ  = ψ  ◦ ψ. (Theorem 4.6.44). We fix compatible almost complex structures J, J  , J  . We also take countable sets of smooth singular simplices Xa (L), Xb (L ), Xc (L ) which generate complexes Ca (L; Λ0,nov ), Cb (L ; Λ0,nov ), Cc (L ; Λ0,nov ), respectively. By using multisections s, s , s of appropriate moduli spaces we obtain filtered A∞ algebras (Ca (L; Λ0,nov ), mJ,s k ),

(Cb (L ; Λ0,nov ), mJk



,s

),

(Cc (L ; Λ0,nov ), mJk



,s

)

respectively. By Theorem 4.6.1 and Corollary 4.6.2 we can find A∞ homomorphisms top(ρ)

gψ,{Jρ }ρ ,s g gψ

ψ  ,{Jρ }ρ ,stop(ρ)



,{Jρ }ρ ,stop(ρ)

J  : (Ca (L; Λ0,nov ), mJ,s k ) → (Cb (L ; Λ0,nov ), mk

: :



,s

)

    (Cb (L ; Λ0,nov ), mJk ,s ) → (Cc (L ; Λ0,nov ), mJk ,s ) J  ,s  (Ca (L; Λ0,nov ), mJ,s ), k ) → (Cc (L ; Λ0,nov ), mk





where {Jρ }ρ , {Jρ }ρ , {Jρ }ρ are homotopies from J ψ to J  , from J ψ to J  , from  J ψ ◦ψ to J  respectively. We remark that the notations C1 (L; Λ0,nov ), C2 (L ; Λ0,nov ), and C3 (L ; Λ0,nov ) are used in Subsection 4.6.2. In this subsection, we use suffix a, b, c in place of 1, 2, 3. top(ρ) This is because when we consider gψ,{Jρ }ρ ,s , the module Cb (L ; Λ0,nov ) plays

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258

4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS 



top(ρ)

the role of C2 (L ; Λ0,nov ), on the other hand, when we consider gψ ,{Jρ }ρ ,s module Cb (L ; Λ0,nov ) plays the role of C1 (L ; Λ0,nov ). 



top(ρ)

Theorem 4.6.44. The composition gψ ,{Jρ }ρ ,s   top(ρ) topic to gψ ,{Jρ }ρ ,s .

top(ρ)

◦ gψ,{Jρ }ρ ,s

the

is homo-

Proof. We have already proved that g does not depend on ψ, {Jρ }ρ , stop(ρ) etc. in the last subsection. So we may assume that {Jρ }ρ is given by    ψ ) ρ ≤ 1/2, (J2ρ  Jρ =  ρ > 1/2. J2ρ−1 



top(ρ)

  We recall that gψ ,{Jρ }ρ ,s is defined by using Mmain k+1 (M, L , {Jρ }ρ : β; top(ρ)). Then we show the following lemma.    Lemma 4.6.45. Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) is isomorphic to the union of the fiber products       Mmain m+1 (M , L , {Jρ }ρ : βlast ; top(ρ)) 1 ,··· ,m

m

i=1

i =k βlast +



i

βi =β

×

0 m

  Mmain i +1 (M , L , {Jρ }ρ

 : βi ; top(ρ)) .

i=1 

Here fiber products are taken over L .    Proof. Let ((Σ, z), (uα ), (ρα )) ∈ Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)). We consider the component Σα such that ρα > 1/2 and denotes by Σ0 the union of such components. By (4.6.7.2) Σ0 is connected. We restrict u and ρ to D0 . It is easy to    see that this restriction defines an element of Mmain m+1 (M , L , {Jρ }ρ : βlast ; top(ρ)), here m is the number of the connected components of Σ \ Σ0 . We thus find the first factor of the fiber produce. Let Σi (i = 1, · · · , m) be the connected component of   Σ \ Σ0 . Restriction of u and τ to Σi defines an element of Mmain i +1 (M , L , {Jρ }ρ :  βi ; top(ρ)). The lemma follows.

P4

P3

0.1

0.1

0.3 P2 P40.2

0.6 P1 ρ = 0.8 0

P5 0.4 0.3

ev0

P7

P6

   ψ ψ ψ ψ(P Figure 4.6.6 fψ ), ψ(P ), f (P , P ), f (1), f (P , P ), ψ(P )) 1 2 3 4 5 6 7 6 2 0 2

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4.6. GEOMETRIC REALIZATION

259

We can choose multisection stop(ρ) so that it is compatible with the isomorphism in Lemma 4.6.45. Then we have 



top(ρ)

fψ ,{Jρ }ρ ,s

(4.6.46)

top(ρ)

◦ fψ,{Jρ }ρ ,s

= fψ



,{Jρ }ρ ,stop(ρ)

.

More precisely we have the following commutative Diagram 4.6.3. In Diagram 4.6.3, the subcomplex Ca,b (L , Λ0,nov ) is C3 (L , Λ0,nov ) if we use the notation of Subsection 4.6.2. Ca,c (L , Λ0,nov ) ←−−−− Cc (L , Λ0,nov ) ⏐ ⏐ #  3 ⏐ ψ ,{J  }ρ ,stop(ρ) ρ C(a,b),c (L1 , Λ0,nov ) ←−−−− Cb,c (L , Λ0,nov ) ⏐f 3 3  } ,stop(ρ) ⏐ ψ  ,{J  }ρ ,stop(ρ) ⏐ ψ  ,{Jρ ρ ⏐ ⏐ f fext ρ Ca (L, Λ0,nov )

−−−−−−−−−−→ top(ρ)

Ca,b (L , Λ0,nov )

fψ,{Jρ }ρ ,s

←−−−− Cb (L , Λ0,nov )

Diagram 4.6.3 By definition, the composition top(ρ)

fψ,{Jρ }ρ ,s

Ca (L, Λ0,nov ) −−−−−−−−−−→ Ca,b (L , Λ0,nov ) −−−−→ Cb (L , Λ0,nov ) top(ρ)

. (Here Ca,b (L , Λ0,nov ) → Cb (L , Λ0,nov ) is the homotopy inverse is gψ,{Jρ }ρ ,s of the inclusion.) In the same way the composition of the two arrows in the third column is   top(ρ) gψ ,{Jρ }ρ ,s . (Here we invert the direction of one of the arrows.) Also the composition of the arrow in the first column and the homotopy inverse of the first line, namely the composition:  } ,stop(ρ) ψ  ,{Jρ ρ

f

Ca (L, Λ0,nov ) −−−−−−−−−−−−→ Ca,c (L , Λ0,nov ) −−−−→ Cc (L , Λ0,nov ) 



top(ρ)

. is gψ ,{Jρ }ρ ,s (4.6.46) implies that we can find C(a,b),c (L1 , Λ0,nov ) ⊃ Ca,c (L1 , Λ0,nov ) ∪ Cb,c (L1 , Λ0,nov ) ψ  ,{J  }ρ ,stop(ρ)

and fext ρ such that Diagram 4.6.3 commutes. (Compare Lemma 7.2.299.) Theorem 4.6.44 follows.  We can also prove that we may replace ‘homotopic’ by ‘hu-homotopic’ in Theorem 4.6.44. This point will follow from a combination of the argument above and one in Section 7.3. 4.6.4. Homotopy equivalence and the operator q I: changing the cycle in the interior. In the remaining subsections of Section 4.6, we will discuss deformation of our filtered A∞ algebra (C(L; Λ0,nov ), m) by a cocycle b in + 0 C 2 (M ; Λ+ 0,nov ) = C(M ; Λ0,nov )[2] using the operator q we introduced in Section 3.8. We defined the filtered A∞ algebra (C(L; Λ0,nov ), m(b,0) ) deformed by b in Subsection 3.8.5. Hereafter we simply write mb in place of m(b,0) . In this subsection, we show that (C(L; Λ0,nov ), mb ) depends only on the cohomology class of b up to the

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260

4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

strict homotopy equivalence (Theorem 4.6.47). We remark that for each cochain b b ∈ C 1 (L; Λ+ 0,nov ), we defined the deformed filtered A∞ algebra (C(L; Λ0,nov ), m ) in Subsection 3.6.1. (See Proposition 3.6.10). It is independent of b up to the (non-strict) homotopy equivalence. Namely it does not even depend on the gauge equivalence class of b, (Remark 4.3.17). This subsection is of purely algebraic nature. The results hold for any filtered A∞ algebra C that has an operator q satisfying the conclusion of Theorem 3.8.32. (However we state the results only for the case of our geometric example.) In the following three subsections (Subsections 4.6.4-4.6.6), we assume R ⊇ Q and use the notations of Section 3.8. Theorem 4.6.47. Let b0 , b1 ∈ C 2 (M ; Λ+ 0,nov ) be cycles representing the same 2 cohomology class in H (M ; Λ0,nov ). Then, the filtered A∞ algebra (C, mb0 ) is strictly homotopy equivalent to (C, mb1 ). Theorem 4.6.47 is proved in Subsection 7.4.6. See Theorem 7.4.118 there. We also need the following proposition which we use in Definition 4.6.50. Let bi ∈ C(M ; Λ+ 0,nov ) i = 0, 1, 2 be cycles representing the same homology class. Let (4.6.48)

fij : (C, mbi ) → (C, mbj )

be the homotopy equivalence given by Theorem 4.6.47. Proposition 4.6.49. f12 ◦ f01 is shu-homotopic to f02 . Proposition 4.6.49 is also proved in Subsection 7.4.6. See Theorem 7.4.134. Using Proposition 4.6.49 we can generalize Definition 4.3.26 as follows. def (L). We say that Definition-Corollary 4.6.50. Let (b0 , b0 ), (b1 , b1 ) ∈ M they are gauge equivalent to each other and write (b0 , b0 ) ∼ (b1 , b1 ) if [b0 ] = [b1 ] ∈ H 1 (L; Λ0,nov ) and f∗ (b0 ) ∼ b1 where f : (C(L; Λ0,nov ), mb0 ) → (C(L; Λ0,nov ), mb1 ) is as in Theorem 4.6.47. Proposition 4.6.49 implies that gauge equivalence is an equivalence relation. weak,def (L) in the same way. We can define an equivalence relation ∼ on M weak,def (L) coincides with def (L) and of M The set of ∼ equivalence classes of M Mdef (L) and Mweak,def (L) in Definition 4.3.26, respectively. weak,def (L) and (b0 , b0 ) ∼ (b1 , b1 ), then Lemma 4.6.51. If (bi , bi ) ∈ M PO(b0 , b0 ) = PO(b1 , b1 ). See Subsection 7.4.6 for its proof. There is a natural projection πamb : Mdef (L) → H 2 (M ; Λ+ 0,nov ). Lemma 4.6.51 implies that we have the potential function (0)

POi : Mweak,def (L(i) ) → Λ0,nov . together with πamb,i : Mweak,def (L(i) ) → H 2 (M ; Λ+ 0,nov ). See Subsection 4.3.3. Here Mweak,def (L(i) ) = {(b, b) | b ∈ H 2 (M ; Λ+ 0,nov ), b ∈ Mweak (C(L(i) ; Λ0,nov ), mb )}.

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4.6. GEOMETRIC REALIZATION

261

In Subsection 5.2.4 Propositions 5.2.37 (3) and 5.2.38 (2), we will prove that the Floer cohomology defined in Subsection 3.8.7 does not change when we change the bounding cochains in a given gauge equivalence class. It follows that, the Floer cohomology group HF ((L(1) , (b1 , b1 )), (L(0) , (b0 , b0 )); Λ0,nov ) is parametrized by ([b1 , b1 ], [b0 , b0 ]) ∈ Mweak,def (L(1) ) ×H 2 (M ;Λ+

(0) 0,nov )×Λ0,nov

Mweak,def (L(0) )

= {([b1 , b1 ], [b0 , b0 ]) | πamb,1 [b1 , b1 ] = πamb,0 [b0 , b0 ], PO1 [b1 , b1 ] = PO0 [b0 , b0 ]}. Hereafter we also denote by the same symbol the gauge equivalence class of (bi , bi ) in the sense of Definition 4.6.50. 4.6.5. Homotopy equivalence and the operator q II: invariance of symplectic diffeomorphisms 1. In this subsection and the next, we will prove the following Theorem 4.6.52. Let (M, ω), (M  , ω  ) be symplectic manifolds and ψ : M → M  a symplectic diffeomorphism. Let L ⊂ M be relatively spin Lagrangian submanifold and L = ψ(L). We fix various choices (compatible almost complex structures, perturbations, and countable sets of smooth singular simplices), for M, L and M  , L to define  (C(L; Λ0,nov ), m) and (C(L ; Λ0,nov ), m). For each [b] ∈ H 2 (M ; Λ+ 0,nov ) and [b ] ∈ + H 2 (M  ; Λ0,nov ), we have obtained the filtered A∞ algebras (C(L; Λ0,nov ), mb ) and  (C(L ; Λ0,nov ), mb ) whose homotopy types depend only on the cohomology classes of b and b respectively. Theorem 4.6.52. For each [b] ∈ H 2 (M ; Λ+ 0,nov ), there exists a homotopy equivalence of filtered A∞ algebras ψ∗b : (C(L; Λ0,nov ), mb ) −→ (C(L ; Λ0,nov ), mψ∗ b ). It is independent of the choice of symplectic isotopy in the same sense as in Theorem 4.1.2. Before proving Theorem 4.6.52, we derive several consequence of it. We begin with the following corollary which is nothing but Theorem B (B.3) in Introduction. Corollary 4.6.53. Any symplectic diffeomorphism ψ : M → M  with L = ψ(L) as above induces a map ψ∗ with the following commutative diagrams: ψ∗

Mweak,def (L) −−−−→ Mweak,def (L ) ⏐ ⏐ ⏐ ⏐π πamb   amb ψ∗

+ 2  H 2 (M ; Λ+ 0,nov ) −−−−→ H (M ; Λ0,nov )

ψ∗

Mweak,def (L) −−−−→ Mweak,def (L ) ⏐ ⏐ ⏐ ⏐PO PO  (0)

Λ0,nov

(0)

Λ0,nov

Diagram 4.6.4 In these diagrams, the horizontal arrows are isomorphisms. The bottom arrow in Diagram 4.6.4 is just given by ψ −1∗ , the pull back by ψ −1 .

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262

4.

HOMOTOPY EQUIVALENCE OF A∞ ALGEBRAS

Proof of Corollary 4.6.53. We can write Mweak,def (L) = {(b, b) | b ∈ H 2 (M ; Λ+ 0,nov ), b ∈ Mweak (C(L; Λ0,nov ), mb )}/ ∼ . From Theorem 4.6.52, we have a homotopy equivalence ψ∗b : (C(L; Λ0,nov ), mb ) −→ (C(L ; Λ0,nov ), mψ∗ b ) for [b] ∈ H 2 (M ; Λ+ 0,nov ). By Theorem 4.3.22, the functor Mweak is homotopy invariant and the homotopy equivalence map ψ∗b induces the bijection (ψ∗b )∗ : Mweak (C(L; Λ0,nov ), mb ) → Mweak (C(L ; Λ0,nov ), mψ∗ b ) defined by (ψ∗b )∗ (b) = ψ∗b (eb ). See also Lemma 3.6.36. We now put ψ∗ (b, b) = (ψ∗ b, (ψ∗b )∗ (b)), where ψ∗ b is the pull back of b by ψ −1 . Then by using Lemma 4.3.23, we can find that this map ψ∗ has the required properties.  Proof of Theorem B (B.4). We use the notation of Theorem B. Let bi = (b, bi ), (i = 0, 1). By Theorem 4.6.52 we have a homotopy equivalence ψ∗b : (C(L; Λ0,nov ), mb ) → (C(L ; Λ0,nov ), mψ∗ b ). Hence we have (ψ∗b )∗ : M(C(L; Λ0,nov ), mb ) → M(C(L ; Λ0,nov ), mψ∗ b ). We define ψ∗ (bi ) = (ψ∗b )∗ (bi ). We put δb1 ,b0 (x) = mb (eb1 xeb0 ),

δψ∗ b1 ,ψ∗ b0 (y) = mψ∗ b (eψ∗ b1 yeψ∗ b0 ).

We define ψ∗b1 ,b0 : (C(L; Λ0,nov ), δb1 ,b0 ) → (C(L ; Λ0,nov ), δψ∗ b1 ,ψ∗ b0 ) by

ψ∗b1 ,b0 (x) = ψ∗b (eb1 xeb0 ).

It is easy to see that ψ∗b1 ,b0 is a chain map. Hence we obtain (4.6.54)

(ψ∗b1 ,b0 )∗ : HF ((L, b1 ), (L, b0 ); Λ0,nov )

→ HF ((L , ψ∗ b1 ), (L , ψ∗ b0 ); Λ0,nov ).

We remark that ψ∗b1 ,b0 ≡ ψ∗ mod Λ+ 0,nov induce an isomorphism on m1 cohomology. Proposition 5.2.38 (2) (which is proved in Section 5.2) implies that (4.6.54) is an isomorphism. (Alternatively we can use the spectral sequence which will be established in Chapter 6 to prove that (4.6.54) is an isomorphism.) Theorem B (B.4) is now proved.  To prove Theorem 4.6.52 we need to combine the construction in the proof of Theorem 4.6.1 with the construction in Subsection 3.8.6. For this purpose we first state the result carefully specifying the choice of countably generated subcomplex

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4.6. GEOMETRIC REALIZATION

263

etc. Let us first take, as in Theorem 3.8.32, countable sets of smooth singular simplices X1+ (L) ⊃ X1 (L) of L and X (M ) of M such that for any Q ∈ X (M ) is transversal to L and that Q ∩ L ∈ X1 (L) ⊂ X2+ (L), by taking appropriate triangulation of Q ∩ L. Moreover we can define the operator q,k : E C(M ; Λ0,nov )[2] ⊗ Bk C1+ (L; Λ0,nov )[1] −→ C1+ (L; Λ0,nov )[1], where C(M ; Λ0,nov ), C1+ (L; Λ0,nov ) are generated by X (M ), X1+ (L) respectively. We can also choose X (M  ), X2+ (L ) so that q,k : E C(M  ; Λ0,nov )[2] ⊗ Bk C2+ (L ; Λ0,nov )[1] −→ C2+ (L ; Λ0,nov )[1] is well defined, where C(M  ; Λ0,nov ), C2+ (L ; Λ0,nov ) are generated by the sets X (M  ), X1+ (L ) respectively. We may choose X  (M  ) which contains ψ(X (M )) and X (M  ) such that H ∗ (C  (M  ; Λ0,nov )) ∼ = H ∗ (M ; Λ0,nov ), where C  (M  ; Λ0,nov ) is generated by X  (M  ). Under the situation above, we can show the following. Proposition 4.6.55. There exists a countable set of smooth singular simplices X3+ (L ) (which generates C3+ (L ; Λ0,nov )) containing X3 (L ) in Theorem 4.6.1 and X2+ (L ), and a sequence of operators f,k : E C(M ; Λ0,nov )[2] ⊗ Bk C1+ (L; Λ0,nov )[1] −→ C3+ (L ; Λ0,nov )[1] of degree 0 with the following properties. (4.6.56.1) f0,k = fk , where fk (k = 0, 1, 2, · · · ) is the filtered A∞ homomorphism as in Definition 4.6.12. (4.6.56.2) q,k : E C(M  ; Λ0,nov )[2] ⊗ Bk C2+ (L ; Λ0,nov )[1] −→ BC2+ (L ; Λ0,nov )[1] is extended to q,k : E C  (M  ; Λ0,nov )[2] ⊗ Bk C3+ (L ; Λ0,nov )[1] −→ BC3+ (L ; Λ0,nov )[1] which satisfy (3.8.33). (4.6.56.3) We define  f : EC(M ; Λ0,nov )[2] ⊗ BC1+ (L; Λ0,nov )[1] −→ BC3+ (L ; Λ0,nov )[1] so that its Bm C3+ (L ; Λ0,nov )[1] component is given by   (−1) (c1 ,c2 ) f(yc(m;1) ⊗ xc(m;1) ) ⊗ · · · ⊗ f(yc(m;m) ⊗ xc(m;m) ), (πm ◦ f)(y ⊗ x) = 1 2 1 2 c1 ,c2

where (c1 , c2 ) =



(m;i)

1≤j0 as a monoid. (C(L), m) is unobstructed over G in the sense of Definition 3.6.16.

The weak and/or def version of rationally unobstructedness can be defined in a similar way. (For the def version we need to assume that b ∈ H 2 (M ; Λ0,nov ) also  has the form (λi ,μi )∈G bi T λi eμi /2 . (bi ∈ H ∗ (M ; R).) Remark 6.2.4. In some of other references (for example in [Fuk03I]) the rationality of L is defined in a different way: We assume [ω] ∈ H 2 (M ; Z). Take a complex line bundle L → M with U (1) connection ∇ such that its first Chern form is ω. Then L was said to be rational if the holonomy group of the restriction of ∇ to L is of finite order. This definition is equivalent to Definition 6.2.1 in the case when M is simply connected but is more restrictive than Definition 6.2.1 in general. Let G be the monoid given in Definition 6.2.1. We will fix G throughout this section. Now we introduce the following Novikov ring        (6.2.5.1) Λ(G) = ai T λi eni ∈ Λnov  λi ∈ E(G), ai ∈ R ,  i

and its subring (6.2.5.2)

 + (G) = Λ



 i

ai T λi eni

     ∈ Λ0,nov  λi ∈ E(G) = Λ0,nov ∩ Λ(G). 

We remark that they are related to but different from the ring ΛR (L) defined right after Definition 2.4.17.  The degree of an element of Λ(G) is the same as that in Λnov . The Novikov  + (G) is Noetherian if G satisfies   + (G). Note that Λ ring Λ(G) is a localization of Λ  + (G)(0) is a discrete valuation ring if R is (6.2.3.1). (In fact its degree 0 part Λ a field.) This simplifies our study of the spectral sequence. In this section, we  prove Theorem I by using our spectral sequence over Λ(G) when L is rational and is rationally unobstructed. In this case the convergence of the spectral sequence follows from the standard argument. We also remark that the property (D.3)

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which will be used here can be proved without assuming that the coefficient ring is Noetherian. (See Subsection 6.4.2.) By (6.2.3.1) we have E(G) = Z>0 · λ0 , (λ0 > 0). Then we indeed have  + (G) ∼ Λ = R[[T λ0 ]][e, e−1 ].

(6.2.6)

 + (G). We put The filtration of Λ0,nov defined by (6.1.1) induces a filtration on Λ  + (G) = F mλ0 Λ  + (G). Fm Λ  + (G) module. Therefore a spectral sequence We remark that Λ0,nov is flat as a Λ  in Theorem D is induced. To prove Theorem I in our case, we will work with Λ(G)  instead of with Λ+ (G), as we noted in the previous section. In this case, we have (6.2.7)

∼  Λ(G) = R[[T λ0 ]][T −λ0 ][e, e−1 ].

 The filtration of Λnov in (6.1.2) also induces a filtration on Λ(G). We put   Fm Λ(G) = F mλ0 Λ(G). Then it follows that we have  + (G)) ∼ grq (FΛ = R[e, e−1 ]. We take a countably generated submodule C k (L; R) of the free R module S k (L; R)  generated by smooth singular (n − k)-simplices on L. We denote by C(L; Λ(G)) ∗  the completion of C(L, R) ⊗R Λ(G). The filtration F Λ(G) induces a filtration on   The degree is the sum of the deC(L; Λ(G)) which we denote by F∗ C(L; Λ(G)).  grees of C(L; R) and Λ(G). Since L is unobstructed over G, there exists a bounding  cochain b ∈ M(L). Moreover we can choose b so that it is contained in C(L; Λ(G)). b b  Then, we have the deformed A∞ algebra (C(L; Λ(G)), m ) with m0 = 0 from Propo to enable us sition 3.6.10. Thus mb1 defines the coboundary operator on C(L; Λ(G))  to obtain the deformed Floer cohomology HF ((L, b), (L, b); Λ(G)) with this coefficient ring. We omit b from our notation in this section whenever no confusion can  occur. The filtration on C(L, Λ(G)) induces the spectral sequence we are defining. We first note that there is an isomorphism (6.2.8)

∼   gr∗ (FC(L; Λ(L))) = C(L; R) ⊗ gr∗ (FΛ(G))

 as gr∗ (F∗ Λ(G)) modules. By construction of the differential (Definition 3.5.6), the differential induced on (6.2.8) is (−1)n ∂ ⊗ id where ∂ is the classical boundary operator on C(L; R). Therefore we obtain the spectral sequence Erp,q where E2 term is given by (6.2.9)

E2p,q =

n

(p−k)  H k (L; R) ⊗ grq (FΛ(G) ).

k=0 (p−k)  Here Λ(G) denotes the degree p − k part. We can also show that there exists  such that a filtration on HF p ((L, L; Λ(G))

(6.2.10)

p,q ∼ q   E∞ Fq+1 HF p (L, L; Λ(G)). = F HF p (L, L; Λ(G))/

We remark that the proof of convergence of the above spectral sequence works for an arbitrary commutative ring R.

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6.2. A TOY MODEL: RATIONAL LAGRANGIAN SUBMANIFOLDS

365

Note that multiplications by T λ0 and e give the following isomorphisms respectively:   T λ0 · : Fq HF p (L, L; Λ(G)) −→ Fq+1 HF p (L, L; Λ(G)) q p q p+2   e· : F HF (L, L; Λ(G)) −→ F HF (L, L; Λ(G)).  Now consider a graded Λ(G) module D that has a filtration F∗ D compatible with ˆ that on Λ(G) in that  · Fk D ⊆ Fk+ D. F Λ(G) (0)   be the degree 0 part of Λ(G). We have Let Λ(G) (0) ∼  Λ(G) = R[[T λ0 ]][T −λ0 ] (0)  which shows that, if R is a field, then Λ(G) is a field and so the degree p part (0)  . Dp of D is a vector space over Λ(G)

Definition 6.2.11. We define the t-rank (the topological rank or tentative rank) by p p+1 t- rank D = dimΛ(G) . (0) D + dimΛ(G) (0) D   ∼ Dp+2 , it follows Since the multiplication by e defines an isomorphism Dp = that the right hand side is independent of p. Hereafter we put R = Q. (The argument of the rest of this section works for arbitrary field R in case Floer cohomology is defined over Λ0,nov (R) and L is rational.) Lemma 6.2.12. For any q, we have:  t- rank HF (L, L; Λ(G)) q p q+1   = dimQ F HF (L, L; Λ(G))/F HF p (L, L; Λ(G)) q+1   + dimQ Fq HF p+1 (L, L; Λ(G))/F HF p+1 (L, L; Λ(G)) p,q p+1,q + dimQ E∞ . = dimQ E∞

The proof is easy and so omitted. The following lemma easily follows from Theorem 3.7.21 and Theorem 4.1.3. Lemma 6.2.13. If ψ is a Hamiltonian diffeomorphism and if L is transversal to ψ(L), then  #(L ∩ ψ(L)) ≥ t- rank HF ((L, b), (L, b); Λ(G)) for any b ∈ M(L). p,q p+1,q We now estimate dimQ E∞ + dimQ E∞ from below. Let δr : Er → Er be the coboundary operator of the spectral sequence. We have δr δr = 0 and

Er+1 = Ker δr / Im δr . ∼    Er is a gr∗ (FΛ(G)) module and gr∗ (FΛ(G)) Therefore we can define = Λ(G). t- rank Er by Definition 6.2.11. Lemma 6.2.14. t- rank Er − 2 t- rank Im δr = t- rank Er+1 ≤ t- rank Er .

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 Proof. There exist short exact sequences of Λ(G) modules: 0 → Ker δr → Er → (Im δr )[−1] → 0, 0 → Im δr → Ker δr → Er+1 → 0. We obtain t- rank Er = t- rank Ker δr + t- rank Im δr from the first exact sequence. The second exact sequence then implies the lemma. Now we give the proof of Theorem I, when L is (weakly) unobstructed (after bulk deformation) over G. From (6.2.9) and from the fact that e is of degree 2, we derive  t- rank E2 = t- rankQ H ∗ (L; Q). We put  K2 = P D(Ker(H∗ (L; Q) → H∗ (M ; Q))) ⊗Q gr∗ (F∗ Λ(G)), Kr ∩ Ker(δr ) Kr+1 = Kr ∩ Im(δr ) as in (D.3). Then as we will prove in Subsection 6.4.2, we can use the operator p to show Im(δr ) ⊂ Kr .

(6.2.15) Therefore we have

t- rank Kr+1 ≤ t- rank Kr − t- rank Im(δr ). Combining this and Lemma 6.2.14, we obtain  t- rank Er = t- rank E2 − 2 t- rank Im(δi ) = ≥

 

2≤i . By changing a part of the generators v1 , · · · , vk by the matrix (ai,i ), we find a generating set {vi } of W that satisfies (6.3.3.1), where the condition (6.3.3.n) is defined as follows: (6.3.3.n)

When we define integer ke ’s by λ1 = · · · = λk1 < λk1 +1 = · · · = λk2 < λk2 +1 = · · · ,

σ(vj )’s for j = 1, · · · , kn are linearly independent. Now we will modify the given elements vi ’s inductively on n so that (6.3.3.n) is satisfied. Let us assume (6.3.3.n). By applying a filtered automorphism of V , we may assume that vj = T λj σ(vj )

(6.3.4)

for j = 1, · · · , λkn . Let X ⊂ V be the R linear subspace generated by σ(vj ), j = 1, · · · , λkn . We consider the following two cases separately: (6.3.5.1) (6.3.5.2)

There exists i ∈ {kn + 1, · · · , kn+1 } such that σ(vi ) ∈ / X. σ(vi ) ∈ X for all i ∈ {kn + 1, · · · , kn+1 }.

We first consider the case where (6.3.5.1) is satisfied. Then we have an invertible matrix (ai,i ) (i, i ∈ {kn + 1, · · · , kn+1 }), ai,j ∈ R and bi,j ∈ R (i ∈ {kn + 1, · · · , kn+1 }, j ∈ {1, · · · , kn }) satisfying the condition (6.3.6) below, if we put kn+1 kn    vi,1 = bi,j σ(vj ) + ai,i σ(vi ) : i =kn +1

j=1

(6.3.6.1)

There exists kn <  ≤ kn+1 such that the union  }kn +1≤i≤ {σ(vj )}1≤j≤kn ∪ {vi,1

is linearly independent.  (6.3.6.2) vi,1 are zero for i > . Now for i = kn + 1, · · · , kn+1 , we replace vi by   bi,j T λkn +1 −λj vj + ai,i vi . vi = i

j

 for kn + 1 ≤ i ≤ . Hence, Note that the leading coefficient of vi is equal to vi,1  (6.3.6.2) implies E(vi ) > λkn +1 for  < i ≤ kn+1 . Now (6.3.3.n + 1) follows from (6.3.6.1). Thus the induction works for the first case.

In case (6.3.5.2) is satisfied, for all kn + 1 ≤ i ≤ kn+1 we have  = vi,1

kn 

bi,j σ(vj )

j=1

for some bi,j . We replace vi for i = kn + 1, · · · , kn+1 by vi = vi −

kn 

bi,j T λkn +1 −λj vj .

j=1

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6.3. THE ALGEBRAIC CONSTRUCTION OF THE SPECTRAL SEQUENCE

369

It then follows from the above that the coefficient of vi for the energy level λkn +1 becomes kn  σ(vi ) − bi,j σ(vj ) = σ(vi ) − σ(vi ) = 0 j=1

for all kn + 1 ≤ i ≤ kn+1 . Therefore we have E(vi ) > λkn +1 = λkn+1 for all kn + 1 ≤ i ≤ kn+1 . We now re-number vi ’s for i = kn + 1, · · · according to their energy. We again divide the cases into (6.3.5.1) and (6.3.5.2) and repeat the same process. If we end up with the (6.3.5.1), we go to the next step of the induction. Otherwise we continue. This process will continue infinitely many times only if all the components of vi , i = kn + 1, · · · are contained in X. (We use (6.3.4) to prove it.) In this case it is easy to see that we can eliminate all vi , i = kn + 1, · · · from the given generating set without changing the module generated by them. The proof of Lemma 6.3.2 is now complete.  The following is an immediate corollary of Lemma 6.3.2 and the remark right before it. Corollary 6.3.7. Any finitely generated submodule W of V is free and closed. We remark that in the course of the proof of Lemma 6.3.2 the following is also proved. Lemma 6.3.2bis. Let W be as in Lemma 6.3.2 and v1 , · · · , vm be a generator of W such that E(vi ) ≤ E(vi+1 ) and {σ(v1 ), · · · , σ(vk )} is linearly independent over R.   Then we may choose a standard basis v1 , · · · , vm  such that vi = vi for i = 1, · · · , k. 6.3.2. d.g.c.f.z. (differential graded c.f.z.). Now we consider the case of graded Λ0,nov modules.  be a graded Λ0,nov module. We assume that C  k is Definition 6.3.8. Let C a c.f.z. for each k. A differential graded c.f.z. (abbreviated as d.g.c.f.z) is a pair  δ) with a degree 1 operator δ : C →C  such that (C, δ ◦ δ = 0,

 ⊆ F λ C.  δ(F λ C)

 k is a finite c.f.z. We call the pair a finite d.g.c.f.z. if each C  k is a Λ(0) module. We remark that C 0,nov We now prove the following proposition. This will be essential for the proof of some convergence properties of the spectral sequence to be constructed in this section. (0) k . Proposition 6.3.9. Let W be a finitely generated Λ0,nov submodule of C Then there exists a constant c depending only on W but independent of λ such that

 k+1 ⊂ δ(W ∩ F λ−c C  k ). δ(W ) ∩ F λ C

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 k+1 and so has Proof. Obviously δ(W ) is a finitely generated submodule of C a standard basis by Lemma 6.3.2. Let {vi } be a standard basis of δ(W ) and let wi ∈ W satisfy δwi = vi . Suppose u ∈ δ(W ) with E(u) = λ and  u= a i vi . (0)

We put σ(ai ) ∈ R to be the leading coefficient of ai ∈ Λ0,nov and let E(vi ) = λi and E(ai ) = μi ≥ 0. Note that λ ≥ λi for all i with ai = 0. Lemma 6.3.10. If λ > λi for all i, then we have λi + μi ≥ λ. Proof. Suppose the contrary that there exists some i with λi + μi < λ. Let λ − a = inf i {λi + μi } with a > 0. We will then have  σ(ai )σ(vi ) = 0, i:λ−a=λi +μi

because E(u) = λ > λ − a. This contradicts to the fact that the leading coefficients of vi ’s are linearly independent. This finishes the proof of Lemma 6.3.10.  It follows from Lemma 6.3.10 that if λ > λi for all i with ai = 0, E(ai ) = μi ≥ λ − λi . Now we put c = max{λi }. Note that c depends only on W and is independent of λ. (0) If λ > λi , then there exists bi ∈ Λ0,nov such that T λ−c bi = ai . Hence   k ). u = δ( T λ−c bi wi ) ∈ δ(W ∩ F λ−c C On the other hand, if λ = λi for some i, we have λ = c and hence Proposition 6.3.9 trivially holds. Proposition 6.3.9 is proved.  Lemma 6.3.11. Let V be a finite c.f.z. and W its submodule such that there exists w0 ∈ W satisfying T λ w0 = w for any w ∈ W with E(w) = λ. Then W is finitely generated. Moreover it has a standard basis all of whose elements have energy 0. Proof. Let W00 be the set of elements of V which are leading coefficients of some elements of W . W00 ∪ {0} is obviously a finite dimensional R linear subspace of V , which we write as W0 . Let {wi } be a basis of W0 . For each wi , we choose an element wi ∈ W whose leading coefficient is wi . By the hypothesis of the lemma, we may assume E(wi ) = 0. We will prove that {wi } generates W , which will in turn imply the lemma. By applying a filtered automorphism on V , we may assume that (6.3.12)

wi = wi ∈ V .

Now let v ∈ W and put v=



T λi vi .

 We have v1 ∈ W0 by definition. (6.3.12) implies v1 ∈ W and so i=2 T λi vi is contained in W . Hence we have v2 ∈ W0 . Repeating the same argument, we prove (0) that vi ∈ W0 for all i. It is then easy to see that v is contained in the Λ0,nov submodule generated by wi ’s, since each of vi is a linear combination of wi ’s. 

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371

Corollary 6.3.13. Let (C, δ) be a finite d.g.c.f.z. Then Ker δk = Ker δ∩C k is finitely generated for all k. Furthermore we can take their standard bases consisting of elements of energy 0. Proof. It is easy to see that Ker δ satisfies the assumption of Lemma 6.3.11. Corollary 6.3.13 and Lemma 6.3.2 immediately imply the following structure theorem on the cohomology of a finite d.g.c.f.z. Proposition 6.3.14. Let (C, δ) be a finite d.g.c.f.z. Then for each integer p, there exist a finite number of constants λp,i > 0, i = 1, · · · , kp and p with p ≤ rank C p − kp and p such that (6.3.15)

H (C, δ) ∼ = p

kp    (0) (0) (0) Λ0,nov /F λp,i Λ0,nov (Λ0,nov )⊕p . i=1

If we choose λp,i so that λp,i+1 ≥ λp,i , then λp,i are uniquely determined. Proof. First note that since C p is finite, Im δp is finitely generated and so has a p +(0) standard basis, say, {T μp,1 u1 , T μp,2 u2 , · · · , T μp,kp ukp } for ui ∈ C = C p /Λ0,nov C p . p Since we have Im δp−1 ⊂ Ker δp and Ker δp has a basis in C consisting of energy zero, ui ∈ Ker δp . Therefore we may extend {u1 , · · · , ukp } to a standard basis {u1 , u2 , · · · , ukp , ukp+1 , · · · , ukp +p }. Now (6.3.15) immediately follows from this. To prove the uniqueness we only need to remark dimR

T λ H p (C, δ) +(0)

T λ Λ0,nov H p (C, δ)

= p + #{i | λp,i > λ}. 

The authors do not know whether Proposition 6.3.9 (and hence the convergence of the spectral sequence) holds when we replace the field R by an arbitrary Noetherian ring.  δ) be a d.g.c.f.z. and 6.3.3. Construction and convergence. Now let (C, (0) k k k  C a completion of C . We assume that C is free over Λ0,nov . We put +(0)  +(0) C,  C = C/Λ0,nov C ∼ = C/Λ 0,nov

 as the energy and let δ be the induced derivation on C. We again embed C ⊆ C ⊆ C  0 part. In general C is not a differential graded subalgebra of C. Let {ei } be a basis (0) (0) +(0) of C (over Λ0,nov ) and ei be the corresponding basis of C over R = Λ0,nov /Λ0,nov . We put  δ(ei ) = δ0,ij ej , →C  by and define δ0 : C δ0 e i =



δ0,ij ej .

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 δ) satisfies the gapped condition if there Condition 6.3.16. We say that (C,  exists λ > 0 such that for any λ we have   δv − δ0 v ∈ F λ+λ C

 for all v ∈ F λ C. Remark 6.3.17. If a filtered A∞ algebra (C, m) with m0 = 0 is G-gapped in the sense of Definition 3.2.26, (C, m1 ) satisfies the gapped condition above. Assuming Condition 6.3.16, we take a constant λ0 with 0 < λ0 < λ and define a filtration by  = F nλ0 C.  Fn C

(6.3.18)

We use this filtration to define our spectral sequence. Since construction of the spectral sequence from a filtered module over a filtered ring is not so standard, we present its construction here for reader’s convenience. We put

(6.3.19.2)

 = {x ∈ Fq C  p | δx ∈ Fq+r−1 C  p+1 } + Fq+1 C p , Zrp,q (C)  = (δ(Fq−r+2 C  p−1 ) ∩ Fq C  p ) + Fq+1 C p , B p,q (C)

(6.3.19.3)

p,q   = Z p,q (C)/B  Erp,q (C) r r (C).

(6.3.19.1)

r

We denote (0)

(0)

Λ(0) (λ) = Λ0,nov /F λ Λ0,nov . (0)

(0)

(0)

We define a filtration of Λ0,nov by Fn Λ0,nov = F nλ0 Λ0,nov . Then its associated graded module is given by (0)

gr∗ (FΛ0,nov ) =



(0)

grn (FΛ0,nov ),

n∈Z≥0 (0)

where each grn (FΛ0,nov ) is naturally isomorphic to Λ(0) (λ0 ). We also have

gr∗ (FΛ0,nov ) = gr∗ (FΛ0,nov )[e, e−1 ] = (0)

grn (FΛ0,nov )[e, e−1 ]. (0)

n∈Z≥0

Recall that Erp,q has a natural structure of Λ(0) (λ0 ) module. The multiplication by e±1 ∈ Λ0,nov defines a map  → Erp±2,q (C),  e±1 : Erp,q (C)  := Erp,q (C)  into a gr∗ (FΛ0,nov ) module and which turns Er (C) p,q p,q

δrp,q :



 → Erp,q (C)

p,q

p,q

is a gr(F∗ Λ0,nov ) module homomorphism.

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 Erp+1,q+r−1 (C)

6.3. THE ALGEBRAIC CONSTRUCTION OF THE SPECTRAL SEQUENCE

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Lemma 6.3.20. There exists a Λ(0) (λ0 ) module homomorphism  → E p+1,q+r−1 (C)  δrp,q : Erp,q (C) r such that (6.3.21.1) (6.3.21.2) (6.3.21.3)

δrp+1,q+r−1 ◦ δrp,q = 0, p,q  Ker(δrp,q )/ Im(δrp−1,q−r+1 ) ∼ = Er+1 (C), ±1 p,q p±2,q ±1 ◦e . e ◦ δr = δ r

 The proof of (6.3.21) is Proof. We define δrp,q [x] = [δx] ∈ Erp+1,q+r−1 (C). standard.   is quite standard. One difference from Of course, the construction of Erp,q (C) the standard case is that our filtration is not bounded. Namely we do not have  = 0 for large n. Hence the convergence property of our spectral sequence is Fn C  = C.  far from being trivial in general. However it is stable from below in that F0 C As a consequence we have: Lemma 6.3.22. There exists an injection p,q   Er+1 (C) → Erp,q (C)

if q − r + 2 ≤ 0.  p−1 ) ∩ Fq C  p + Fq+1 C  p in the definition Proof. The denominator δ(Fq−r+2 C  is independent of r if q − r + 2 ≤ 0.  of Erp,q (C) Definition 6.3.23. We define p,q   E∞ (C) = lim Erp,q (C) ←−

as the projective limit which exists by Lemma 6.3.22. Lemma 6.3.24. We have an isomorphism  ∼ E2∗,∗ (C) = H(C; δ) ⊗R gr∗ (FΛ0,nov ) as gr∗ (FΛ0,nov ) modules. Proof. By definition we have  ∼ E1∗,∗ (C) = C ⊗R gr∗ (FΛ0,nov ). It follows from Condition 6.3.16 that δ1 = δ. Hence it finishes the proof.



 δ) to be the image of H(Fq C,  δ) in Definition 6.3.25. We define Fq H(C,  δ). H(C, p,q  δ), we need some To relate the limit E∞ of the spectral sequence and Fq H(C, finiteness assumption which we now describe. Let (C, δ) and (C  , δ  ) be d.g.c.f.z’s satisfying the gap condition. Let ϕ : C → C  be a map such that ϕδ = δ  ϕ and   +(0) +(0) let ϕ : C → C be the map induced on C = C/Λ0,nov C and C = C  /Λ0,nov C  respectively. The induced map ϕ lifts to ϕ0 : C → C  using the basis on C and C  .

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Definition 6.3.26. Under the situation above, we say that ϕ : C → C  satisfies a gapped condition, or is a gapped cochain map, if there exists λ such that  ⊂ F λ+λ C.  (ϕ − ϕ0 )(F λ C) As in Remark 6.3.17, we note that if a filtered A∞ homomorphism is G-gapped in the sense of Definition 3.2.29, the induced map satisfies the gapped condition above.  δ) is said to be weakly finite if there exists a finite Definition 6.3.27. (C,    δ) → (C  , δ  ) which d.g.c.f.z. (C , δ ), and a gapped filtered cochain map ϕ : (C, induces isomorphisms on cohomologies    δ) ∼ ϕ∗ : H(C, = H(C , δ ).

Using these definitions, we prove Theorem 6.3.28. If C is weakly finite, then there exists r0 such that: p,q p,q  ∼ q p   ∼  ∼  δ) Erp,q (C) (C) = F H (C, δ)/Fq+1 H p (C, = Er0 +1 (C) = ··· ∼ = E∞ 0 (0)

(0)

as Λ(0) (λ0 ) = Λ0,nov /F λ0 Λ0,nov modules. Proof. We first consider the case when C k is finite c.f.z. (In this case we have  k .) By Proposition 6.3.9 there exists a positive constant c such that C =C k

δ(C k ) ∩ F λ C k+1 ⊂ δ(F λ−c C k ) for any λ. We choose r0 so that (r0 −1)λ0 −c > λ0 . Now let r ≥ r0 and x ∈ Zrp,q (C). We have x = x0 + x1 such that δx0 ∈ Fr+q−1 C = F (r+q−1)λ0 C,

x1 ∈ Fq+1 C.

Therefore, by Proposition 6.3.9, we have y such that δx0 = δy,

y ∈ F (r+q−1)λ0 −c C ⊂ F (q+1)λ0 C = Fq+1 C.

In follows that [x] = [x − y] = [x0 − y] ∈ Erp,q (C), and δr [x] = δr [x0 − y] = 0,

p,q p,q ∼ Erp,q . = Er0 +1 ∼ = ··· ∼ = E∞ 0

p,q ∼ q p To prove E∞ = F H (C, δ)/Fq+1 H p (C, δ), we first construct a map p,q . πp,q : Fq H p (C, δ) → E∞

Let x ∈ Fq C p with δx = 0. Then [x] ∈ Erp,q (C) for any r ≥ r0 . Such [x] defines an p,q element of E∞ (C). Suppose that x − x = δy. If r > q, we have y ∈ Fq−r V p−1 = p−1 V . Hence [x] = [x ] in Erp,q (C). Therefore this assignment defines a well-defined map which we denote by πp,q . p,q We next prove that πp,q is surjective. Let [x] ∈ Erp,q (C) ∼ (C). Then, as = E∞ 0 q+1 we proved before, there exists y ∈ F C such that [x − y] ≡ [x] ∈ Erp,q (C),

δ(x − y) = 0.

Therefore x−y defines a cocycle in Fq C p and so [x−y] ∈ Fq H p (C, δ) with πp,q [x−y] = [x] and hence πp,q is surjective.

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6.4. THE SPECTRAL SEQUENCE ASSOCIATED TO A LAGRANGIAN SUBMANIFOLD 375

We next prove the equality Ker πp,q = Fq+1 H p (C, δ). Let [x] ∈ Fq H p (C, δ) and πp,q ([x]) = 0. Then for r > q + 2 we have x ∈ δ(Fq−r+2 C p−1 ) + Fq+1 C p ⊂ δ(C p−1 ) + Fq+1 C p . Hence [x] ∈ Fq+1 H p (C, δ). This proves Ker πp,q ⊂ Fq+1 H p (C, δ). On the other hand, Fq+1 H p (C, δ) ⊂ Ker πp,q is obvious.  is finitely generated. Let (C  , δ  ) be a Finally we relax the condition that C  → C  a gapped cochain map which induces isomorphisms finite d.g.c.f.z. and ϕ : C on cohomologies as in Definition 6.3.27. Then ϕ induces a morphism of spectral sequence. It is an isomorphism in the E2 term and hence an isomorphism at E∞ term. On the other hand, ϕ is assumed to induce an isomorphism H(C, δ) →   H(C , δ ). We can then prove that ϕ induces an isomorphism on the cohomology  δ) → H(C  , δ  ), in the same way as the proof of Proposition 4.3.18. Therefore H(C, πp,q is an isomorphism for C because it is so for C  . That completes the proof of Theorem 6.3.28.  In the case of Floer homology of periodic Hamiltonian systems, we can show that the modified Floer homology in [Ono95] is isomorphic to the (usual) Floer homology based on an analogous statement as Proposition 6.3.9. 6.4. The spectral sequence associated to a Lagrangian submanifold 6.4.1. Construction. So far, we have constructed the spectral sequence in a purely algebraic manner. In this section we will prove Theorem D by applying the results in the previous section to our geometric situation. In this subsection we prove (D.1) and (D.2). Let L be a relatively spin Lagrangian submanifold of a symplectic manifold M and (C(L, Λ0,nov ), m) the associated filtered A∞ algebra. Let b = (b, b) ∈ Mweak,def (L). By the construction in Section 3.5, C(L, Λ0,nov ) is a c.f.z. in the sense of Subsection 6.3.1. (C(L, Λ0,nov ), mb 1 ) is a chain complex. Moreover we already found that the gapped condition for (C(L, Λ0,nov ), mb 1 ) follows from Gromov’s compactness theorem. (See Theorem 3.5.11.) Furthermore, (C(L, Λ0,nov ), mb 1 ) is weakly finite in the sense of Definition 6.3.27.  In fact it is chain homotopy equivalent to (H(L; Λ0,nov ), mcan,b,b ), where b ∈ 1 Mweak (H(L; Λ), mcan,b ). This is a consequence of Theorem 5.4.2. Therefore Theorem 6.3.28 implies that we have a spectral sequence as in Theorem D (D.1), (D.2). The construction of the spectral sequence as in Theorem 6.1.4 satisfying (6.1.5.1), (6.1.5.2) is similar by using Corollary 5.4.21. 6.4.2. A condition for degeneration: proof of (D.3). In this subsection, we will prove (D.3) by using the operators p constructed in Subsection 3.8.9. We recall that we have the sequence of operators p,k : E C(M ; Λ0,nov )[2] ⊗ Bkcyc C1+ (L; Λ0,nov )[1] → S ∗ (M ; Λ0,nov ) defined in Subsection 3.8.9. See Proposition 3.8.78. Then for (b, b) ∈ Mdef (L), we define (6.4.1)

b,b b b b b b b pb,b 1 (x) = p0,1 (x) := p(e , e xe ) = p(e ⊗ e xe )

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for x ∈ C2 (L; Λ0,nov )[1]. Note that b b b pb,b 1 (x) = p (e xe ),

where the operator pb is already defined in Subsection 3.8.9. We also recall that δ(b,b) (x) = q(eb , eb xeb ) = q(eb ⊗ eb xeb ). See Definition 3.8.58. Then we can show the following. Lemma 6.4.2. Let δM be the coboundary operator on S ∗ (M ; Λ0,nov ) in Proposition 3.8.78. Then we have (6.4.3.1)

pb,b 1 ≡ i!

(6.4.3.2)

b,b pb,b 1 ◦ δ(b,b) + δM ◦ p1 = 0,

mod Λ+ 0,nov ,

where i! : H ∗ (L; Q) → H ∗+n (M ; Q) is the Gysin homomorphism. b,b + Proof. As b ≡ b ≡ 0 mod Λ+ 0,nov , we have p1 ≡ p1 mod Λ0,nov . Then (6.4.3.1) follows from (3.8.87.1). Since db (eb xeb ) = eb mb (eb xeb )eb = eb q(eb , eb xeb )eb , we have by (3.8.87.2)

0 = pb ◦ db,cyc (eb xeb ) + δM ◦ pb (eb xeb ) = pb (eb q(eb , eb xeb )eb ) + δM ◦ pb,b 1 (x). 

This proves (6.4.3.2).

When L is weakly unobstructed after bulk deformation, we use the operation : C1+ (L; Λ0,nov )+ [1] → S ∗ (M ; Λ0,nov ) p+b,b 1 defined by (6.4.4)

+ b b b (x) = p+b,b p+b,b 1 0,1 (x) := p (e , e xe )

for (b, b) ∈ Mweak,def (L). Here p+ is defined in Proposition 3.8.78. Then we can show the following lemma analogous to Lemma 6.4.2. We omit the proof of the lemma. Lemma 6.4.5. Under the situation above, we have (6.4.6.1)

p+b,b ≡ i! 1

(6.4.6.2)

+ ◦ δ(b,b) + δM ◦ p+b,b = 0. p+b,b 1 1

mod Λ+ 0,nov ,

+ (x) = q+ (eb , eb xeb ). See Definition 3.8.58. Here δ(b,b)

Proof of Theorem D (D.3). We put C = C(L; Λ0,nov ), δ = δb,b . Let  Namely we assume x = y+z such that δy ∈ Fq+r−1 C  p+1 , z ∈ Fq+1 C p . x ∈ Zrp,q (C). Note

(6.4.7)

p+1,q+r−1  (C)  = Zr δr ([x]) = [δy] ∈Erp+1,q+r−1 (C) p+1,q+r−1  Br (C) p+1,q+r−1  (C) Zr . = q+1 p q+r−1 p+1    p+1 (δ(F C ) ∩ F ) + Fq+r C C

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6.4. THE SPECTRAL SEQUENCE ASSOCIATED TO A LAGRANGIAN SUBMANIFOLD 377

By (6.4.3.2) we have (6.4.8)

b,b pb,b 1 (δy) = −δM (p1 (y)).

By (6.4.3.1), we may choose λ0 in (6.3.18) so that (6.4.9)

λ0 pb,b . 1 − i! ≡ 0 mod T

 p+1 = T (q+r−1)λ0 C  p+1 , it follows from (6.4.7) and (6.4.8) that Since δy ∈ Fq+r−1 C (6.4.10)

(q+r)λ0 p+1 i! (δy) + δM (pb,b C (M ). 1 (y)) ∈ T

 p+1 and Therefore we can write δy = w1 + w2 such that w2 ∈ Fq+r C (6.4.11)

i! (w1 ) ∈ Im δM .

(We use that fact that i! and δM do not involve T to prove this.) We remark that the Gysin homomorphism i! : H k (L; Q) → H n+k (M ; Q) is identified with i∗ : Hn−k (L; Q) → Hn−k (M ; Q) by Poincar´e duality. Hence (6.4.7), (6.4.11) and the definition of Kr imply δr ([y]) ∈ Kr+1 . The property (D.3) is proved.  The proof of (6.2.15) is similar. Remark 6.4.12. We remark that (D.3) does not hold for the Floer cohomology HF ((L; (b, b1 )), (L; (b, b0 )); Λ0,nov ) in case b1 = b0 , in general. Let us consider L = S 1 = S 1 × {0} ⊂ S 1 × S 1 = M . If b = 0, b1 = 0, b0 = T P D([pt]), then δb1 ,b0 (P D([S 1 ])) = m2 (P D([S 1 ]), b0 ) = T · P D([pt]), which is not contained in Ker(i! : H 1 (L; Λ0,nov ) → H 2 (M ; Λ0,nov )). 6.4.3. Non-vanishing theorem: proof of Theorem 6.1.9. To each pair + (b, bi ) ∈ Mweak,def (L), we can associate the coboundary operator δ(b,b (x) = 1 ),(b,b0 ) + b b1 b0 q (e , e xe ). (Definition 3.8.58 and Lemma 3.8.60.) Suppose PO(b1 ) = PO(b0 ). We start from proving (6.1.10.1). We recall that C(L; Λ0,nov )+ = C(L; Λ0,nov ) ⊕ Λ0,nov e+ ⊕ Λ0,nov f as in Section 3.3 and e = P D([L]) is the homotopy unit in (C(L; Λ0,nov ), m). We show the following. + Lemma 6.4.13. δ(b,b (f) = e+ − e + h+M (eb , eb1 ⊗eb0 ). 2 1 ),(b,b0 )

Proof. From Definition 3.8.38 we have the filtered A∞ algebra (C(L; Λ0,nov )+ , m+b ) deformed by b. In this case (3.8.35.4) implies +M b b + (e , 1⊗1). q+ ∗,1 (e , f) = e − e + h2

This is a ‘def’ version of (3.3.5.2). Then by (3.8.35.3), we have + (f) = q+ (eb , eb1 feb0 ) δ(b,b 1 ),(b,b0 ) +M b b1 b (e , e ⊗eb0 ) − h+M (eb , 1⊗1) = q+ ∗,1 (e , f) + h2 2

= e+ − e + h+M (eb , 1⊗1) + h+M (eb , eb1 ⊗eb0 ) − h+M (eb , 1⊗1) 2 2 2 = e+ − e + h+M (eb , eb1 ⊗eb0 ). 2

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 + (e+ ) = 0, we obtain Then since δ(b,b 1 ),(b,b0 ) + (e − h+M (eb , eb1 ⊗eb0 )) = 0. δ(b,b 2 1 ),(b,b0 )

This proves (6.1.10.1). We next prove (6.1.10.2). Since the statement over Λnov coefficient is sharper than that over Λ0,nov , we work over Λnov coefficient. For the proof of (6.1.10.2), we use the operator p. Note that we assumed b = b1 = b0 in (6.1.10.2). Suppose to + the contrary that P D([pt]) ∈ Im(δr ). Then we can put T λ P D([pt]) = δ(b,b) (x) for some x ∈ C(L; Λ0,nov ) and λ ≥ 0. Lemma 6.4.5 shows that T λ i! (P D[pt]) ≡ T λ p+b,b (P D[pt]) mod T λ Λ+ 0,nov 1 + = p+b,b (δ(b,b) (x)) = −δM (p+b,b (x)). 1 1

Hence i! (P D[pt]) = 0 in H 2n (M ; Q). (Note δM is the usual boundary operator and hence does not contain T .) On the other hand P D([pt]) = 0. This contradicts the fact that the Gysin homomorphism i! : H n (L; Q) → H 2n (M ; Q) is injective. This proves (6.1.10.2). We now turn to the proof of (6.1.12). Here we use the assumption of Maslov index. We will also use the fact that H  (L; Q) is nonzero only for  ∈ {0, · · · , n}. Note that C  (L; Q) ⊂ Sn− (L; Q) may be nonzero for  < 0. In fact there exists a nonzero singular chain on L of dimension > n = dim L. By this reason we work on canonical model to prove (6.1.12). We consider (6.4.14)

b=

∞ 

emi /2 T λi bi ∈ H 2 (M ; Λ+ 0,nov ),

i=1

with bi ∈ H(M ; Q). We assume that deg bi = 0. (We will remove this assumption later.) Then mi = 2 − deg bi ≤ 0.

(6.4.15)

We put G(L) = {(ω(β), μ(β)) ∈ R × 2Z | M1 (β) = ∅} and G(L, b) = {(ω(β) + λj , μ(β) + mj ) ∈ R × 2Z | (ω(β), μ(β)) ∈ G(L), j = 1, 2, · · · }, where mj , λj are as in (6.4.14). We consider the filtered A∞ algebra (C(L; Λ0,nov )+ , m+,b ). It is unital and G(L; b) gapped. We take its canonical model and denote it by (H(L;Λ0,nov ),mcan,b ). It is also unital and G(L; b) gapped. There exists a unital and G(L; b) gapped homotopy equivalence f : (H(L; Λ0,nov ), mcan,b ) → (C(L; Λ0,nov )+ , m+,b ). It induces a map weak (H(L; Λ0,nov ), mcan,b ) → M weak (C(L; Λ0,nov )+ , m+,b ), f∗ : M which induces an isomorphism between gauge equivalence classes. We consider the weak (C(L; Λ0,nov )+ , m+,b ). following condition for b ∈ M

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6.4. THE SPECTRAL SEQUENCE ASSOCIATED TO A LAGRANGIAN SUBMANIFOLD 379

Condition 6.4.16. b=

(6.4.17)

∞ 

eki /2 T λi bi

i=1

with bi ∈ C(L; Q)+ and ki ∈ 2 Z≤0 . Lemma 6.4.18. We assume that ni ≤ 0 for any (λi , ni ) ∈ G(L). Then any weak (C(L; Λ0,nov )+ , m+,b ) is gauge equivalent to one satisfying Conelement of M dition 6.4.16. Proof. The assumption implies that m ≤ 0 for any (λ, m) ∈ G(L; b). (In fact mi ≤ 0 where mi is as in (6.4.14).) Therefore, since f is G(L; b) gapped, it suffices weak (H(L; Λ0,nov ), mcan,b ). to prove the same statement for M Let ˜b =

(6.4.19)

∞ 



eki /2 T λi ˜bi

i=1

weak (H(L; Λ0,nov ), mcan,b ) is contained in the first with ˜bi ∈ H(L; Q). Since M 1 cohomology group H (L; Λ0,nov ), it follow that deg bi = 1 − ki . On the other hand, H  (L; Q) is nonzero only for  ∈ {0, · · · , n}. Moreover ki is even. Therefore ki ≤ 0 as required.  weak (H(L; Λ0,nov ), mcan,b ) and put b = f∗ (˜b). Then b satisfies We take ˜b ∈ M Condition 6.4.16 and any weak bounding cochain is gauge equivalent to such b. We put b=

(6.4.20)

∞ 



eni /2 T λi bi

i=1

and define (6.4.21)

G(L; b, b) = {(ω(β) + λi , μ(β) + ni ) ∈ R × 2Z | (ω(β), μ(β)) ∈ G(L, b), i = 1, 2, · · · }. ˜

We define mcan,b,b , m+,b,b by deforming mcan,b , m+,b using ˜b, b, respectively. Then ˜ we have G(L; b, b)-gapped and unital filtered A∞ algebras (H(L; Λ0,nov ), mcan,b,b ) ˜

(1) are proportional to the and (C(L; Λ0,nov )+ , m+,b,b ) such that m0can,b,b (1), m+,b,b 0 ˜

+ can = m1can,b,b , δ(b,b) = m+,b,b . Then we have unit. We put δ(b, 1 ˜ b) can can ◦ δ(b, = 0, δ(b, ˜ ˜ b) b)

+ + δ(b,b) ◦ δ(b,b) = 0.

We define can can ) → (C(L; Λ0,nov )+ , δ(b, ), ϕ : (H(L; Λ0,nov ), δ(b, ˜ ˜ b) b)

by ϕ(x) =



fk1 +k2 +1 (˜b⊗k1 , x, ˜b⊗k2 ).

k1 ,k2

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Then it is easy to see that ϕ is a chain map which induces an isomorphism on cohomology group. It is G(L; b, b)-gapped in an obvious sense. We use the unitality of f to show (6.4.22)

ϕ([P D[L]]) = e+ .

can ).) (Note [P D[L]] is the unit of (H(L; Λ0,nov ), δ(b, ˜ b) Now we are in the position to complete the proof of Theorem 6.1.9. The Poincar´e dual P D([pt]) in C(L; Λ0,nov )+ gives an element of (cohomology) degree n. We remark C  (L; Λ0,nov )+ = 0 for  > n. we have  + + = δ(b,b),β T λi eni /2 δ(b,b) (λi ,ni )∈G(L;b,b) + and δ(b,b),β : C(L; Q) → C(L; Q) is of degree 1 − ni . By assumption ni ≤ 0. Hence + δ(b,b),β (P D[pt]) = 0. This proves (6.1.12.2). We next prove (6.1.12.1). We first prove that the unit [P D[L]] ∈ H 0 (L; Q) is can not in the image of δ(b, . We put ˜ b)  can can δ(b, = δ(b,b),β T λi eni /2 . ˜ b) (λi ,ni )∈G(L;b,b) can Here δ(b,b),β : H(L; Q) → H(L; Q) is of degree 1 − ni . Since ni ≤ 0, and since  H (L; Q) is zero for  < 0, it follows that the unit [P D[L]] ∈ H 0 (L; Q) is not in the can image of δ(b,b),β . can Therefore [P D[L]] ∈ H 0 (L; Q) is nonzero in δ(b,b),β cohomology. Therefore by + (6.4.22), e+ is not zero in H(C(L, Λ0,nov )+ , δ(b,b) ). Since

P D([L]) − h+M (eb , eb1 ⊗eb0 ) 2 + is cohomologous to e+ , it follows that it is not zero in δ(b,b) cohomology. Hence the proof of (6.1.12.1).

The linear independence of these classes follows from the fact that the filtration by the degree is preserved by the coboundary operator. We assumed deg bj = 0 so far. Using (3.8.36) and Proposition 3.8.54, we can remove this assumption in the following way. During the above proof we take a linear subspace of C 2 (M ; Λ0,nov ) which represents H 2 (M ; Λ0,nov ). We can use C 2 (M ; Λ0,nov )+ in place of C 2 (M ; Λ0,nov ). (C 2 (M ; Λ0,nov )+ is defined in (3.8.36).) Then we may choose H 2 (M ; Λ0,nov ) ⊂ C 2 (M ; Λ0,nov )+ such that P D[M ] is represented by [˜ e+ ], by virtue of (3.8.36.3). Now, let bj be as in (6.4.14). We put  emi /2 T λi bi ∈ H 2 (M ; Λ+ b = 0,nov ). deg bi =0

Since b − b = cP D[M ] = c[˜ e+ ], Proposition 3.8.54 implies  0 unless x ∈ B0 C(L, Λ0,nov ),  (6.4.23) q(eb , x) − q(eb , x) = + if x = 1 ∈ B0 C(L, Λ0,nov ). cee

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6.4. THE SPECTRAL SEQUENCE ASSOCIATED TO A LAGRANGIAN SUBMANIFOLD 381

We can use it to show weak (C(L; Λ0,nov )+ , m+,b ) weak (C(L; Λ0,nov )+ , m+,b ) = M M + + + +,b  and δ(b,b) = δ(b ). Therefore we can always  ,b) for b ∈ Mweak (C(L; Λ0,nov ) , m  replace b by b . Therefore we have finished the proof of Theorem 6.1.9. 

We remark that (6.1.12.2’) does not hold for the case b1 = b0 . The example in Remark 6.4.12 will be a counter example. Proof of Theorem L. We assume L ⊂ M is relatively spin and H 2 (L; Q) = 0, and μL = 0. Since the obstruction for M(L; Q) lies in H 2 (L; Q) = 0 we have bounding cochain b ∈ M(L; Q). Then it follows from Theorem D (D.1) and (6.1.12.1) that the element P D([L]) − h+M (eb , eb1 ⊗eb0 ) defines a nontriv2 ial element of the Floer cohomology HF ((L, b), (L, b); Λnov ). Therefore for any Hamiltonian diffeomorphism ψ we have ψ(L) ∩ L = ∅, as required. Suppose L is transversal to ψ(L). We note that since μL = 0 we can define Floer cohomology (0) (0) HF ((L, b), (L, b); Λnov ) over Λnov . (See Subsection 5.4.6.) In other words the inte(eb , eb1 ⊗eb0 ) is zero ger degree is well defined. Since the degree of P D([L]) − h+M 2 (before shifted), there should be an intersection point ψ(L) ∩ L of Maslov index 0. Theorem L is proved.  6.4.4. Application to the Maslov class conjecture: proofs of Theorems 6.1.15 and 6.1.17. Proof of Theorem 6.1.15. From Theorems 3.1.11 and 3.8.50, the obstruction classes are in H 2−μL (βk ) (L; Q)/ Im H 2−μL (βk ) (M ; Q). Since μL = 0 and the map H 2 (M, Q) → H 2 (L; Q) is surjective by the assumption, all the obstruction classes vanish. Therefore L is unobstructed after bulk deformation. Now Theorem 6.1.9 concludes we have HF ((L, b), (L, b); Λnov ) = 0.  Proof of Theorem 6.1.17. We first prove (1). We assume L is weakly unobstructed after bulk deformation and ΣL = 0. Then there exists b ∈ Mweak,def (L). Using the assumption ΣL = 0 and Theorem 6.1.9, we have HF ((L; b), (L; b); Λnov ) = 0. This contradicts to our assumption that L is displaceable. weak,def (L) as in the assumption of (2). We next prove (2). Let (b, b) ∈ M Lemma 6.4.24. π2 (G(L; b, b)) = π2 (G(L)), where π2 : R × 2Z → 2Z is the projection. Here G(L; b, b) is defined by (6.4.21).    Proof. If we put b = emi /2 T λi bi , b = eki /2 T λi bi then mi = 2 − deg bi = 0, ki = 1 − deg bi = 0. The lemma follows from definition.  We put can = mcan,b,b : H(L; Λ0,nov ) → H(L; Λ0,nov ). δ(b,b) 1 can (H(L; Λ0,nov ), δ(b,b) ) is a chain complex whose cohomology is Floer cohomology HF ((L; b, b), (L; b, b)). By Lemma 6.4.24, have  can = eμi /2 T λi δi , δ(b,b) (μi ,λi )∈G(L;b,b)

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where δi : H(L; C) → H(L; C). We now assume ΣL ∈ / {2, · · · , n + 1} and will deduce a contradiction. By this assumption either (6.4.25.1)

ki = 0, or

(6.4.25.2)

|ki | > n + 1.

In case (6.4.25.2), we have deg δi = 1 − ki ∈ / {−n, · · · , n}. Since δi is a map between ordinary cohomology group of L (which is n dimensional), it follows that δi = 0. This contradicts to the displacability of L. Thus we may assume ki = 0. can Therefore, P D[L] ∈ H 0 (L; C) is not in the image of δ(b,b) . On the other hand, can since P D[L] is a unit, we have δ(b,b) (P D[L]) = 0. Thus the Floer cohomology HF ((L; b, b), (L; b, b); Λnov ) is nontrivial. This contradicts to the displaceability of L. The proof of Theorem 6.1.17 is complete.  Remark 6.4.26. In the proof of Theorem 6.1.17, we use P D([L]) but do not use P D([pt]). Therefore the proof also works over Λ0,nov (Z) (resp. Λ0,nov (Z/2Z)) by considering P D([L]) − h2 (eb ⊗eb ), in the case Floer cohomology over Z or Z2 is defined. (See [FOOO09I].) 6.4.5. Compatibility with the product structure. We now prove Theorem D (D.4) and which will complete the proof of Theorem D. In Subsection 3.7.7, we defined the product structure m2 : HF ((L, b2 ), (L, b1 )) ⊗ HF ((L, b1 ), (L, b0 )) → HF ((L, b2 ), (L, b0 )) 2 ,b1 ,b0 ˆ ˆ on the Floer cohomology. It is induced by the operator mb : C(L) ⊗ C(L) → 2 ˆ C(L), where 2 ,b1 ,b0 mb (x, y) = mb (eb2 , x, eb1 , y, eb0 ) 2

(6.4.27)

and bi = (b, bi ). (See (3.7.90).) By (6.4.27) it is easy to see that (6.4.28)

2 ,b1 ,b0 ˆ ˆ ˆ mb (Fq1 C(L) ⊗ Fq2 C(L)) ⊆ Fq1 +q2 C(L). 2

This implies that the filtration F on HF ((L, bj ), (L, bi )) are preserved by m2 . To find a product structure on Er it suffices to show the following: Lemma 6.4.29. 























(6.4.30.1)

2 ,b1 ,b0 (Zrp,q ⊗ Zrp ,q ) ⊆ Zrp+p ,q+q , mb 2

(6.4.30.2)

2 ,b1 ,b0 mb (Brp,q ⊗ Zrp ,q ) ⊆ Brp+p ,q+q , 2

(6.4.30.3)

2 ,b1 ,b0 mb (Zrp,q ⊗ Brp ,q ) ⊆ Brp+p ,q+q . 2









Note that we use δb2 ,b1 , δb1 ,b0 , δb2 ,b0 to define Zrp,q , Zrp ,q , Zrp+p ,q+q in (6.4.30.1), respectively. (Similarly for (6.4.30.2), (6.4.30.3).) Proof. We prove (6.4.30.2) only. (The other formulae are proved in the same way.) We recall (6.4.31)

2 ,b1 ,b0 δb2 ,b0 (mb (x, y)) 2 2 ,b1 ,b0 2 ,b1 ,b0 = −mb (δb2 ,b1 (x), y) + (−1)deg x mb (x, δb1 ,b0 (y)). 2 2

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6.4. THE SPECTRAL SEQUENCE ASSOCIATED TO A LAGRANGIAN SUBMANIFOLD 383

(See (3.7.91).) Let x ∈ δb2 ,b1 x1 + x2 ∈ Brp,q where x1 ∈ Fq−r+2 Cˆ p−1 , δb2 ,b1 x1 ∈     Fq Cˆ p , x2 ∈ Fq+1 Cˆ p . Let y = y1 + y2 ∈ Zrp ,q where δb1 ,b0 y1 ∈ Fq +r−1 Cˆ p +1 ,     y1 ∈ Fq Cˆ p , y2 ∈ Fq +1 Cˆ p . (See (6.3.19).) It is easy to see that 







2 ,b1 ,b0 (δb2 ,b1 x1 , y2 ) ∈ Fq+q +1 Cˆ p+p ⊆ Brp+p ,q+q mb 2     2 ,b1 ,b0 mb (x2 , y1 ) ∈ Fq+q +1 Cˆ p+p ⊆ Brp+p ,q+q 2     mb2 ,b1 ,b0 (x2 , y2 ) ∈ Fq+q +1 Cˆ p+p ⊆ Brp+p ,q+q .

2

On the other hand, (6.4.31) implies 2 ,b1 ,b0 mb (δb2 ,b1 x1 , y1 ) 2 2 ,b1 ,b0 2 ,b1 ,b0 = −δb2 ,b0 (mb (x1 , y1 )) + (−1)deg x mb (x1 , δb1 ,b0 (y1 )). 2 2

We have 2 ,b1 ,b0 mb (x1 , y1 ) ∈ Fq+q 2



−r+2

 Cˆ p+p −1 .

This implies       2 ,b1 ,b0 (x1 , y1 )) ∈ δb2 ,b0 (Fq+q −r+2 Cˆ p+p −1 ) ∩ Fq+q Cˆ p+p ⊆ Brp+p ,q+q . δb2 ,b0 (mb 2

Moreover 2 ,b1 ,b0 mb (x1 , δb1 ,b0 (y1 )) ∈ Fq+q 2



+1

   Cˆ p+p ⊆ Brp+p ,q+q .

Thus (6.4.30.2) is proved.



2 ,b1 ,b0 Lemma 6.4.29 implies mb induces a product structure m2 on Erp,q . We 2 b2 ,b1 ,b0 remark that m2 ≡ m2 mod T λ0 Λ0,nov . Compatibility of m2 with isomorphisms in (D.1) follows. The compatibility with the isomorphism (D.2) is obvious from definition. The formula

(6.4.32)

δr (m2 (x, y)) = −m2 (δr (x), y) + (−1)deg x m2 (x, δr (y))

is an immediate consequence of (6.4.31). The proof of Theorem D is now complete. Remark 6.4.33. The fact that the spectral sequence relating ordinal cohomology and Floer cohomology preserves product was suggested by P. Biran [Bir04]. Problem 6.4.34. In which sense our spectral sequence is compatible with (higher) Massey product mk , k ≥ 3 ? The above compatibility (6.4.32) of the spectral sequence with the induced m2 has the following improvement of the second named author’s result Theorem III [Oh96I] concerning the Maslov class of monotone Lagrangian tori (Audin’s question): he proved the same consequence under the rather peculiar dimensional restriction dim L ≤ 24. The idea of using the compatibility of spectral sequence in the current proof is due to P. Seidel [Sei01]. L. Buhovsky proved Theorem 6.4.35 independently in [Buh06] where he showed the compatibility of the multiplicative structures of Oh’s spectral sequence for the monotone case. His proof of the compatibility is different from ours given here in which we use the filtered A∞ formula in a crucial way.

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SPECTRAL SEQUENCES

Theorem 6.4.35. Suppose that L ⊂ (M, ω) is a monotone Lagrangian torus such that there exists a Hamiltonian diffeomorphism φ with φ(L) ∩ L = ∅. Then we have its minimal Maslov number ΣL satisfies ΣL = 2. Proof. We first remark that T n is orientable and spin, and the monotonicity implies ΣL > 0 by definition. We will prove the theorem by contradiction. Suppose to the contrary that ΣL ≥ 3. (Since T n is orientable, this in fact implies ΣL ≥ 4. This fact will not be used in the proof.) The monotonicity then implies that L is unobstructed by the same argument as in [Oh96I] or in Subsection 2.4.4. It is easy to check that b = 0 lies in M(L) and so the Floer’s undeformed δ satisfies δδ = 0. Then a simple degree counting argument using the monotonicity provides the decomposition of the boundary map into

where N ≤

7

n+2 ΣL

8

δ = δ(0) + δ(1) + · · · + δ(N ) =: δ(0) + δ  and δ(k) has the form δ(k) = δ (k) ⊗ T kλ0 ekΣL .

(6.4.36)

Here δ (k) : Ek → Ek has the degree 1 − kΣL . Since ΣL ≥ 3, this degree of δ (k) is smaller than equal to −2 for k ≥ 1. Now we consider the action of δ on the E2 -term E2 = H ∗ (T n ; Q) ⊗ Λ0,nov . We recall that the cohomology ring H ∗ (T n ; Q) is generated by the one-dimensional cohomology classes α1 , · · · , αn ∈ H 1 (L; Q). By a simple degree counting, we derive δ2 (αk ) = δ (k) (αk ) · T kλ0 ekΣL = 0 for k = 1, · · · , n. From (6.4.32) we derive (6.4.37)

δ2 (m2 (αi , αj )) = m2 (δ2 (αi ), αj ) + m2 (αi , δ2 (αj )) = 0.

On the other hand, from the energy consideration, we have m2 (αi , αj ) ≡ αi ∪ αj . Therefore δ2 (αi ∪ αj ) = 0 for all i, j = 1, · · · n. Inductively applying the above arguments to all possible products of αi ’s, we derive that the spectral sequence degenerates in the E 2 -term and hence we conclude HF ∗ (L; Λ0,nov ) ∼ = H ∗ (T n ; Q) ⊗ Λ0,nov which is in particular a free module over Λ0,nov . Therefore (6.4.38)

HF ∗ (L; Λnov ) ∼ = H ∗ (T n ; Q) ⊗ Λnov = 0.

(So far we have not used the assumption that L ⊂ M is displaceable, which enters in the following last stage of the proof.) On the other hand, it follows from the invariance of HF ∗ (L; Λnov ) and the existence of a Hamiltonian diffeomorphism φ with φ(L) ∩ L = ∅ implies HF ∗ (L; Λnov ) = {0}. This contradicts to (6.4.38) and

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6.5. APPLICATIONS TO LAGRANGIAN INTERSECTIONS

385

hence the proof of ΣL ≤ 2. Now the theorem follows since T n is orientable and so the non-zero integer ΣL must be even.  We can prove the following in the same way. Theorem 6.4.35bis. Suppose that L ⊂ (M, ω) is a monotone Lagrangian submanifold such that there exists a Hamiltonian diffeomorphism φ with φ(L) ∩ L = ∅. We assume also either (1) or (2) below. (1) H ∗ (L; Z2 ) is generated by H 1 (L; Z2 ) as a ring. (2) L is orientable and relatively spin. There exists a commutative ring R such that H ∗ (L; R) is generated by H 1 (L; R) and H 2 (L; R) as a ring. Then we have its minimal Maslov number ΣL satisfies ΣL = 2. Proof. Since L is monotone we can define Floer homology of arbitrary coefficient ring R. In case (1) the proof is exactly the same as the proof of Theorem 6.4.35. In case (2) we use orientability to show that if ΣL = 2 then ΣL ≥ 4. Hence the proof of Theorem 6.4.35 works under the milder assumption that H ∗ (L; R) is generated by H 1 (L; R) and H 2 (L; R) as a ring.  Proof of Theorem 6.1.4. The proof is similar to the proof of Theorem D. In fact we can construct the spectral sequence satisfying (6.1.5.1), (6.1.5.2) by the results of Subsections 3.7.5 and 6.3.3. The proof of (6.1.5.3) is similar to the proof of (D.4) above. We leave the detail for the reader.  6.5. Applications to Lagrangian intersections In this section we give applications of Floer cohomology to the problems of Lagrangian intersections. One is the estimate of the number of the intersection points of L and ψ(L), where ψ is a Hamiltonian diffeomorphism, which is given in Subsections 6.5.1, 6.5.2, and the other is related to the Hofer distance of the Hamiltonian isotopy, which is discussed in Sections 6.5.3-5. 6.5.1. Proof of Theorem H. We prove Theorem H. Assume that i∗ : H∗ (L; Q) → H∗ (M ; Q) is injective. Then by Corollary 3.8.43, L is unobstructed after bulk deformation. Thus we can define the spectral sequence satisfying Theorem D. Moreover, from (D.3) the spectral sequence collapses at the E2 -term level. It follows that   #(L ∩ ψ(L)) ≥ rank HF ((L, (b, b)), ψ; Λnov ) = rank H k (L; Q). k

Here HF ((L, (b, b)), ψ; Λnov ) is as in (6.1.8).



6.5.2. Proof of Theorem I. In this subsection, we prove Theorem I. The proof uses the algebraic materials presented in Subsections 6.3.1, 6.3.2, but does not use the spectral sequence in Theorem D. In the next proposition we consider the Novikov ring Λ0,nov = Λ0,nov (R) for a field R. Proposition 6.5.1. Let (C, δ) be a finite d.g.c.f.z. We put C {0,1} = C 0 ⊕ C 1 . There exist elements zi (i = 1, · · · , b), hi (i = 1, · · · , a), bi (i = 1, · · · , b), of C {0,1} with the following properties.

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386

6.

(6.5.2.1) (6.5.2.2) (6.5.2.3) (6.5.2.4) (6.5.2.5)

SPECTRAL SEQUENCES

E(zi ) = E(hi ) = E(bi ) = 0. {z1 , · · · , zb , b1 , · · · , bb , h1 , · · · , ha } is a standard basis of C {0,1} . There exists λi ≥ 0 such that δzi = T λi epi bi . (Here pi = deg zi ∈ {0, 1}.) {b1 , · · · , bb , h1 , · · · , ha } is a standard basis of Ker δ ∩ C {0,1} . {T λ1 b1 , · · · , T λb bb } is a standard basis of Im δ ∩ C {0,1} .

Proof. Let b1 , · · · , bb be a standard basis of Im δ ∩C {0,1} . We put E(bi ) = λi . We define bi = T −λi bi ∈ C. We can choose zi ∈ C {0,1} such that δzi = edeg zi bi = T λi edeg zi bi . Note bi ∈ Ker δ and {σ(bi ) = σ(bi ) | i = 1, · · · , b} is linearly independent. (Here σ(bi ) is the leading coefficient of bi .) Therefore by Corollary 6.3.13 and Lemma 6.3.2bis, we can find h1 , · · · , ha such that {b1 , · · · , bb , h1 , · · · , ha } is a standard basis of Ker δ ∩ C {0,1} . Lemma 6.5.3. The set {z1 , · · · , zb , b1 , · · · , bb , h1 , · · · , ha } generates C as a Λ0,nov module. Proof. Let x ∈ C. We have ci ∈ Λ0,nov such that δx =

b 

ci bi .

i=1

Therefore we obtain

 δ x−

b 

 − deg zi

e

ci zi

= 0.

i=1

Since {b1 , · · · , bb , h1 , · · · , ha } is a standard basis of Ker δ ∩ C {0,1} , there exists di , ei ∈ Λ0,nov such that x−

b  i=1

e− deg zi ci zi =

b 

di bi +

i=1

a 

ei hi .

i=1

This proves that x is a linear combination of the elements z1 , · · · zb , b1 , · · · , bb , h1 , · · · , ha . Hence the proof.  Lemma 6.5.4. The set {σ(z1 ), · · · , σ(zb ), σ(b1 ), · · · , σ(bb ), σ(h1 ), · · · , σ(ha )} is linearly independent. Proof. Recall that our choices of bi , hj and zk imply that E(b1 ) = · · · = E(bb ) = E(z1 ) = · · · = E(zb ) = E(h1 ) = · · · = E(ha ) = 0 and {σ(b1 ), · · · , σ(bb ), σ(h1 ), · · · , σ(ha )} is linearly independent. Therefore, if the conclusion is false, we may assume without loss of any generality that σ(z1 ) can be written as a linear combination of the set {σ(z2 ), · · · , σ(zb ), σ(b1 ), · · · , σ(bb ), σ(h1 ), · · · , σ(ha )}.

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6.5. APPLICATIONS TO LAGRANGIAN INTERSECTIONS

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Lemma 6.5.3 then implies that this set generates C as an R vector space. (Here C is the R vector space such that C ⊗R Λ0,nov = C.) It follows that {z2 , · · · , zb , b1 , · · · , bb , h1 , · · · , ha } generates C as a Λ0,nov module. Therefore {e− deg z2 δz2 , · · · , e− deg zb δzb } = {b2 , · · · , bb } generates Im δ. This is a contradiction to the choice made so that {b1 , · · · , bb } is a standard basis of  Im δ ∩ C {0,1} . 

Proof of Proposition 6.5.1 is complete. Remark 6.5.5. Using the notation of Proposition 6.5.1 we have b Λ0,nov Ker δ (0) = (Λ0,nov )⊕a ⊕ . λi (0) Im δ i=1 T Λ (0)

0,nov

Namely a is the Betti number and λi ’s are the torsion exponents. Now we go back to the geometric situation and prove Theorem I. Let L be a relatively spin Lagrangian submanifold of M . Using the canonical model in the same was as Subsection 6.4.1, we have a finite d.g.c.f.z., denoted by (C, δ), that + is chain homotopy equivalent to (C(L; Λ0,nov )+ , δ(b,b) ). We take R = Q. We may + assume that δ ≡ 0 mod Λ0,nov . We apply Proposition 6.5.1 and obtain a standard basis {z1 , · · · , zb , b1 , · · · , bb , h1 , · · · , ha } of C {0,1} . Here we prove the following lemma on the leading coefficients σ(bi ). Lemma 6.5.6. We have i! (σ(bi )) = 0 for i = 1, 2, · · · , b. Proof. Let

+ ) ϕ : (C, δ) → (C(L; Λ0,nov )+ , δ(b,b)

be a cochain map preserving the filtration and inducing the isomorphisms + H(C, δ) ∼ ) = H(C(L; Λ0,nov )+ , δ(b,b)

and C = H(C, δ) ∼ = H(L; Q). We recall the map : C(L; Λ0,nov )+ → S ∗ (M ; Λ0,nov ) p+b,b 1 from Lemma 6.4.5 which satisfies + δM ◦ p+b,b = −p+b,b ◦ δ(b,b) , 1 1

where δM is the classical coboundary operator with Λ0,nov coefficient δM : S ∗ (M ; Λ0,nov ) → S ∗ (M ; Λ0,nov ). Moreover we also have (6.5.7)

(i! − p+b,b )(F λ C(L; Λ0,nov )+ ) ⊂ F λ+λ0 C(M ; Λ0,nov ) 1

from (6.4.9), where the constant λ0 is defined by λ0 = inf{β ∩ [ω] | β ∈ Π(M ; L), M(β) = ∅, β = β0 }.

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388

6.

SPECTRAL SEQUENCES

By the choice we made for zi in the beginning of the proof of Proposition 6.5.1, we have δzi ≡ T λi edeg zi σ(bi ) mod T λi Λ+ 0,nov . On the other hand (i! − p+b,b )ϕδ(zi ) ∈ F λ+λ0 C 1 by (6.5.7). Therefore we have )ϕδ(zi ) + p+b,b ϕδ(zi ) i! ϕδ(zi ) = (i! − p+b,b 1 1 + = (i! − p+b,b )ϕδ(zi ) ± p+b,b δ(b,b) ϕ(zi ) 1 1

(6.5.8)

≡ ±δM p+b,b ϕ(zi ) mod F λi +λ0 C(M ; Λ0,nov ). 1 We also have δ0 δ(zi ) ≡ 0

(6.5.9)



mod T λi +λ0 C

for some positive λ0 , because we have 0 = δδ(zi ) = (δ0 + δ  )δ(zi ) = δ0 δ(zi ) + δ  δ(zi ) 

and we know that δ  δ(zi ) lies in F λi +λ0 C. Here δ0 is the coboundary operator induced by δ : C → C and δ  is the higher order term of δ. Since ϕ induces an isomorphism H(C, δ) ∼ = H(L; Q), (6.5.8) and (6.5.9) imply that i! (σ(bi )) = ep (i! ◦ ϕ∗ )([T −λi δ(zi )]) = 0 ∈ H p (M ; Q) for some p ∈ Z. The proof of Lemma 6.5.6 is complete.



Now we are ready to complete the proof of Theorem I. Lemma 6.5.6 implies  (6.5.10) b≤ rankQ Ker(i! : H p (L; Q) → H p (M ; Q)). p

On the other hand, Proposition 6.5.1 implies 

rankQ H p (L; Q),

(6.5.11)

a + 2b =

(6.5.12)

a = rankΛnov HF ((L, (b, b)), (L, (b, b)); Λnov ).

p

Combining (6.5.10) – (6.5.12) we have obtained rankΛnov HF ((L, (b, b)), (L, (b, b)); Λnov )  ≥ rankQ Hp (L; Q) − 2 rankQ Ker(i! : H ∗ (L; Q) → H ∗ (M ; Q)). Hence the proof of Theorem I.



6.5.3. Torsion of the Floer cohomology and Hofer distance: proof of Theorem J. Proof of Theorem 6.1.25 ⇒ Theorem J. We use the notation of Theorem 6.1.25. Theorem 6.1.25 immediately implies b ≥ b(μ). On the other hand it is easy to see that #(ψ (0) (L(0) ) ∩ ψ (1) (L(1) )) ≥ a + 2b .  This fact and a = a imply Theorem J. 

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389

Example 6.5.13. Let Σ be a Riemann surface and L(0) , L(1) be two circles on it. We assume that there exists a unique disc D2 ⊂ Σ contributing a component δ[p], [q] of the coboundary operator and there is no such disc for δ[q], [p] . Let μ be the area of D2 . It is easy to see that HF ((L(1) , 0), (L(0) , 0); Λ0,nov ) ∼ =

Λ0,nov . T μ Λ0,nov

It is easy to find an Hamiltonian isotopy ψ such that ψ(L(0) ) ∩ L(1) = ∅ and the Hofer distance of ψ from identity is μ + for any > 0. Theorem J is thus optimal in this case. (See Figure 6.5.1).

M

Figure 6.5.1 Proof of Theorem 6.1.25. For the simplicity of the notation, we will restrict ourselves to the case of a transversal pair (L(0) , L(1) ) where both L(0) and L(1) are unobstructed. The general cases of the clean intersection pair of (L(0) , L(1) ) where L(i) are weakly unobstructed or weakly unobstructed after bulk deformations can be treated in the same way, except that we need to carefully choose the countably generated subcomplex of the singular chain complex of the intersection manifold L(0) ∩ L(1) . We leave this modification to the interested readers. (j) Let {ψρ }ρ (j = 0, 1) and μ be as in (6.1.24). We put L(j) = ψ (j) (L(j) ). We use the notation as Subsection 5.3.2. We consider the weakly filtered A∞ bimodule homomorphism J  ,s

(1) ϕ : (C(L(1) , L(0) ; Λnov ), nJk1t ,s , L(0) ; Λnov ), nk1t ,k0 ) ,k0 ) → (C(L

(6.5.14) (1)

(1)

top(ρ)

(0)

(0)

top(ρ)

, g0,ψ ,{Jρ }ρ ,s0 ) with energy loss μ. (See Definition over (g1,ψ ,{Jρ }ρ ,s1 5.2.1 and (5.3.15)). (Here we write C(L(1) , L(0) ; Λnov ) for C2 (L(1) , L(0) ; Λnov ) since we assumed that L(0) is transversal to L(1) .) We define a bimodule homomorphism φ(x) = ϕ(eb1 ⊗ x ⊗ eb0 ). We have (6.5.15) (j)

φ ◦ δb1 ,b0 = δψ(1) b1 ,ψ(0) b0 ◦ φ. ∗

∗

top(ρ)

j,ψ (j) ,{J (j) }ρ ,s

ρ j (bj ) ∈ M(ψ (j) (L(j) )). Since the energy loss of ϕ is Here ψ∗ bj = g∗ μ it follows that φ induces a bimodule homomorphism

(6.5.16)

φ : T λ+μ C((L(1) , L(0) ); Λ0,nov ) → T λ C((L(1) , L(0) ); Λ0,nov ).

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390

6.

SPECTRAL SEQUENCES (j)

(j) −1 We consider the isotopy {ψ (j) }−1 ) }ρ which is a Hamiltonian ρ = {ψ1−ρ ◦ (ψ (j) −1 isotopy from identity to (ψ ) . We remark that

{ψρ(j) }ρ  = {ψ (j) }−1 ρ . Here {ψρ }ρ  is the Hofer length of the family of Hamiltonian isotopy. (5.3.40).) Therefore in the same way we have the induced homomorphism:

(See

φ : T μ+λ C((L(1) , L(0) ); Λ0,nov ) → T λ C((L(1) , L(0) ); Λ0,nov )

(6.5.17) defined by

(1)

φ (x) = ϕ (eψ∗

(6.5.18)

b1

(0)

⊗ x ⊗ eψ∗

b0

).

Here ϕ is a weakly filtered A∞ bimodule homomorphism associated to {ψ (j) }−1 ρ . We have the identity δ((ψ(1) )−1 )

(6.5.19)

(1) (0) )−1 ) (ψ (0) (b )) ∗ (ψ∗ (b1 )),((ψ ∗ 0 ∗

◦ φ = φ ◦ δψ(1) b1 ,ψ(0) b0 . ∗

∗

(j) ((ψ (j) )−1 )∗ (ψ∗ (bi ))

is gauge equivalent to bi . It follows from Theorem 4.1.3, that Therefore Proposition 5.2.37 implies the existence of a homomorphism i : C((L(1) , L(0) ); Λ0,nov ) → C((L(1) , L(0) ); Λ0,nov ) such that i ◦ δ((ψ(1) )−1 )∗ (ψ(1) (b1 )),((ψ(0) )−1 )∗ (ψ(0) (b0 )) = δb1 ,b0 ◦ i

(6.5.20)

∗

∗

and i ≡ id mod Λ+ 0,nov .

(6.5.21)

If we put φ = i ◦ φ , then (6.5.19) and (6.5.20) imply φ ◦ δψ(1) b1 ,ψ(0) b0 = δb1 ,b0 ◦ φ .

(6.5.22)

∗

∗

Lemma 6.5.23. For each λ ≥ 0, the composition φ ◦ φ : T 2μ+λ C((L(1) , L(0) ); Λ0,nov ) → T λ C((L(1) , L(0) ); Λ0,nov ) is cochain homotopic to the inclusion T 2μ+λ C((L(1) , L(0) ); Λ0,nov ) → T λ C((L(1) , L(0) ); Λ0,nov ). Proof. For s ∈ [0, 1] and T > 0, we consider a smooth function χs,T with the following properties. (6.5.24.1) (6.5.24.2) (6.5.24.3)

χs,T is non decreasing on (−∞, 0] and non increasing on [0, ∞). χs,T (t) ≡ s for t ∈ [−T, T ]. χs,T (t) ≡ 0 for |t| sufficiently large. (j)

For each s, T , we define a Hamiltonian deformation {ψσ,s,T }σ by (6.5.25)

(j)

(j)

ψσ,s,T = ψχs,T (σ) .

We remark (6.5.26)

(j)

{ψσ,s,T }σ  ≤ 2μ.

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6.5. APPLICATIONS TO LAGRANGIAN INTERSECTIONS

391 (j)

For each s, T we use time ordered product with respect to σ of the family {ψσ,s,T }σ to obtain a weakly filtered A∞ bimodule homomorphism. (See Subsection 5.3.2). It induces φs,T : C((L(1) , L(0) ); Λnov ) → C((L(1) , L(0) ); Λnov ). We use Proposition 5.3.45 and (6.5.26) to find that φs,T (T μ+λ C((L(1) , L(0) ); Λ0,nov )) ⊆ T λ C((L(1) , L(0) ); Λ0,nov ).

(6.5.27)

We put s = 1. By a well established gluing argument (see the proof of Theorem 5.3.39), we find that lim φ1,T = φ ◦ φ

(6.5.28)

T →∞

(j)

on T 2μ C((L(1) , L(0) ); Λ0,nov ). We consider the family {ψσ,s,T }s,σ for T large. We remark that χ0,T (t) ≡ 0. We use the time-ordered product with respect to σ and time-wise product with respect to s to obtain a homotopy between two filtered A∞ bimodule homomorphisms ϕ1,T and ϕ0,T . (See Subsection 5.3.3.) We remark that φ0,T is an identity and φ1,T converges to ϕ ◦ ϕ. Here ϕ is as in (6.5.14) and ϕ is as in (6.5.18). It is easy to see that this implies Lemma 6.5.23, using the definitions of φ, φ , (6.5.26) and Proposition 5.3.45.  Remark 6.5.29. In fact the slight modification of the energy estimate shows that the composition φ ◦ φ induces a homomorphism φ ◦ φ : T μ+λ C((L(1) , L(0) ); Λ0,nov ) → T λ C((L(1) , L(0) ); Λ0,nov ) which is cochain homotopic to the inclusion. We will not use it however. Let λ ≥ 2μ. We have chain maps φ : (T λ C((L(1) , L(0) ); Λ0,nov ), δb1 ,b0 )

(6.5.30.1)

−→ (T λ−μ C((L(1) , L(0) ); Λ0,nov ), δψ(1) b1 ,ψ(0) b0 ) ∗

∗

and φ : (T λ−μ C((L(1) , L(0) ); Λ0,nov ), δψ(1) b ,ψ(0) b ) 1 0 ∗

(6.5.30.2)

−→ (T

λ−2μ

C((L

(1)

,L

(0)

∗

); Λ0,nov ), δb1 ,b0 )



such that φ ◦ φ is chain homotopy equivalent to the inclusion φ : (T λ C((L(1) , L(0) ); Λ0,nov ), δb1 ,b0 ) −→ (T λ−2μ C((L(1) , L(0) ); Λ0,nov ), δb1 ,b0 ). Here we use the following: (0)

Lemma 6.5.31. Let D be a finitely generated Λ0,nov module that is isomorphic to (0) D∼ = (Λ0,nov )A ⊕

B

Λ0,nov

i=1

T ci Λ0,nov

(0)

(0)

.

(1) For each λ > 0 the minimum number of generators of T λ D as Λ0,nov module is A + #{i | ci > λ}. (2) Any finitely generated submodule D of T λ D is generated by the less than A + #{i | ci > λ} elements.

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+(0) Proof. (1) follows from T λ D/T λ Λ0,nov D ∼ = QA+#{i|ci >λ} . To prove (2), we first remark that it suffices to consider the case when λ = 0, ˜ = (Λ(0) )A+B and since we can apply the same argument to T λ D. We put D 0,nov ˜ → D. We lift the generators of D to D ˜ and consider the natural projection π : D ˜  be the submodule generated by them. Then D ˜  is finitely generated and let D ˜  ) = D . Using standard basis (Lemma 6.3.2) we find that D ˜  is generated by π(D  A + B elements. It follows that D is generated by A + B elements, as required.

(6.5.30) induces the homomorphisms φ∗ : T λ HF ((L(1) , L(0) ); Λ0,nov ) → T λ−μ HF ((L(1) , L(0) ); Λ0,nov ) and φ∗ : T λ−μ HF ((L(1) , L(0) ); Λ0,nov ) → T λ−2μ HF ((L(1) , L(0) ); Λ0,nov ) whose composition φ∗ ◦ φ∗ is the same as the natural inclusion. Since the Λ0,nov module T λ−μ HF ((L(1) , L(0) ); Λ0,nov ) is generated by a + b (λ − μ) elements, it follows from Lemma 6.5.31 (2) that the image of T λ HF ((L(1) , L(0) ); Λ0,nov ) in T λ−2μ HF ((L(1) , L(0) ); Λ0,nov ) is generated by a + b (λ − μ) elements. Therefore using Lemma 6.5.31 (1), we find a + b(λ) ≤ a + b (λ − μ).

(6.5.32)

Now we prove Theorem 6.1.25 under the additional assumption λ↓i > 3μ. (Note in Theorem 6.1.25, we enumerate the torsion exponents so that λ↓i ≥ λ↓i+1 .) We remark that “λ↓i ≥ λ and i ≤ b” is equivalent to b(λ) ≥ i. We use (6.5.32) for λ = λ↓i to find that b ≥ b (λ↓i − μ) ≥ b (λ↓i ) ≥ i and λ↓i ≥ λ↓i −μ. This proves λ↓i ≥ λ↓i −μ > 3μ−μ = 2μ. We obtain λ↓i ≥ λ↓i −μ by a similar argument with L(j) and L(j) exchanged, using λ↓i > 2μ. That proves Theorem 6.1.25 in case λ↓i > 3μ. To prove the general case we partition the Hamiltonian isotopy into small pieces so that we can apply the above arguments to each of the pieces. The detail of this argument is in order. We use the same notation as Theorem 6.1.25. We take a partition ρ0 = 0 < ρ1 < · · · < ρN = 1 with the following properties (6.5.33). (6.5.33)

λ↓i −

j 

μk > 3μj+1 ,

j = 0, 1, 2, · · · , N − 1.

k=1

Here μi = dist(ψρ(1) , ψρ(1) ) + dist(ψρ(0) , ψρ(0) ) i−1 i i−1 i  where dist stands for Hofer distance. Since j μj = μ < λ↓i , we can find such ρi . In fact, we may take     j j   1 λ↓i − μk , μ − μk , μj+1 = min 4 k=1

(0) ψρ ,

k=1

(1) ψρ

slightly, we may assume that inductively. By perturbing the isotopies (0) (1) (0) (1) ψρj (L ) is transversal to ψρj (L ) for each j = 0, · · · , N .

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6.5. APPLICATIONS TO LAGRANGIAN INTERSECTIONS (0)

393

(1)

Let λ↓i (j) be the i-th torsion exponent of HF (ψρj (L(0) ), ψρj (L(1) ); Λ0,nov ). (We enumerate them so that λ↓i (j) ≥ λ↓i+1 (j).) We will then prove the following by induction on j λ↓i (j) ≥ λ↓i −

(6.5.34)

j 

μk .

k=1

In fact, if we assume (6.5.34) for j then (6.5.33) implies that λ↓i (j) ≥ 3μj+1 . Therefore we can apply Theorem 6.1.25 (since it is already established under this additional assumption) to show (6.5.34) for j + 1. Since λ↓i (N ) = λ↓i , it follows that  λ↓i ≥ λ↓i − μk = λ↓i − μ. The proof of the inequality in the opposite direction is similar. The proof of Theorem 6.1.25 is now complete.  Partitioning Hamiltonian isotopies to control the energy loss was also used in [Ono95] to show that the modified Floer homology is well-defined, i.e., independent of the choice of filtrations. 6.5.4. Floer cohomologies of Lagrangian submanifolds that do not intersect cleanly. We next apply Theorem 6.1.25 to define Floer cohomology over Λ0,nov in the case when L(0) and L(1) are not necessarily of clean intersection. Assume that a pair (L(0) , L(1) ) is relatively spin and weakly unobstructed after bulk deformation. Let bj ∈ Mweak,def (L(j) ) for j = 0, 1 such that πamb (b0 ) = πamb (b1 ) and PO(b0 ) = PO(b1 ). We then can define the Floer cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λnov ) over the Λnov coefficients. This is because the Floer cohomology over Λnov is invariant of Hamiltonian isotopy and we can always perturb L(i) so that they intersect transversely and the resulting cohomology is canonically isomorphic. We would like to define the Floer cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) over the Λ0,nov -coefficients, using Theorem 6.1.25. (0) (1) We consider two sequences of Hamiltonian diffeomorphisms ψm , ψm , (m = 1, 2, 3, · · · ) such that (6.5.35.1) (6.5.35.2)

(0)

(1)

ψm (L(0) ) has clean intersection with ψm (L(1) ). (i) limm→∞ dist(ψm , id) = 0 where dist is the Hofer distance.

Proposition 6.5.38 below says that the Floer cohomology (1)

(0)

(1) (0) HF ((ψm (L(1) ), ψm∗ b1 ), (ψm (L(0) ), ψm∗ b0 ); Λ0,nov )

converges as m → ∞ in the sense we define below. Definition 6.5.36. Let λm,↓1 , · · · , λm,↓bm (m = 1, 2, · · · , ∞) be sequences of positive numbers such that λm,↓i ≥ λm,↓i+1 . We assume bm < ∞ for m = ∞. (But b∞ itself could be infinity.) If b∞ = ∞ we assume that limi→∞ λ∞,↓i = 0 in addition. We say lim (λm,↓1 , · · · , λm,↓bm ) = (λ∞,↓1 , · · · , λ∞,↓b∞ ),

m→∞

if lim bm = b∞ and limm→∞ λm,↓i = λ∞,↓i for each i.

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Remark 6.5.37. In the situation of Proposition 6.5.38 we may have infinitely many λ∞,↓i such that limi→∞ λ∞,↓i = 0. For this case we need to enumerate λ∞,↓i so that λ∞,↓i ≥ λ∞,↓i+1 . (0)

(1)

Proposition 6.5.38. Let ψm ,ψm be two sequences of Hamiltonian diffeomorphisms satisfying (6.5.35). We put (1)

(0)

(1) (0) HF ((ψm (L(1) ), ψm∗ b1 ), (ψm (L(0) ), ψm∗ b0 ); Λ0,nov )

∼ = (Λ0,nov )a ⊕

bm i=1

Λ0,nov . T λm,↓i Λ0,nov

Then there exists λ∞,↓1 , · · · , λ∞,↓b∞ as in Definition 6.5.36, such that lim (λm,↓1 , · · · , λm,↓bm ) = (λ∞,↓1 , · · · , λ∞,↓b∞ )

m→∞

in the sense of Definition 6.5.36. Proposition 6.5.38 is an immediate consequence of Theorem 6.1.25. Definition 6.5.39. We put HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) = (Λ0,nov )a ⊕

b∞

Λ0,nov . λ ∞,↓i T Λ0,nov i=1

We remark that limm→∞ (λm,↓1 , · · · , λm,↓bm ) converges without taking subsequence. Hence the right hand side of Definition 6.5.39 is independent of the choice (j) of ψm . We remark that we proved Theorem G in Chapter 5 in case when L(1) has clean intersection with L(0) . The general case is a consequence of Proposition 6.5.38 and Definition 6.5.39. The fact that the Floer cohomology in Definition 6.5.39 satisfies (G.3),(G.4) follows from the definition. We can check (G.5) using the proof of Theorem 6.1.25. We omit the detail of the check of (G.5), since we do not use it. Example 6.5.40. Let X = {x +

√

−1e−1/|x| sin 1/|x| | |x| ≤ 1} ⊂ C,

Y = [−1, 1] × {0} ⊂ C.

We can find Lagrangian submanifolds L(1) , L(0) ⊂ T 2 , p ∈ L(1) ∩ L(0) , a neighborhood U of p and a symplectic diffeomorphism F : U → F (U ) ⊂ C onto its image such that F (p) = 0, and F (L(0) ∩ U ) = X, F (L(1) ∩ U ) = Y . (See Figure 6.5.2). It is easy to see that HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) is not finitely generated. (In other words b∞ = ∞.) We remark that Floer cohomology HF ((L(1) , b1 ), (L(0) , b0 ); Λ0,nov ) is finitely generated if L(1) has clean intersection with L(0) . We can find a similar example in higher dimension, for example, by taking a direct product of the above example with (T 2n , T n ).

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6.5. APPLICATIONS TO LAGRANGIAN INTERSECTIONS

395

Figure 6.5.2 We remark that Theorems J and 6.1.25 both hold in the case when L(1) does not have clean intersection with L(0) using Definition 6.5.39. Combined with Example 6.5.40 it implies the following. Corollary 6.5.41. In each dimension, there exists a pair L(1) , L(0) of Lagrangian submanifolds and N ( ) such that lim →0 N ( ) = ∞, and if ψ is a Hamiltonian isotopy whose Hofer distance from the identity is smaller than then #(ψ(L(0) ) ∩ L(1) ) ≥ N ( ). 6.5.5. Unobstructedness modulo T E . In Theorem J, we assumed that L , L(1) are weakly unobstructed after bulk deformations. However, it is easy to see that a similar result holds under a slightly milder assumption. (0)

Definition 6.5.42. A relatively spin Lagrangian submanifold L ⊂ M is said to be unobstructed modulo T E , if there exists a cochain b ∈ C 1 (L; Λ+ 0,nov ) such that (6.5.43.1)

ˆ b ) ≡ 0 mod T E . d(e

L is said to be weakly unobstructed modulo T E , if there exists a cochain b ∈ + C 1 (L; Λ+ 0,nov ) such that (6.5.43.2)

ˆ b ) ≡ 0 mod (T E , e+ ). d(e

L is said to be unobstructed modulo T E after bulk deformation, if there exist + 2 b ∈ C 1 (L; Λ+ 0,nov ) and b ∈ H (L; Λ0,nov ) such that (6.5.43.3)

q(eb , eb ) ≡ 0 mod T E .

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6.

SPECTRAL SEQUENCES

L is said to be weakly unobstructed modulo T E after bulk deformation, if there + + 2 exist b ∈ C 1 (L; Λ+ 0,nov ) and b ∈ H (L; Λ0,nov ) such that (6.5.43.4)

q(eb , eb ) ≡ 0 mod (T E , e+ ).

When (6.5.43.2) or (6.5.43.4) is satisfied, we define the potential function PO(b, b) ∈ Λ0,nov /T E Λ0,nov by q(eb , eb ) − PO(b, b)e+ ≡ 0 mod T E . We remark that for any L, there always exists some positive constant E such that L is unobstructed modulo T E . The proof of the following lemma is a straightforward modification of the proof of Proposition 3.7.17. Lemma-Definition 6.5.44. We assume that L(0) has clean intersection with + 2 L . If bi ∈ C 1 (L(i) ; Λ+ 0,nov ) and b ∈ H (M ; Λ0,nov ) satisfy (6.5.43.4), then we define (b,b ),(b,b0 ) m1 1 : C ∗ (L(1) , L(0) ; Λ0,nov ) → C ∗+1 (L(1) , L(0) ; Λ0,nov ) by δ(b,b1 ),(b,b0 ) (x) = r(eb , eb1 xeb0 ). If PO(b, b0 ) = PO(b, b1 ), we have (1)

δ(b,b1 ),(b,b0 ) ◦ δ(b,b1 ),(b,b0 ) ≡ 0 mod T E . Definition 6.5.45. Under the same condition as Lemma-Definition 6.5.44 we define a Λ0,nov /T E Λ0,nov module HF ((L(1) , (b, b1 )), (L(0) , (b, b0 )); Λ0,nov /T E Λ0,nov ) by Ker δ(b,b1 ),(b,b0 ) / Im δ(b,b1 ),(b,b0 ) . In a way similar to the proof of Proposition 6.3.14 we can prove that Λ0,nov (6.5.46) HF ((L(1) , (b, b1 )), (L(0) , (b, b0 )); Λ0,nov /T E Λ0,nov ) ∼ = T λi Λ0,nov i for some 0 < λi ≤ E. In this situation, Theorem J can be generalized as follows. Theorem 6.5.47. Under the assumption of Definition 6.5.45 we define λi by (6.5.46). We put b(μ) = #{i | E > λi ≥ μ},

a(i) = #{i | E = λi }.

If μ is the Hofer distance of an Hamiltonian diffeomorphism ψ : M → M from the identity, then we have #(ψ(L(0) ) ∩ L(1) ) ≥ a(μ) + 2b(μ). The proof is a minor modification of the proof of Theorem J and is omitted. We can also generalize Theorem 6.1.25 in the same way. We remark that Theorem 6.5.47 slightly generalizes Chekanov’s theorem in [Chek96,98].

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https://doi.org/10.1090/amsip/046.2/01

CHAPTER 7

Transversality In this chapter, we provide the details of constructions used in the various parts of the book. A section-wise outline of contents is in order. In Section 7.1, we give the analytic details in the study of various moduli spaces of pseudo-holomorphic discs. More specifically we equip them with Kuranishi structures. For our purpose, we need to work in the chain level and to construct virtual fundamental chains on them. It is also crucial to consider the moduli spaces Mmain k+1 (β) simultaneously for various k and β. This is because the algebraic structure we obtain is well-defined only as a whole. The moduli spaces Mmain k+1 (β) for various k and β are inter-related to one another in such a way that the boundary of one of them is decomposed to a union of the fiber product of the others. We construct our Kuranishi structures on Mmain k+1 (β) so that they are compatible with this decomposition whose precise meaning is stated in Subsection 7.1.1. Analytic details of the construction of Kuranishi structure are then given in Subsections 7.1.2 -7.1.4 for the moduli space Mmain k+1 (β). In Subsection 7.1.2 we construct a Kuranishi neighborhood of each point in the interior of the moduli space. The main task to be done for this purpose is to make a proper Fredholm setting to apply the implicit function theorem. In Subsection 7.1.3, we construct a Kuranishi neighborhood of each point in the boundary of our moduli space. The main ingredient needed here is a Taubes’ type gluing argument in a setting of the Bott-Mores type. In Subsection 7.1.4 we combine the results of these two subsections and complete the construction of Kuranishi structure on Mmain k+1 (β). In Subsection 7.1.5 we consider the cases of various other moduli spaces involved in the construction of the earlier sections. One novel geometric point in this regard is Theorem 7.1.51 which gives a geometric construction of the space appearing in the definition of A∞ maps. The heart of this chapter is Section 7.2, where we discuss the transversality problem. We emphasize that after the year 1996 when the virtual fundamental chain technique was introduced, the transversality issue mainly hinges on the problem of finite dimensional topology rather than one on infinite dimensional analysis. Therefore in this book we treat the transversality problem in Section 7.2 in the way that is separated both from the Fredholm theory and the gluing theorem discussed in Section 7.1. There are two new difficulties arising in dealing with the transversality required in the current circumstances, for which we need to develop some new techniques beyond those used in [FuOn99II]. One difficulty in our transversality problem arises from the fact that the situation we deal with is of the ‘Bott-Morse’ type. Namely, various moduli spaces involved are inter-related to one another via the fiber product. For the ‘Morse’ case, in which various moduli spaces are related via the direct product, the method of virtual fundamental chains laid out e.g., in Theorem 3.11, Lemma 3.14 [FuOn99II] 397

398

7.

TRANSVERSALITY

immediately implies the required transversality at each finite stage. We will explain this point more in Subsections 7.2.1-7.2.2. Another difficulty arises from the fact that we need to handle infinitely many moduli spaces simultaneously. This point is explained in Subsection 7.2.3. After the explanation on these two main points to take care of, details of the solutions to the transversality problem are presented in Subsections 7.2.4-7.2.14. It turns out that we need to heavily use the language of A∞ algebra to resolve the two difficulties. Since the homological algebra and the schemes of various proofs employed in these subsections do not use the special feature of the current problem of ours the same method can be used to deal with the transversality problem in various other similar situations. In Section 7.3 we give the details of our construction of the homotopy unit. In Section 7.4 we provide the details of the construction of the operators p, q, r introduced in Section 3.8. There are two main issues to discuss. One is to clarify the relation of Gromov-Witten invariants to the operator p. This is related to the fact that stable map ‘compactification’ is not actually compact for the moduli space of pseudo-holomorphic discs without boundary marked points. This phenomenon is studied in Subsection 7.4.1. Another main issue handled in Section 7.4 is a homological algebra that we use in the description of the operators q, p and r. In Section 3.8 we gave formulas that describe the main properties of these operators. In Section 7.4, we show that those formulas we gave in Section 3.8 can be interpreted in terms of L∞ structures. Such interpretations not only clarify the meaning of those formulas but also useful for the purpose of applying the arguments used in Section 7.2 also to the construction of q, p, r. In order to apply the homological techniques of Section 7.2 for the construction of these operators, we need to interpret them systematically using the language of homological algebra which are provided in Subsections 7.4.2-7.4.11. The main purpose of Section 7.5 is to prove Theorem V, which states our filtered A∞ algebra is homotopy equivalent to the de Rham complex after reducing the coefficient to R. The main new idea to prove this equivalence is the usage of continuous family of perturbations. 7.1. Construction of the Kuranishi structure 7.1.1. Statement of the results in Section 7.1. In this section, we construct the Kuranishi structure of the moduli spaces we used in earlier chapters. (See Section A1 for the definition of Kuranishi structures.) Constructions of the Kuranishi structures of various moduli spaces are almost the same. So in Subsections 7.1.1-4 we focus on the proof of Proposition 3.4.2. We will mention the other cases in Subsection 7.1.5. In fact, we will prove Propositions 7.1.1 and 7.1.2 which indeed imply the precise statement of Proposition 3.4.2. See Section A1 for the definition of the notion used in Propositions 7.1.1 and 7.1.2. Proposition 7.1.1. Mmain k+1 (β) has a topology with respect to which it is compact and Hausdorff. Furthermore, it has an oriented Kuranishi structure with boundary and corners of dimension μL (β) + k − 2 + n with respect to which the k+1 evaluation map ev : Mmain is a smooth strongly continuous map and is k+1 (β) → L weakly submersive. In Proposition 7.1.1, we need to construct a Kuranishi structure so that it satisfies certain compatibility conditions. We precisely describe those compatibility

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

399

conditions as Proposition 7.1.2 below. To state the proposition, we use the notion of fiber product of Kuranishi structures which is discussed in Section A1. We compare the induced Kuranishi structure on Sm Mmain k+1 (β) with the one obtained as the fiber   (β ) for ω(β ) < ω(β) or ω(β  ) = ω(β) and k < k. product of Mmain  k +1 main We denote by Sm Mmain k+1 (β) the codimension m corner of Mk+1 (β). According −1 to the notation of (A1.1) it is the union of ψp (Sm (Vp ) ∩ sp (0)) where Sm (Vp ) is the codimension m corner of Vp . Let ((Σ, z), w) ∈ Sm Mmain k+1 (β). Note that an irreducible component of Σ can be either a disc or a sphere, and the union of disc components is connected still of genus zero and with k + 1 marked points on the boundary. We decompose this latter curve into irreducible disc components. After that we attach a tree of sphere components of Σ to a unique irreducible disc component. This gives a decomposition Σ = Σ1 ∪ · · · ∪ Σm ∪ Σ0 where each Σi intersects the boundary of Σ. (We can show by construction that codimension m corner consists of ((Σ, z), w) such that Σ has m + 1 disc components.) Let β(i) = w∗ ([Σi ]) ∈ Π(M, L) and Σi contain bi +1 singular or marked points on its boundary. (See Definition 2.4.17 for Π(M, L).) (β(i)). We Then, the restriction wi of w to Σi determines an element of Mmain,reg bi +1 assume z0 ∈ Σ0 and order Σi ’s in the same way as Definition 4.6.6. For each i = 0, there exists unique j(i) such that Σj(i) > Σi and Σj(i) ∩ Σi = ∅ (recall that Σ has genus 0). Then Σj(i) ∩ Σi is a marked point of (Σi , wi ) regarded as an element of Mmain,reg (β(i)). We order marked points so that this marked point becomes the bi +1 0-th one on Σi . Let (z0i , · · · , zbii ) be the marked point on Σi . We associate an element (T, i, v0 , Vtad , η) of G+ z ), w) as follows (See k to ((Σ,  .): T is a tree. Its interior vertex corresponds one to one to Subsection 5.4.4 for G+ k the disc component of Σ. Its exterior vertex corresponds one to one to a marked point zi . We join two interior vertex if and only if the corresponding disc components intersect. We join an exterior vertex to an interior vertex if and only if the corresponding marked point is on the corresponding disc component. For each interior vertex v corresponding to Σi we define η(v) = η(i) if β(i) = w∗ [Σi ] is the η(i)-th element βη(i) of Π(L). (Note Σi includes the sphere component rooted on this disc component.) The embedding i : T → D2 is determined by the obvious ribbon structure of T . The first (exterior) vertex v0 is the exterior vertex corresponding to the 0-th marked point. The interior vertex corresponding to Σi is in Vtad (the set of tad-poles) if and only if Σi has one (boundary) singular point and has no marked point. + We denote by G+ k,m (β) the set of all elements of Gk which has exactly m interior  vertex and βη(v) = β. We define the map  2m ev + = (ev1+ , ev2+ ) : Mmain × Lk+1 bi +1 (βη(i) ) → L as follows: Take (i) so that Σj(i) ∩ Σi is the (i)-th marked point of Σj(i) . Let bi +1 ev i : Mmain bi +1 (βη(i) ) → L

be the evaluation map. Note that Σ has m singular points on the boundary. The first factor of ev + (w0 , w1 , · · · , wm ) is defined by   j(1) j(m) ev1+ (w0 , w1 , · · · , wm ) = w1 (z01 ), wj(1) (z(1) ), · · · , wm (z0m ), wj(m) (z(m) ) .

400

7.

TRANSVERSALITY j(i)

Here zij is the i-marked point of Σj . (Note z0i and z(i) will become the same point of Σ.)

Σi

Σj(i) Σ0

j(i) z0i = z(i)

z0 Figure 7.1.1

The second factor is the usual evaluation map. Namely if the i-th marked point of (Σ, z) corresponds to the b(i)-th marked point of Σa(i) , it is given by a(1)

a(k)

ev2+ (w0 , w1 , · · · , wm ) = (w0 (z00 ), wa(1) (zb(1) ), · · · , wa(k) (zb(k) )). (Note a(0) = b(0) = 0.) Note that ev + is induced from ev i and hence is a smooth strongly continuous map by Proposition 7.1.1. Proposition 7.1.1 also implies that ev1+ is weakly submersive. The map ev is determined by the associated combina+ + + torial data Γ = (T, i, v0 , Vtad , η) ∈ G+ k . So we denote evΓ = (evΓ,1 , evΓ,2 ). m 2m Let L → L be the map (x1 , · · · , xm ) → (x1 , x1 , · · · , xm , xm ). We consider the fiber product  m Mmain bi +1 (βη(i) )ev + ×L2m L . Γ,1

We have a continuous and surjective map   m main Mmain bi +1 (βη(i) )ev + ×L2m L → S m Mbi +1 (β) Γ∈G+ k,m (β)

Γ,1

main to the closure S m Mmain bi +1 (β) of codimension m corner Sm Mbi +1 (β). It is finite m to one and is injective on each of the Mmain (Γ ∈ G+ bi +1 (βη(i) )ev + ×L2m L k,m (β)). Γ,1

main m + ×L2m L Both S m Mmain have Kuranishi structure. (The k+1 (β) and Mbi +1 (βη(i) )evΓ,1 fiber product as Kuranishi structure is well defined in Definition A1.37, since the evaluation map is assumed to be weakly submersive.)

Proposition 7.1.2. The Kuranishi structure on the fiber product  m Mmain bi +1 (βη(i) )ev + ×L2m L Γ,1

coincides with the pull-back of the induced Kuranishi structure on S m Mmain k+1 (β). The proofs of Propositions 7.1.1 and 7.1.2 given in this section are a combination of the techniques developed in [FOh97, FuOn99II, Fuk96II].

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401

7.1.2. Kuranishi charts on Mmain,reg (β): Fredholm theory. In Subseck+1 tions 7.1.2 and 7.1.3 we will carry out the analytic detail of the proof of Proposition 7.1.1. For this purpose we need to fix a metric on elements Σ of Mmain,reg . We use k+1 main,reg the map Θ : Grk → Mk introduced in Section 3.4. We there fixed a smooth family of Riemannian metrics of elements of Mmain,reg . k Now we fix an almost complex structure compatible with ω and denote the corresponding Cauchy-Riemann operator by ∂. Let L ⊂ (M, ω) be a Lagrangian submanifold. Let ((Σ, z), w) be a pair such that (Σ, z) ∈ Mmain,reg and w : (Σ, ∂Σ) → (M, L) k be a smooth map. We denote by β ∈ Π(M, L) the equivalence class of the homotopy class of w. We identify Σ\{z1 , · · · , zk } ∼ = X(t, ) as in Section 3.4 and regard w as a map from (X(t, ), ∂X(t, )). On the other hand, if we are given a map w : (X(t, ), ∂X(t, )) → (M, L), we need to impose some condition on the asymptotic behavior of w to compactify it to a map : (Σ, ∂Σ) → (M, L). This leads us to considering the following Banach manifold. We take and fix δ > 0 sufficiently small and choose p > 0 sufficiently large. Definition 7.1.3. Wk1,p (M, L) is the set of all ((t, ), w, p) with the following properties: (7.1.4.1) (t, ) ∈ Grk . 1,p , i.e., locally of W 1,p (7.1.4.2) w : (X(t, ), ∂X(t, )) → (M, L) is a map of Wloc class. (7.1.4.3) p = (p1 , · · · , pk ), pi ∈ L. 1 (7.1.4.4) Let e ∈ Cext (t). We have Le (−∞, 0] × [0, 1] ⊂ X(t, ). We use τ for the parameter of (−∞, 0] and t for [0, 1]. Then we have  δ|τ | p e |∇w| dτ dt + eδ|τ | dist(w(τ, t), pi )p dτ dt < ∞. Le

Le

Here dist is the Riemannian distance. Here and hereafter, | · | : R → [0, ∞) is a smooth function such that |τ | = |τ | for |τ | ≥ 1 and |τ | = 0 for |τ | ≤ 1/2. In the next lemma and thereafter we take a unitary connection of T M such that T L is preserved by the parallel transport and use it to define the isomorphism Pal : Tpi M → Tw(τ,t) M in (7.1.6.5) and etc.. Lemma 7.1.5. The space Wk1,p (M, L) is a Banach manifold. Its tangent space T((t,),w, p) Wk1,p (M, L) at ((t, ), w, p) is identified with the set of (ξ, V, v) such that (7.1.6.1) (7.1.6.2) (7.1.6.3) (7.1.6.4) (7.1.6.5)

ξ ∈ T(t,) Grk . 1,p V is a section of w∗ (T M ) in Wloc . v = (v1 , · · · , vk ), vi ∈ Tpi L. V (z) ∈ Tw(z) L if z ∈ ∂X(t, ). 1 (t). We have Le (−∞, 0] × [0, 1] ⊂ X(t, ). We then have Let e ∈ Cext  eδ|τ | (|∇(V − Pal vi )|p + |V − Pal vi |p )dτ dt < ∞. Le

402

7.

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Here Pal : Tpi M → Tw(τ,t) M is the parallel transport. The norm on Wk1,p (M, L) is defined by

p p p (ξ, V, v)1,p =ξ + |v | + (|∇V |p + |V |p )dτ dt +

1 (t) ei ∈Cext



1 (t) e∈Cint

Le



eδ|τ | (|∇(V − Pal vi )|p + |V − Pal vi |p )dτ dt.

Lei

The proof is standard and omitted. We remark that v above is determined uniquely from V . We next define a Banach bundle E 0,p on Wk1,p (M, L). Let ((t, ), w, p) be an element of Wk0,p (M, L). We consider the section V of Λ0,1 ⊗ w∗ (T M ) such that (7.1.7.1) (7.1.7.2)

0,p V is in Wloc . 1 Let e ∈ Cext (t). We have Le (−∞, 0] × [0, 1] ⊂ X(t, ). Then  eδ|τ | |V |p dτ dt < ∞. Le 0,p E((t,),w, p)

We define to be the totality of such V ’s satisfying (7.1.7.1) - (7.1.7.2). It is a Banach space with its norm defined by



 |V |p dτ dt + eδ|τ | |V |p dτ dt. (7.1.7.3) V p0,p = 1 (t) e∈Cint

Le

With respect to the obvious topology on  E 0,p :=

1 (t) ei ∈Cext

Lei

0,p E((t,),w, p) ,

((t,),w, p)

it follows that E 0,p defines a Banach bundle over Wk1,p (M, L) (In fact, this bundle can be extended over Wk0,p (M, L).) The following is easy to show. 0,p Lemma 7.1.8. For each ((t, ), w, p) ∈ Wk1,p (M, L), we have ∂w ∈ E((t,),w, p) .

Furthermore the assignment ((t, ), w, p ) → ∂w defines a smooth section of E 0,p . Using the parallel transport along the minimal geodesics, we define a local trivialization of E 0,p . (We choose our metric so that L is totally geodesic.) Here and hereafter we take a unitary connection of T M such that T L is preserved by the parallel transport and use it to define the above isomorphism. With this trivialization, we define the covariant derivative of ∂: (7.1.9)

0,p D((t,),w, p) ∂ : T((t,),w, p) Wk1,p (M, L) → E((t,),w, p) .

Lemma 7.1.10. If δ > 0 is sufficiently small, then D((t,),w, p) ∂ becomes a Fredholm operator with Fredholm index μL (β) + k − 3 + n. Here β ∈ π2 (M, L) is the homotopy class of w and μL (β) is its Maslov index. 

The proof is by now standard and omitted. Note that we need the weight eδ|τ | to make D((t,),w, p) ∂ a Fredholm operator, because our operator is degenerate at the ends.

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

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We also have the following which is a consequence of the exponential decay estimates. (See Lemma 7.1.37 stated later.) Lemma 7.1.11. Let ((Σ, z), w) be a pair such that (Σ, z) ∈ Mmain,reg and w : k (Σ, ∂Σ) → (M, L) be locally of L1,p -class. We assume that ∂w has compact support. We identify Σ\{z1 , · · · , zk } with X(t, ) and put pi = w(zi ). Then ((t, ), w, p ) ∈ Wk1,p (M, L) if δ is sufficiently small. Now we let ((t, ), w, p ) ∈ Wk1,p (M, L) such that ∂w = 0. It follows from Lemma (β) for some β. 29.11 that this element corresponds to an element of Mmain k Proposition 7.1.12. There exists a Kuranishi neighborhood of ((t, ), w, p) in the space Mmain (β). k Proof. We consider the operator (7.1.9). By Lemma 7.1.10, we can find a 0,p finite dimensional subspace E((t,),w, p) of E((t,),w, p) such that the map (7.1.13)

0,p π ◦ D((t,),w, p) ∂ : T((t,),w, p) Wk1,p (M, L) −→ E((t,),w, p) p) /E((t,),w,

is surjective, where π is the obvious projection. In view of the unique continuation theorem [Aro57], we may choose E((t,),w, p) so that each V ∈ E((t,),w, p) has the following properties: (7.1.14.1) (7.1.14.2)

V is smooth. V has compact support.

We next take a small neighborhood U of (t, ) in Grk . For each (t ,  ) ∈ U we can find a diffeomorphism I(t,),(t , ) : X(t, ) −→ X(t ,  ) with the following properties: (7.1.15.1) I(t,),(t , ) depends smoothly on (t ,  ). (We use the universal family of open Riemann surfaces over U to define smoothness of I(t,),(t , ) with respect to (t ,  ).) (7.1.15.2) I(t,),(t , ) is the identity outside a compact subset. Here we remark that X(t, ) minus a compact set is identified with the union of (−∞, −T ] × [0, 1] ⊂ 1 Le over e ∈ Cext (t). Now let ((t ,  ), w , p  ) ∈ Wk1,p (M, L) be in a small neighborhood of ((t, ), w, p ). We can find a canonical isomorphism 0,p 0,p E((t,),w, p) E((t , ),w , p )

by the parallel transport along the minimal geodesics. (We remark that we take a unitary connection of T M such that T L is preserved by the parallel transport and use it to define the above isomorphism. We do not repeat this remark any more.) Now we consider the equation (7.1.16)

∂w ≡ 0 mod E((t,),w, p) .

0,p Here the right hand side is identified with a subspace of E((t,),w, p) by the above isomorphism.

404

7.

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Lemma 7.1.17. Let U ((t, ), w, p) be the set of all elements ((t ,  ), w , p  ) satisfying (7.1.16). Then U ((t, ), w, p ) is a smooth manifold of dimension μL (β) + k − 3 + n + dim E((t,),w, p) . The proof is immediate from Lemma 7.1.10 and the implicit function theorem. We now define a map s((t,),w, p) : U ((t, ), w, p) −→ E((t,),w, p) by

s((t,),w, p) ((t ,  ), w , p  )) = ∂w . The right hand side is an element of E((t,),w, p) by the definition of U ((t, ), w, p). We recall that the automorphism group of (Σ, z) ∈ Mmain,reg (β) is torsion free if k k > 0. So the automorphism of ((t, ), w, p ) is trivial. Hence we can simply take Γ((t,),w, p) = {1}. (We need nontrivial Γ’s only in case when there is a sphere bubble in the interior.) This finishes the proof of Proposition 7.1.12.  However we need to put one more condition on our Kuranishi neighborhood in relation to the evaluation map ev : U ((t, ), w, p ) −→ Lk defined by

ev((t ,  ), w , (p1 , · · · , pk )) = (p1 , · · · , pk ).

Lemma 7.1.18. We can take our Kuranishi neighborhood in Proposition 7.1.12 so that ev becomes a submersion. Proof. We consider the subspace of T((t,),w, p) Wk1,p (M, L) consisting of elements (ξ, V, v) with v = 0. We denote it by (d ev)−1 (0). We consider the restriction 0,p D((t,),w, p) ∂|(d ev)−1 (0) : (d ev)−1 (0) → E((t,),w, p)

of the operator (7.1.9). We remark that (d ev)−1 (0) is of finite codimension in T((t,),w, p) Wk1,p (M, L). Hence D((t,),w, p) ∂|(d ev)−1 (0) is also a Fredholm operator. We can then choose E((t,),w, p) so that 0,p π ◦ D((t,),w, p) ∂ : (d ev)−1 (0) → E((t,),w, p) p) /E((t,),w,

is surjective. It is straightforward to see that the conclusion of Lemma 7.1.18 holds for this choice of E((t,),w, p) .  (β): gluing. In 7.1.3. Kuranishi charts in the complement of Mmain,reg k+1 the previous subsection, we have constructed a Kuranishi neighborhood of each point in Mmain,reg (β). We next construct Kuranishi neighborhoods of points in the k+1 main,reg main set Mk+1 (β)\Mk+1 (β). We need to perform Taubes’ type gluing construction for this purpose. We first define a gluing map of graphs Compk1 ···km ;m ;j1 ,··· ,jm : Grk1 +1 × · · · Grkm +1 × Grm+m +1 → Grk1 +···km +m +1 where 0 < j1 < j2 < · · · < jm < m + m + 1. Let ti ∈ Gki +1 , i = 1, · · · , m and t0 ∈ Gm+m +1 . We identify the 0-th exterior edge of ti with the ji -th edge of t0

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

405

and obtain an element t ∈ G ki +m +1 . For given i ∈ Gr(ti ) and  ∈ Gr(t0 ) we 1 (t) → (0, ∞] as follows: define a map  : Cint  1 1 1 (t) = Cint (ti ) ∪ Cint (t0 ) ∪ {e1 , · · · , em }, Cint i

where e1 , · · · , em are new edges corresponding to the 0-th exterior edge of ti . We put ⎧ 1 if e ∈ Cint (ti ) ⎪ ⎨ i (e) 1 (e) = 0 (e) if e ∈ Cint (t0 ) ⎪ ⎩ ∞ if e = ei , i = 1, · · · , m and Compk1 ···km ;m ;j1 ,··· ,jm ((t1 , 1 ), · · · , (tm , m ); (t0 , 0 )) =: (t, ). Each such element (t, ) is obtained by applying Comp iteratively to the elements of Gr∗ ’s (the interior elements of the moduli space) in a unique way. (See Figure 7.1.2.) In the rest of this section, we only consider the case when (t, ) = Compk;m;j ((t1 , 1 ), (t0 , 0 )) and (t1 , 1 ) ∈ Grk+1 , (t0 , 0 ) ∈ Grm+1 .

j

t0 e (new )

t1

Figure 7.1.2 In other words, we only consider the case where under the map Θ : Grk+1 → Mmain k+1 , the element (t, ) corresponds to a marked semi-stable curve of genus 0 with one singular point (i.e., one nodal point) on the boundary. The other cases are similar except that the notation will become more complicated. We also remark that we can handle sphere bubble by the same way as in [FuOn99II]. (See Subsection 7.1.4.) Let w : (X(t, ), ∂X(t, )) → (M, L) be a pseudo-holomorphic map that extends to a stable map: (Σ, ∂Σ) → (M, L) when X(t, ) is biholomorphic to the complement Σ\{marked points} (see Section 3.4). Then w determines the maps w1 : X(t1 , 1 ) → M,

w0 : X(t0 , 0 ) → M,

406

7.

TRANSVERSALITY

and the points p 1 = (p10 , · · · , p1k ) ∈ Lk+1 ,

p 0 = (p00 , · · · , p0m ) ∈ Lm+1

such that 1,p (M, L), ((t1 , 1 ), w1 , p 1 ) ∈ Wk+1

1,p ((t0 , 0 ), w0 , p 0 ) ∈ Wm+1 (M, L), and p10 = p0j .

We have chosen E((t1 ,1 ),w1 , p 1 ) , E((t0 ,0 ),w0 , p 0 )) as in Subsection 7.1.2. We define a subspace X of (7.1.19.1)

1,p 1,p X + := T((t1 ,1 ),w1 , p 1 ) Wk+1 (M, L) ⊕ T((t0 ,0 ),w0 , p 0 )) Wm+1 (M, L)

by (7.1.19.2)

X := {((ξ1 , V1 , v 1 ), (ξ0 , V1 , v 0 )) ∈ X + | v01 = vj0 ∈ Tp10 L = Tp0j L}.

We consider the direct sum D((t1 ,1 ),w1 , p 1 ) ∂ ⊕ D((t0 ,0 ),w0 , p 0 )) ∂ and denote the restriction of it to X by 0,p 0,p D(w1 ,w0 ) ∂ : X → E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0). 1 ,1 ),w1 ,

Lemma 7.1.20. The composition 0,p 0,p π ◦ D(w1 ,w0 ) ∂ : X → E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0) p 1 ) /E((t1 ,1 ),w1 , p 0 ) /E((t0 ,0 ),w0 , 1 ,1 ),w1 ,

is surjective. Here π be the obvious projection map. Proof. This is a consequence of Lemma 7.1.18 and the “Mayer-Vietoris principle” of Mrowka [Mro89] (Compare it also with Figure 6 in [Fuk96II]).  Using Lemma 7.1.20, we now perform Taubes’ type gluing construction. To carry this out, we need to modify the pseudo-holomorphic curve equation to one similar to (7.1.16), and to construct the moduli space of its solutions that is parameterized by the kernel of π ◦ Du ∂ plus a neighborhood of (t, ) in Mmain k+m+1 . This neighborhood of (t, ) is parameterized by V × (C, ∞] where V is the neighborhood of 0 in main T(t1 ,1 ) Mmain k+1 ⊕ T(t0 ,0 ) Mm+1 . Here (C, ∞] corresponds to the set of parameters which controls “stretching” of the attaching discs at the boundary singular point. (We take a sufficiently large positive number for C.) As in other cases of the gluing construction, we first use partitions of unity to construct approximate solutions (of (7.1.22) which we will define later). Then we study their linearizations and check if they are surjective. Because of the degeneracy at infinity, we need to use the weighted Sobolev norms with some unusual weight to achieve uniformity of the estimates involved. (This method is similar to the one used in Section 8 [Fuk96II]). Let (t1 , 1 ) ∈ Grk+1 and (t0 , 0 ) ∈ Grm+1 be in small neighborhoods of (t1 , 1 ) and (t0 , 0 ) respectively. For each sufficiently large positive number T and an element ((ξ1 , V1 , v 1 ), (ξ0 , V1 , v 0 )) of Ker π ◦ Dw ∂, we will construct an approximate solution of the pseudo-holomorphic curve equation. We start with gluing the domains of maps by considering (t , ∞ ) = Compk;m;j ((t1 , 1 ), (t0 , 0 )).

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

407

Note that (t , ∞ ) has one interior edge of infinite length, which is the edge where we glued t0 and t1 . We write this edge e(new). Replace this edge by an edge of length 10T and denote by (t , T ) = Comp((t1 , 1 ), (t0 , 0 ); T ) ∈ Grk+m the resulting graph. More precisely, X(t , T ) is constructed from X(t1 , 1 ) and X(t0 , 0 ) as follows: Let e1,0 be the 0-th exterior edge of t1 and e0,j be the j-th exterior edge of t0 . In order to make the exposition easier to understand, we change the parameterizations of Le1,0 and Le0,j so that Le1,0 [−5T, ∞) × [0, 1] ⊂ X(t1 , 1 ) Le0,j (−∞, 5T ] × [0, 1] ⊂ X(t0 , 0 ), in place of identifying Le1,0 [0, ∞) × [0, 1], Le0,j (−∞, 0] × [0, 1]. We remove [0, ∞) × [0, 1] ⊂ Le1,0 from X(t1 , 1 ) and (−∞, 0] × [0, 1] ⊂ Le0,j from X(t0 , 0 ) and glue them at {0} × [0, 1]. Noting that Le(new) [−5T, 5T ] × [0, 1] ⊂ X(t0 , 0 ), it is easy to see that X(t0 , T ) is obtained in this way. −5T

0

5T

X(t1 , 1 )

X(t0 , 0 )

−5T

5T

X(t0 , T )

Figure 7.1.3 By the construction, we have the natural maps: I(t0 ,0 ),(t ,T ) : X(t0 , 0 ) \ ((−∞, −5T ) × [0, 1]) → X(t , T ) I(t1 ,1 ),(t ,T ) : X(t1 , 1 ) \ ((5T, ∞) × [0, 1]) → X(t , T ),

408

7.

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where (5T, ∞) × [0, 1] is regarded as a subset of Le1,0 and (−∞, −5T ) × [0, 1] as that of Le0,j respectively. It is obvious that I(t0 ,0 ),(t ,T ) and I(t1 ,1 ),(t ,T ) are diffeomorphisms to their images. Furthermore the union of their images is X(t , T ) and the intersection of the images is Le(new) . (See Figure 7.1.3.) We need to use cut-off functions of various kinds, which we now introduce. (See Figure 7.1.4.)

1 S−1 S S+1

0

τ χ+ S

1

S−1 S S+1

τ χ− S

χ1S S t1

χ0S t0 S

χ+ S S

χ− S t

Figure 7.1.4

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

409

For each S ∈ [−5T + 1, 5T − 1], we take a smooth function χ+ S : [−5T, 5T ] → [0, 1] such that  0 if τ < S − 1 χ+ (τ ) = S 1 if τ > S + 1 and (7.1.21)

|∇χ+ S | < C.

+ We put χ− S = 1 − χS . Regarding them as the functions defined on Le(new) =   [−5T, 5T ] × [0, 1], we extend χ± S to X(t ,  ) by putting them to be 0 or 1 on the two   components of X(t , T ) − Le(new) respectively. We denote the resulting functions by the same notations. Note that  χ1S = χ− S ◦ I(t1 ,1 ),(t ,T ) ,

 χ0S = χ+ S ◦ I(t0 ,0 ),(t ,T ) .

This time regarding χ− S as a function on Le1,0 [−5T, ∞) × [0, 1], we similarly extend it to X(t1 , 1 ) and denote the resulting function by χ1S . Similarly, regarding χ+ S as a function on Le0,j (−∞, 5T ] × [0, 1], we extend it to X(t0 , 0 ) and denote the resulting function by χ0S . We will use these cut off functions to glue sections on X(t0 , 0 ) and X(t1 , 1 ) to obtain sections on X(t , T ). Using the exponential convergence of w0 ∈ Mmain k+1 (β0 ) (β ) at infinity, we can glue them in a small coordinate neighborand w1 ∈ Mmain 1 m+1 hood of p10 = p0j and obtain a map wT : X(t , T ) → M . We now define several maps S J0,∗ : Γ(X(t0 , 0 ), Λ0,1 ⊗ w0∗ T M ) → Γ(X(t , T ), Λ0,1 ⊗ wT∗ T M ), S J1,∗ : Γ(X(t1 , 1 ), Λ0,1 ⊗ w1∗ T M ) → Γ(X(t , T ), Λ0,1 ⊗ wT∗ T M ), S J∗,0 : Γ(X(t , T ), Λ0,1 ⊗ wT∗ T M ) → Γ(X(t0 , 0 ), Λ0,1 ⊗ w0∗ T M ), S J∗,1 : Γ(X(t , T ), Λ0,1 ⊗ wT∗ T M ) → Γ(X(t1 , 1 ), Λ0,1 ⊗ w1∗ T M ),

which we will use in the gluing construction of sections. We define a map w0 : Im(I(t0 ,0 ),(t ,T ) ) → M by −1 w0 = w0 ◦ I(t   0 ,0 ),(t ,

T)

and define w1 : Im(I(t1 ,1 ),(t ,T ) ) → M in a similar way. Let V ∈ Γ(X(t0 , 0 ), Λ0,1 ⊗ w0∗ T M ). The section χ0S V is supported in the set X(t0 , 0 )\((−∞, −5T ) × [0, 1]). Hence we can use I(t0 ,0 ),(t ,T ) to obtain a section of Γ(X(t, ), Λ0,1 ⊗ w0∗ T M ). We then use the parallel transport along the minimal S geodesics to obtain an element J0,∗ (V ) in Γ(X(t , T ), Λ0,1 ⊗wT∗ T M ). The definition S 1 of J1,∗ is similar by using χS . Next let V ∈ Γ(X(t , T ), Λ0,1 ⊗ wT∗ T M ). The support of χ+ S V lies in the image of I(t0 ,0 ),(t ,T ) . We use the parallel transport along the minimal geodesics to obtain an element of Γ(X(t , T ), Λ0,1 ⊗ w0∗ T M ). We then pull it back by I(t0 ,0 ),(t ,T ) to S S obtain an element J∗,0 (V ) of Γ(X(t0 , 0 ), Λ0,1 ⊗ w0∗ T M ). The definition of J∗,1 is − 1,p 0,p similar by using χS . These maps extend to the W or W closures of the spaces involved.

410

7.

TRANSVERSALITY

We next recall that elements of E((t1 ,1 ),w1 , p 1 ) , E((t0 ,0 ),w0 , p 0 ) are of compact support. Using this fact, we can choose E((t ,T ),wT , p  ) so that S S J1,∗ + J0,∗ : E((t1 ,1 ),w1 , p 1 ) ⊕ E((t0 ,0 ),w0 , p 0 ) → E((t ,T ),wT , p  )

is injective. So we may assume that E((t1 ,1 ),w1 , p 1 ) ⊕ E(t0 ,0 ),w0 , p 0 ) ⊆ E((t ,T ),wT , p  ) . We now consider the equation (7.1.22)

∂w ≡ 0 mod E((t1 ,1 ),w1 , p 1 ) ⊕ E((t0 ,0 ),w0 , p 0 ) .

Here w is C 0 close to wT . We first describe an approximate solution of (7.1.22). For this purpose, we introduce more notations. Define the “small” tangent space 1,p small T((t p 1 ) Wk+1 (M, L) 1 ,1 ),w1 , 1,p (M, L) | ξ1 = 0} := {(ξ1 , V1 , v 1 ) ∈ T((t1 ,1 ),w1 , p 1 ) Wk+1 1,p small and T((t p 0 ) Wk+1 (M, L) in a similar way. By using these small tangent spaces 0 ,0 ),w0 ,

1,p (M, L) (i = 0, 1), we obtain in place of the full tangent spaces T((ti ,i ),wi , p i ) Wk+1 + Xsmall , Xsmall (See (7.1.19)). We use Xsmall rather than X since we want to handle the moduli parameters ξ1 , ξ2 separately in a way different from the other parameters V1 , v01 , · · · , vk1 etc. We next define a map 1,p I S : Xsmall → T((t ,T ),wT , p  ) Wk+1 (M, L)

for S > 2 as follows. Here p  = (p1 , · · · , pk+m ) is obtained by re-ordering p11 , · · · , p1k , p00 , · · · , p0j , · · · , p0m . Let ((0, V0 , v0 ), (0, V1 , v1 )) ∈ Xsmall . We put v = v01 = vj0 . (See (7.1.19.2).) Then we define I S by (7.1.23) I S ((0, V0 , v0 ), (0, V1 , v1 ))(x) ⎧ V0 (x) if x ∈ I(t0 ,0 ),(t ,T ) (X(t0 , 0 ) \ ((−∞, +5T ) × [0, 1])), ⎪ ⎪  ⎪     ⎪ ⎪ ⎨ Pal v + χ+ (τ ) Pal(V (τ, t)) − v + χ− (τ ) Pal(V (τ, t)) − v 0 1 −S +S = ⎪ ⎪ ⎪ if x = (τ, t) ∈ [−5T, 5T ] × [0, 1], ⎪ ⎪ ⎩ if x ∈ I(t1 ,1 ),(t ,T ) (X(t1 , 1 ) \ ((−5T, ∞) × [0, 1])). V1 (x) By restricting I T , we define small I T : Ker (π ◦ DwT ∂|Xsmall ) → T((t  ,

 p ) T ),wT ,

1,p Wk+m (M, L).

We take a diffeomorphism E = (E1 , E2 ) : {(x, v) ∈ T M | |v| < } → M × M

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

411

to its image such that E1 (x, v) = x,

 dE2 (x, tv)  =v  dt t=0

and E(x, v) ∈ L × L,

for x ∈ L, v ∈ Tx L.

(For example, we can choose E defined by E(x, v) = exp(x, v) for the exponential map exp with respect to the Levi-Civita connection of a metric g such that L becomes totally geodesic.) We use the map E to define a map small Exp : T((t  ,

 p ) T ),wT ,

1,p 1,p Wk+m (M, L) → Wk+m (M, L)

by Exp(0, V, v )(x) = E2 (wT (x), V (x)). Here x is a point on the domain X(t , T ). (More precisely, Exp is defined for V, v in a neighborhood of 0 only.) Now, for each (t0 , 0 ),(t1 , 1 )), T and ((0, V1 , v 1 ), (0, V0 , v 0 )) ∈ Xsmall , we take (7.1.24)

((t , T ), Exp(I T ((0, V1 , v 1 ), (0, V0 , v 0 )))

as our approximate solution, where (t , T ) = Comp((t1 , 1 ), (t0 , 0 )), T ). The next step is to perturb (7.1.24) and obtain an exact solution of (7.1.22). As we mentioned before, we use somewhat non-standard weighted Sobolev norm for this purpose. The weight function α : X(t , T ) → [1, ∞) we use here is defined as follows: We recall that | · | : R → [0, ∞) is a smooth function such that |x| = |x| if |x| ≥ 1 and |x| = 0 if |x| < 1/2. For each exterior 1 edge e ∈ Cext (t ), we put α(τ, t) = exp(δ|τ | ) on Le (−∞, 0] × [0, 1]. For the edge e(new), we identify Le(new) with [−5T, 5T ] × [0, 1] and put  α(τ, t) =

1 if |τ | > 5T − 1 exp(δ(5T − |τ |)) if 5T − 2 > |τ | > 1.

In the intermediate regions, there is a constant C such that 1 ≤ α(τ, t) < C for 5T − 1 > |τ | > 5T − 2 and |α(τ, t) − exp(5T δ)| < C for |τ | < 1. Finally we put α = 1 on other Le ’s. See Figure 7.1.5.

412

7.

TRANSVERSALITY

eδ(5T + τ)

eδ(5T − τ)

1 τ −5T + 1

−1

5T−1

0 1

Figure 7.1.5 (α(τ, t)) Using this weight function, we now define the Sobolev norm we use. Let w : (X(t, T ), ∂X(t, T )) → (M, L) be the map C 0 -close to wT . We consider the vector space E((t ,T ),w, p  ) but will put a different norm on it. For V ∈ E((t ,T ),w, p  ) , we put V p0,p,α =

(7.1.25)

X(t ,T )

α|V |p .

Since the weight α coincides, outside a compact set Le(new) , with the one we used to define E((t ,T ),w, p  ) (see (7.1.7.3)), the right hand side of (7.1.25) is finite and this norm 0,p,α is equivalent to the previous norm 0,p . However the ratio between the norms 0,p and 0,p,α on E((t ,T ),w, p  ) goes to infinity as T goes to infinity. 1,p small Next we modify the norm on T((t v ) = (V, v )  , ),w, p  ) Wk+m (M, L). Let (0, V,  T

1,p small be an element of T((t  , ),w, p  ) Wk+m (M, L). We remark that V is continuous and T so V (0, 1/2) makes sense, if p is sufficiently large. Here (0, 1/2) ∈ Le(new) [−5T, 5T ] × [0, 1]. We put





(V, v)p1,p,α =

(|∇V | + |V |)p

1 (t),e=e(new) e∈Cint





+

α · (|∇(V − Pal(vi ))|p + |V − Pal(vi )|)p

1 (t) ei ∈Cext



Le

Lei

α · (|∇(V − Pal(V (0, 1/2)))|p + |V − Pal(V (0, 1/2))|)p

+ Le(new)

+



|vi |p + |V (0, 1/2))|p ,

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

413

where Pal is an appropriate parallel transport along minimal geodesic. In the same way as the proof of Lemma 7.1.20, we may assume that the restriction (7.1.26)

π ◦ D(w1 ,w0 ) ∂ : Xsmall −→

0,p E((t p 1) 1 ,1 ),w1 ,

E((t1 ,1 ),w1 , p 1 )

⊕

0,p E((t p 0) 0 ,0 ),w0 ,

E((t0 ,0 ),w0 , p 0 )

of π ◦ D(w1 ,w0 ) ∂ to Xsmall is surjective. Proposition 7.1.27. Assume that (7.1.26) is surjective. Then there exists C > 0 independent of T with the following properties: For each (t , T ) = Comp((t1 , 1 ), (t0 , 0 )); T ) and w sufficiently close to wT , there exists a linear map Q((t ,T ),w) :

E((t ,T ),w, p  ) 1,p small −→ T((t  , ),w, p  ) Wk+m (M, L) T E((t1 ,1 ),w1 , p 1 ) ⊕ E((t0 ,0 ),w0 , p 0 ))

such that (7.1.28.1)

(π ◦ Dw ∂) ◦ Q((t ,T ),w) = id,

(7.1.28.2)

Q((t ,T ),w) (V )1,p,α < CV 0,p,α .

We emphasize that the uniformity of the constant C cannot be achieved if we use 1,p , 0,p in place of 1,p,α , 0,p,α . Before proving Proposition 7.1.27, we state the following lemma whose proof will be postponed to the end of this subsection. Lemma 7.1.29. There exist constants c > 0, C > 0 independent of T such that ∂ Exp(I T ((0, V1 , v 1 ), (0, V0 , v 0 )))0,p,α < C(V1 1,p + V2 1,p + 1) exp(−cT ). Once we have Proposition 7.1.27 and Lemma 7.1.29, it is by now standard to construct a family of solutions of (7.1.22) parameterized by the product of the kernel of π◦D(w1 ,w0 ) ∂ and a neighborhood of (t, ) in Grk+m . Therefore to construct Kuranishi neighborhoods of points at infinity (i.e., at lower dimensional strata), it remains to prove Proposition 7.1.27 and Lemma 7.1.29. Proof of Proposition 7.1.27. By assumption, there exists a bounded linear map 0,p 0,p Q0 : E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0 ) → Xsmall p 1 ) /E((t1 ,1 ),w1 , p 0 ) /E((t0 ,0 ),w0 , 1 ,1 ),w1 ,

such that π ◦ D(w1 ,w0 ) ∂ ◦ Q0 = identity.

(7.1.30)

Here we recall that D(w1 ,w0 ) ∂ is the restriction of (D((t1 ,1 ),w1 , p 1 ) ∂ ⊕ D((t0 ,0 ),w0 , p 0 )) ∂) to Xsmall . We put (7.1.31)

S ˜ ((t , ),w) (V ) = (I T ◦ Q0 )(J −S (χ+ V ) ⊕ J∗,1 (χ− Q ∗,0 0 0 V )), T

where I T is as in (7.1.23), 0,p 0,p V ∈ E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0) 1 ,1 ),w1 ,

414

7.

TRANSVERSALITY

and 0,p 0,p V ∈ E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0) p 1 ) /E((t1 ,1 ),w1 , p 0 ) /E((t0 ,0 ),w0 , 1 ,1 ),w1 ,

is its equivalence class. Note that the support of any element of E((t1 ,1 ),w1 , p 1 ) , E((t0 ,0 ),w0 , p 0 ) is disjoint from Le(new) and hence the equivalence classes 0,p −S (χ+ J∗,0 p 1), 0 V ) ∈ E((t1 ,1 ),w1 , p 1 ) /E((t1 ,1 ),w1 ,

and 0,p S (χ− J∗,1 p 0) 0 V )) ∈ E((t0 ,0 ),w0 , p 0 ) /E((t0 ,0 ),w0 ,

depend only on V . Moreover the right hand side of (7.1.31) is independent of S as far as 2 < S < − 5T − 2, since χ+ 0 V is 0 on [−5T, −2] × [0, 1] and χ0 V is 0 on [2, 5T ] × [0, 1]. ˜ ((t , ),w) . We next need the following two lemmas to study the properties of Q T They are key lemmas used in our estimate. Lemma 7.1.32. We have (Dw ∂ ◦ I T )((0, V0 , v 0 ), (0, V1 , v 1 )) −T +T − ((J0,∗ + J1,∗ ) ◦ D(w1 ,w0 ) ∂)((0, V0 , v 0 ), (0, V1 , v 1 ))

≤ C(exp(−cT ) + dist(w, w0 ) + dist(w, w1 ))((0, V0 , v 0 ), (0, V1 , v 1 )).

Proof. There are two reasons why I and J do not commute with Dw ∂, D(w1 ,w0 )) ∂. One is that we use the parallel transport along the minimal geodesics to go from w0 , w1 to w. The terms caused by the parallel transport are estimated by the second and the third terms of the right hand side of the lemma. The second and a more essential point of the matter is the fact that we use cut-off functions and weighted norms in our estimate. Namely the terms are caused by χ+ −T (τ )(Pal(V0 (τ, t)) − v) and χ− +T (τ )(Pal(V1 (τ, t)) − v) in (7.1.23). Since the argument for χ− +T (τ )(Pal(V1 (τ, t)) − v) is similar, we only discuss χ+ (τ )(Pal(V (τ, t)) − v) here. We remark that χ+ 0 −T −T is locally constant outside [−T − 1, −T + 1] × [0, 1]. Therefore the argument to be integrated in the Sobolev norm  ·  above is supported in [−T − 1, −T + 1] × [0, 1]. The value of the weight we put for the domain of the operator (that is the weight we put for the norm of V0 ) is Ceδ(T +5T ) . (We remark that we identify the end of (X(t0 , 0 ), w0 , p 0 )  with [0, 1] × (−∞, 5T ] in place of [0, 1] × (−∞, 0]. Hence the weight is eδ|5T −τ | in  place of eδ|τ | . See Definition 7.1.3.) On the other hand, the weight we put for the range is Ceδ(5T −T ) . Therefore the norm drops down by the ratio Ce−cT as required. See Figure 7.1.6. 

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

415

weight on X(t0 , 0 )

eδ |5T − τ |

e−cT

eδ |5T + τ |

weight on X(t , T )

Figure 7.1.6 Lemma 7.1.33. ˜ ((t , ),w) (V )1,p,α < CV 0,p,α . Q T Proof. We first show the following: (7.1.34)

I T ((0, V0 , v0 ), (0, V1 , v1 ))1,p,α < C((V0 , v0 )1,p + (V1 , v1 )1,p ),

with C independent of T . To prove (7.1.34), it is enough to estimate the norm of the left hand side on [−5T, 5T ] × [0, 1] ∼ = Le(new) . (The estimate of the other part is trivial.) We put V  = I T ((0, V0 , v0 ), (0, V1 , v1 )). We remark that   V  (τ, t) = Pal Pal(V0 (τ, t)) + Pal(V1 (τ, t)) − v on {(τ, t) | |τ | ≤ 1} (see (7.1.23)). In fact, the above equality holds for −T + 1 < τ < T − 1. (Note that T > 2.) Hence we have |V  (0, 1/2) − v| < Ce−δc ((0, V0 , v0 )1,p + (0, V1 , v1 )1,p ) by the definition of the norm 1,p given in Lemma 7.1.5. Therefore it suffices to estimate (7.1.35) α · (|∇V  | + |V  − v|)p . Le(new)

416

7.

TRANSVERSALITY

It is easy to see that (7.1.35) is estimated by C

−5T



+C +C



5T

dτ 0 5T

−5T 5T −5T



1

α|χ+ −T (V0

1

−5T

dτ 0

1

p α|χ− +T (V1 − v)| dt

p α|∇(χ+ −T (V0 − v))| dt

dτ 0



− v)| dt + C



5T

p

1

p α|∇(χ− +T (V1 − v))| dt.

dτ 0

By observing that the weight α appears above can be estimated by the weight we put to define (V0 , v0 )1,p , (V1 , v1 )1,p , it is easy to estimate this by the right hand side of (7.1.34). To complete the proof of Lemma 7.1.33 we only need to show −T J∗,0 (χ+ 0 V )0,p ≤ V 0,p,α

(7.1.36.1)

T J∗,1 (χ− 0 V )0,p ≤ V 0,p,α .

(7.1.36.2)

To prove (7.1.36.1) it suffices to remark that on the support of X0+ · V the weight α coincides with the weight we used to define  · 0,p . The proof of (7.1.36.2) is the same.  By Lemma 7.1.32, we have the following inequality for 0,p 0,p V ∈ E((t p 1 ) ⊕ E((t0 ,0 ),w0 , p 0) 1 ,1 ),w1 ,

and its equivalence class V . ˜ ((t , ),w) (V ) − V  π ◦ Dw ∂ ◦ Q T −T +T −S − S ≤ π ◦ (J0,∗ ⊕ J1,∗ ) ◦ D(w1 ,w0 ) ∂ ◦ Q0 ((J∗,0 (χ+ 0 V ) ⊕ J∗,1 (χ0 V ))) − V 

+ o(T )V  −T −S +T +S − ≤ π(J0,∗ J∗,0 (χ+ 0 V ) + J1,∗ J∗,1 (χ0 V )) − V  + o(T )V .

Here limT →∞ o(T ) = 0. The first term on the right hand side is actually zero, because −T −S + +T +S − J0,∗ J∗,0 (χ+ J1,∗ J∗,1 (χ− 0 V ) = χ0 V, 0 V )) = χ0 V. − We remark χ+ 0 V is 0 on [−5T, −2] × [0, 1] and χ0 V is 0 on [2, 5T ] × [0, 1]. Therefore ˜ ((t , ),w) − id (π ◦ Dw ∂) ◦ Q T

goes to zero as T goes to infinity. On the other hand, Lemma 7.1.33 implies that Q((t ,T ),w) is uniformly bounded. Therefore, using Lemma 7.1.33, we can find  Q((t ,T ),w) satisfying (7.1.28) by a standard argument. Proof of Lemma 7.1.29. To prove Lemma 7.1.29 we use the following lemma. We consider the situation of Lemma 7.1.29 and use the same notations from there.

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

417

Lemma 7.1.37. Let V be an element of the kernel of D((t,),w, p) ∂ and e ∈ 1 (t). We identity Le (−∞, 0] × [0, 1] as before. Then there exist c > 0 and Cext C > 0 independent of T and δ such that |V (τ, t)| < C exp(−c|τ |)V 0,p,α . This lemma is a standard decay estimate which is a part of the proof of Lemma 7.1.11. The next lemma follows from the removable singularity theorem for pseudoholomorphic discs. See the estimate in p.138 in [Oh92] and translate it in terms of the coordinates (−∞, 0] × [0, 1] near the puncture. Lemma 7.1.38. Let ((t , T ), w, p) ∈ Wk1,p (M, L) 1 satisfy (7.1.22) and e ∈ Cext (t) be an exterior edge. Identify Le (−∞, 0] × [0, 1]. Then there exist positive constants C, C , c, c independent of T and δ such that

dist(w(τ, t), pi ) < C exp(−c|τ |),

∇ w(τ, t) < C exp(−c |τ |).

Lemma 7.1.29 follows immediately from these two lemmas and the construction.  We thus have constructed Kuranishi neighborhoods of ((Σ, z ), w) where Σ has one (boundary) singular point. The case where there are several singular points on the boundary can be handled in the same way. The case where there are interior singular points can be handled in the same way as in [FuOn99II] Chapter 3, as we will explain in the next subsection. We can also prove “surjectivity”, which means that the Kuranishi neighborhood of ((Σ, z), w) we have constructed contains all the elements of the moduli space Mk+1 (β) close to the given point ((Σ, z), w). The proof of this surjectivity is similar to Section 14 [FuOh97] or Section 9 [FuOn99II]. (We use Lemma 7.1.38 for this purpose also.) In [FOOO09II] we prove surjectivity in a case similar to but is more difficult than the case of this section. Before we end this subsection, we remark that the analytic results developed in Subsections 7.1.2 and 7.1.3 can be similarly applied to the Bott-Morse case. In fact, our argument here is based on that in [Fuk96II] where the foundation of analytic results for the Bott-Morse case was established. We remark that there is an alternative method to work out analytic detail of gluing in some of the Bott-Morse cases, which was discovered by DonaldsonKronheimer [DoKr90] in the gauge theory case. The same method was used in the case of pseudo-holomorphic curve (closed Riemann surface) by McDuff-Salamon [McSa94]. The main difference is, we give a cylindrical metric on the ‘neck region’ so that the region is isometric to [−5T, 5T × [0, 1] (or [−5T, 5T ] × S 1 ), while in Donaldson-Kronheimer’s (or McDuff-Salamon’s) approach, ‘neck region’ is regarded as two copies of D4 \ {0} (or two copies of D2 \ {0}) glued. In [FuOn99II] the first and fourth named authors used McDuff-Salamon’s approach to handle sphere bubble. Here we use the approach by [Fuk96II] since our parametrization + ∼ of Mmain k+1 = Grk+1 is more directly related to the cylindrical parametrization of the ‘neck region’.

418

7.

TRANSVERSALITY

7.1.4. Wrap-up of the proof of Propositions 7.1.1 and 7.1.2. In the last two subsections, we described the analytic part of the construction of the Kuranishi structure as in Propositions 7.1.1 and 7.1.2. In this subsection we globally construct a Kuranishi structure and complete the proof. The proof is rather a minor modification of the discussion of [FuOn99II]. We give the precise definition of the topology on Mmain k+1,m (L; β). The topology is an analog to the one defined in Section10 [FuOn99II] and is described as follows. The statement of Propositions 7.1.1 and 7.1.2 concerns only the case m = 0. However to study sphere bubble etc., it is necessary to include the case m = 0. In this section we consider only the case k + 1 > 0. The case k + 1 = 0 will be discussed in Subsection 7.4.1. Let (Σi , zi , zi+ ) ∈ Mmain k+1,m be a sequence of marked bordered stable curves of genus 0, where zi = (z0 , z1 , · · · , zk ) are boundary marked points and zi+ are interior marked points. (See Definition 2.1.17 and Definition 2.1.18.) And let wi : Σi → M be a sequence of pseudo-holomorphic maps. Suppose + lim (Σi , zi , zi+ ) = (Σ∞ , z∞ , z∞ ) ∈ Mmain k+1,m .

i→∞

By taking a subsequence if necessary, we may assume that combinatorial types of (Σi , zi , zi+ ) are independent of i. For each singular point of Σ∞ which does not correspond to a singular point of Σi , we consider its  neighborhood and let V,∞ ⊂ Σ∞ be their union. We take the corresponding union of rectangle V,i ⊂ Σi . We have diffeomorphisms ui, : Σi \V,i ∼ = Σ∞ \V,∞ . (See Section 7 [FuOn99II], of which our case is an analog. We can reduce it to the case discussed in [FuOn99II] by the argument of Subsection 2.1.2 in this book.) (See Figure 7.1.7.) V,i V∞,i

z0

z0

Σ∞

Σi

Figure 7.1.7 Definition 7.1.39. We say + lims ((Σi , zi , zi+ ), wi ) = ((Σ∞ , z∞ , z∞ ), w∞ )

i→∞

if, for each δ, there exists (δ) and I(δ) such that the following holds for i > I(δ) and  < (δ). + (7.1.40.1) limi→∞ (Σi , zi , zi+ ) = (Σ∞ , z∞ , z∞ ) ∈ Mmain k+1,m . (7.1.40.2) sup dist(wi , w∞ ◦ ui, ) < δ. (Here we fix a Riemannian metric on M and dist is the induced metric.) (7.1.40.3) Diameter of each of the image of connected components of V,i , V,∞ by wi , w∞ respectively is smaller than δ.

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

419

Now we want to generalize Definition 7.1.39 to the case when the marked bor+ dered genus zero Riemann surfaces (Σi ,zi ,zi+ ), (Σ∞ ,z∞ ,z∞ ) may not be stable but + + ((Σi , zi , zi ), wi ), ((Σ∞ , z∞ , z∞ ), w∞ ) are stable. The idea is to add marked points so that they will become stable. Let  : {1, · · · , k } → {1, · · · , k} be an order preserving injection and  : {1, · · · , m } → {1, · · · , m} an injection. We define a forgetful map main Forget ,  : Mmain k+1,m (L; β) → Mk +1,m (L; β)

for the case when k ≥ 0 as follows. Let ((Σ, z, z+ ), w) ∈ Mmain k+1,m (L; β). We consider 

  Σ, (z0 , z (1) , · · · , z (k ) ), (z + (1) , · · · , z + (m ) ) , w .

This is not necessarily a stable map. The unstable component D is one of the following: (w should be a constant map on that component.) See Figure 7.1.8 – Figure 7.1.13. (7.1.41.1) (7.1.41.2) (7.1.41.3) (7.1.41.4) (7.1.41.5) (7.1.41.6)

Sphere with one singular point and no marked point. Sphere with two singular points and no marked point. Sphere with one singular point and one marked point. Disc with one boundary singular point and no marked point. Disc with two boundary singular points and no marked point. Disc with one boundary singular point and one boundary marked point.

We remark that, since k + 1 > 0, it follows that any disc component D has either a marked or a singular point. Hence if D also has an interior singular point, then it is stable. In case (7.1.41.1) or (7.1.41.4), we remove D. In case (7.1.41.2) or (7.1.41.5), we remove D and glue two components which used to be attached to D. In case (7.1.41.3) or (7.1.41.6), we remove D and move the marked point on D to the component which used to be attached to D. We repeat this process until there will be no unstable component left. We have thus obtained the map Forget ,  .

D D

Figure 7.1.8 (7.1.41.1)

D

Figure 7.1.9 (7.1.41.2) Figure 7.1.10 (7.1.41.3)

420

7.

TRANSVERSALITY

z1

z2

D

D

z2

z1 z0

D

z1

z0

z3 z0

z2

z1

z0

z2

z2

z1

z1 z0

z3

z2

z3

z0

z3

Figure 7.1.11(7.1.41.4) Figure 7.1.12(7.1.41.5) Figure 7.1.13(7.1.41.6) Definition 7.1.42. Let + ), w∞ ) ∈ Mmain ((Σi , zi , zi+ ), wi ), ((Σ∞ , z∞ , z∞ k+1,m (L; β)

with k ≥ 0. Then we say + lim ((Σi , zi , zi+ ), wi ) = ((Σ∞ , z∞ , z∞ ), w∞ ),

i→∞

if there exist k , m and ,  as above and + ∼ ((Σi , zi , zi+ )∼ , w ˜i ), ((Σ∞ , z∞ , z∞ ) ,w ˜∞ ) ∈ Mmain k +1,m (L; β)

such that

+ ∼ ˜i ) = ((Σ∞ , z∞ , z∞ ) ,w ˜∞ ) lims ((Σi , zi , zi+ )∼ , w

i→∞

and

˜i ) = ((Σi , zi , zi+ ), wi ), Forget ,  ((Σi , zi , zi+ )∼ , w + ∼ + Forget ,  ((Σ∞ , z∞ , z∞ ) ,w ˜∞ ) = ((Σ∞ , z∞ , z∞ ), w∞ ).

(Note that we used Mmain k+1,m (L; β) for positive m in order to define a topology main on Mmain (L; β) = M (L; β).) k+1 k+1,0 We thus have established our moduli space Mmain k+1,m (L; β) as a metrizable space. Metrizability follows from the fact that it has a stratification such that each stratum is a subset of a Banach manifold and that the number of stratum is finite. We defined topology by specifying the limit of sequences. The above fact also justifies it. Namely we can define the topology by specifying the limit of sequences instead of open subsets. Theorem 7.1.43. Mmain k+1,m (L; β) is compact and Hausdorff. Proof. The compactness is well-established in the literature, especially in [Grom85, Pan94, Ye94, IzSh00]. More precisely, since the definition of the topology is given along the same line as [FuOn99II], we can prove the compactness in the same way as [FuOn99II] Sections 10,11 using the analytic results in [Grom85, Pan94, Ye94, IzSh00]. We omit the detail since the compactness of

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the moduli space of pseudo-holomorphic curves has been well established, (although a correct definition of the topology of the moduli space of pseudo-holomorphic disc is hard to find in the literature). The proof of the Hausdorffness is the same as [FuOn99II] Lemma 10.4.  We now are going to globalize Kuranishi structure on Mmain k+1,m (L; β). The analytic tool was already given in Subsections 7.1.2 and 7.1.3. We also use the following Theorem 7.1.44 whose proof was postponed from Section 2.1. Theorem 7.1.44. (See [Sep84,Sep91, SeSi89].) The space Mmain k+1,m defined in Section 2 has a structure of smooth manifolds with corners of dimension k + 2m − 2, if k + 2m − 2 ≥ 0 and k ≥ 0. ∼ k−2 Proof. The proof is by induction on m. In the case m = 0, Mmain k+1,0 = D (the k − 2 dimensional disc) if k ≥ 2. (See Section 2.1.) If k = 1, then we need to main start the induction at Mmain is homeomorphic to the 2,1 . It is easy to see that M2,1 interval [0, 1]. If k = 0, the first case we need to study is Mmain 1,1 , which is a point. Now we discuss the induction step. Suppose k + 2m − 2 ≥ 0, k ≥ 0. We construct a map main Forget : Mmain k+1,m+1 → Mk+1,m as follows. Let (Σ, z, z+ ) represent an element of Mmain z = (z0 , · · · , zk ) k+1,m+1 . Here  + are boundary marked points and z+ = (z1+ , · · · , zm+1 ) are the interior marked points. We put + z+− = (z1+ , · · · , zm ). +− If (Σ, z, z ) is stable, then we define Forget(Σ, z, z+ ) = (Σ, z, z+− ). If not, then one of (7.1.41.1)-(7.1.41.6) occurs. We then shrink the component in the same way to obtain Forget(Σ, z, z+ ). (In fact, the construction is the special main case of Forget ,  : Mmain k+1,m (L; β) → Mk +1,m (L; β) where (M, L) are points.) We describe the fiber of Forget in the next lemma. For each (Σ , z  , z+ ) ∈ main ˜  (which depends only on Σ ) as follows. For each boundary Mk+1,m we define Σ  singular point of Σ we deform the singularity and make it smooth. We do not ˜  is a single disc together with sphere bubbles. deform the interior singularity. So Σ ˜  . Namely we do not consider its complex We only consider the C ∞ structure of Σ ˜  is well-defined. structure. Hence Σ main Lemma 7.1.45. The fiber of the forgetful map Forget : Mmain k+1,m+1 → Mk+1,m ˜ . at (Σ , z  , z+ ) is diffeomorphic to Σ

We remark that in [FuOn99II] Section 23 a similar argument appears where the case of marked closed Riemann surface is discussed. Proof. Let Forget(Σ, z, z+ ) = (Σ , z , z+ ). We consider the following cases separately. (Below we use the same notation as in the definition of Forget.) (Case 1) (Σ, z, z+− ) is stable: In this case (Σ, z, z+− ) = (Σ , z , z+ ). The element (Σ, z, z+ ), in this case, corresponds one to one to Σ \(∂Σ ∪z+ ∪{singular points}). (Case 2) (7.1.41.1) occurs: It is easy to see that this never happens. (Case 3) (7.1.41.2) occurs: Such (Σ, z, z+ ) corresponds one to one to the interior singular points of Σ . (Figure 7.1.14.)

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(Case 4) (7.1.41.3) occurs: Such (Σ, z, z+ ) corresponds one to one to the points of z+ , (i.e., the interior marked points). (Figure 7.1.15.) (Case 5) (7.1.41.4) occurs: Such (Σ, z, z+ ) corresponds one to one to the points of ∂Σ \ (z  ∪ {singular point}) (Figure 7.1.16.) (Case 6) (7.1.41.5) occurs: This corresponds to a boundary singular point of Σ . However for each boundary singular point of Σ , there is a set of (Σ, z, z+ ) parameterized by Mmain that is an interval. (Figure 7.1.17.) 2,1 (Case 7) (7.1.41.6) occurs: Such (Σ, z, z+ ) corresponds one to one to the point of z  , (i.e., the boundary marked points). (Figure 7.1.18.)

z+m+1 z+m+1

z+i z+m+1

z+i

Figure 7.1.14

Figure 7.1.15

Figure 7.1.16 z2 z+m+1 z 1

z+m+1

z0

z3 z4 z2

z1

z3

z0

z4

Figure 7.1.17

Figure 7.1.18

By the above discussion the fiber is obtained from Σ after replacing each of ˜  we defined the boundary singular point by the interval [0, 1]. This is the space Σ above. The proof of Lemma 7.1.45 is complete.  Using Lemma 7.1.45 we are ready to complete the inductive step of the proof of Theorem 7.1.44. Assuming that Mmain k+1,m is a smooth manifold (with corners) we will prove that Mmain is a smooth manifold as well. Let (Σ, z, z+ ) represent k+1,m+1 an element of Mmain z , z+ ) = (Σ , z  , z+ ). By Lemma 7.1.45, k+1,m+1 and Forget(Σ, 

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˜  . If x is not a singular point, then we (Σ, z, z+ ) corresponds to a point x in Σ can construct a coordinate chart around it easily by the induction hypothesis. (We can do it even if x is a boundary point.) Let us assume that x is a singular ˜  has no boundary singular point and hence x is necessarily point. By definition Σ an interior point. Therefore we can use the same argument as in the case of the moduli space of Riemann surface without boundary to find a coordinate chart around (Σ, z, z+ ). More precisely we proceed as follows. There exists a divisor S (namely a subspace of complex codimension one) in a neighborhood of (Σ , z  , z+ ) in Mmain k+1,m such that x remains to be a singular point only for elements on S. We recall the induction hypothesis that Mmain k+1,m is a smooth manifold. Therefore a   + main neighborhood of (Σ , z , z ) in Mk+1,m can be identified with a neighborhood of     zero in Rd≥0 × Rd−d × C where S = Rd≥0 × Rd−d × {0}. (d = k + 1 + 2m − 5 = dim Mmain k+1,m − 2.) We use the coordinate (u1 , · · · , ud , q) under this identification. (ui ∈ R or R≥0 , q ∈ C). Now a neighborhood of (Σ, z, z+ ) in Mmain k+1,m+1 is identified d d−d 2 with a neighborhood of zero in R≥0 × R × C where Forget is given by (u1 , · · · , ud , q1 , q2 ) → (u1 , · · · , ud , q1 q2 ). In fact, the intersection of a neighborhood of the (Σ, z, z+ ) ∈ Mmain k+1,m+1 with the fiber of a point in S = {(u1 , · · · , ud , q) | q = 0} is given by q1 q2 = 0, which has a nodal point q1 = q2 = 0 corresponding to x. The proof of Theorem 7.1.44 is now complete.  Remark 7.1.46. From the proof of Theorem 7.1.44 above and from Lemma 7.1.45, we can describe the homology group of Mmain k+1,m in a way similar to the case of moduli space of genus zero stable marked curve, as done by Knudsen [Knu83] and Keel [Kee92]. For example, if k + 1 ≥ 3, then Mmain k+1,m is contractible for m = 0, 1, 2 but has nontrivial homology group for m = 3. The simplest of those is main main ∼ 4 the nontrivial homology class of H2 (Mmain 3,3 ; Q) = H (M3,3 , ∂M3,3 ; Q) which is + + + the fundamental class of the closure of the set of all (Σ, z, (z1 , z2 , z3 )) such that Σ is a disc with one sphere bubble attached at 0 ∈ D2 and z1+ , z2+ , z3+ are on this sphere bubble. (It is easy to see that this set is a 2 dimensional submanifold of the 6 dimensional manifold Mmain 3,3 .) We can extend the construction of the sequence of the operators q in Section 3.8, by coupling them with homology classes of Mmain k+1,m in a way similar to the case of the usual Gromov-Witten invariant. We leave this for a future research. Now we are ready to complete the proofs of Propositions 7.1.1, 7.1.2. For this purpose, we need to glue the Kuranishi neighborhoods constructed so far to produce a global Kuranishi structure. Using the compactness and Hausdorffness of Mk+1 (β), we may do this gluing only for a finite number of Kuranishi neighborhoods. The details of this gluing argument goes in the same way as in Section 4.2 [FuOn99II]. We review the arguments used in [FuOn99II], in the gluing construction of the Kuranishi neighborhoods. For each σ = ((Σ, z ), w) ∈ Mmain k+1 (β) ∗ we have chosen a finite dimensional subspace Eσ of Γ(Σ, Λ0,1 ⊗ w T M ) such that Σ the equation ∂w ≡ 0 mod Eσ is the equation defining the Kuranishi neighborhood Uσ of σ. (In the last section, we denote E((t,),w, p) for Eσ . In Section A1 we use the notation V for Kuranishi

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neighborhood. But here we use U , because V was used for the different meaning in Subsections 7.1.2, 7.1.3.) We may assume Uσ to be a closed neighborhood of σ by shrinking it a bit. We take a finite number of σ’s, say σ1 , · · · , σN , so that  (7.1.47) Int Uσ i ⊃ Mmain k+1 (β). i

We put σi = ((Σi , z(i) ), wi ). Using the covering (7.1.47), we will modify Eσ to construct a Kuranishi neighborhood of σ. We first consider the case when (Σ, z ) is stable. We collect all i’s such that σ ∈ Uσ i and put  Eσ i . Eσ = i:σ∈Uσ

i

Eσ i

Here we regard as a subspace of Γ(Σ, Λ0,1 ⊗ w∗ T M ) in a way similar to the first half of this section. A Kuranishi neighborhood Uσ is the set of isomorphism classes of pairs ((Σ , z  ), w ) such that (Σ , z  ) is close to (Σ, z ) in Mmain k+1 with respect to the topology given in Definition 2.1.16 and w satisfies ∂w ≡ 0 mod Eσ . Using the fact that Uσ i is closed, we can choose Uσ so that if σ1 ∈ Uσ2 then Eσ1 ⊆ Eσ2 . (Here we use a map Γ(Σ1 , Λ0,1 ⊗ w1∗ T M ) → Γ(Σ2 , Λ0,1 ⊗ w2∗ T M ) (defined by the parallel transport) to compare Eσ1 and Eσ2 . (σi = ((Σi , zi ), wi ).) Therefore we may assume that if σ1 ∈ Uσ2 , then a neighborhood of σ1 in Uσ1 is contained in Uσ2 . We put this neighborhood Uσ1 ,σ2 which will satisfy the conditions (A1.12) in Section A1. For the case where (Σ, z ) is unstable, we use Theorem 7.1.44 and proceed as follows. (The argument below is a copy of that in Appendix [FuOn99II]. There is an alternative argument which will be similar to Section15 [FuOn99II].) We add some interior marked points z+ so that (Σ, z, z+ ) becomes stable. We may also assume the following: (7.1.48.1) (7.1.48.2)

Any point zi+ of z+ lie on a component where w is nontrivial. w is an immersion at each point of z+ .

By the same reason as in the appendix [FuOn99II], we can make this as⊂ M (a submanifold of sumption without loss of generalities. We choose Q2n−2 i intersects with w(Σ) transvercodimension 2) for each zi+ ∈ z+ such that Q2n−2 i + sally at w(zi ). Using Theorem 7.1.44, we can construct a Kuranishi neighborhood of ((Σ, z, z+ ), w) in Mk+1,m (β), in the same way as in the case m = 0 which we discussed already. (Here m is the order of z+ .) We have the evaluation map evint : Mk+1,m (β) → M m at the interior marked points. We may assume that evint is weakly submersive. Hence we have a Kuranishi structure on (7.1.49)

Mk+1,m (β)

evint

×M m

m 

Qi .

i=1

By (7.1.48.2) it is easy to see that the Kuranishi neighborhood of the point + (((Σ, z, z+ ), w), (w(z1+ ), · · · , w(zm )))

in the space (7.1.49) can be identified with a Kuranishi neighborhood of ((Σ, z), w) in Mmain k+1 (β). Thus we have constructed a Kuranishi neighborhood of the points

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425

((Σ, z), w) where (Σ, z ) is not necessarily stable. The rest of the argument is the same as the stable case. The proofs of Propositions 7.1.1 and 7.1.2 are now complete.    7.1.5. The Kuranishi structure of Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)): A∞ map analog of Stasheff cells. By now, we have equipped the moduli space Mmain k+1 (β) with a Kuranishi structure. Construction of the Kuranishi structures on other types of the moduli spaces that appear in this book can be carried out in the same fashion. We, however, would like to point out that we have carried out a gluing construction in the Bott-Morse setting using the cylindrical model. This gluing construction requires a delicate choice of a family of weighted Sobolev norms on the function spaces, which is essential to obtain various uniform estimates needed in the gluing construction as the moduli space degenerates. This kind of analysis is not new but was used in the literature of the gauge theory (See [Mro89, DoKr90, Fuk96II]) and of the symplectic geometry (See [McSa94, FuOh97]) and become a well-established discipline among the experts in the field for some time ago. Because these analytical details do not seem to be widely known in the symplectic geometry community and cannot be directly borrowed from the above mentioned literature, we have provided an essentially self-contained account of this analytic construction in Subsections 7.1.2 and 7.1.3 for the completeness’ sake and for the reader’s convenience. On the other hand, as far as topology and geometry concerns, there is something novel to show in order to prove that various moduli spaces involved has Kuranishi structures. Those points are not well established in the literature, so we provide detailed explanation on those points. To be more specific, we prove a result which is analogous to Theorem 7.1.44, that is, we show that the moduli space     Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ)) is a smooth manifold in the case when M and L main   are one point. Recall that Mk+1 (M , L , {Jρ }ρ : β; top(ρ)) is the moduli space we used to construct the filtered A∞ homomorphism in Section 4.6. (See Definition 4.6.8, where

(M  , L ) = (ψ(M ), ψ(L)) for a symplectic diffeomorphism ψ. When we do not need to specify the domain and the target of the symplectic diffeomorphism ψ, we sometimes write M, L instead of M  , L . ) We remark that the moduli space Mmain k+1 can be identified with the Stasheff cell [Sta63], which was used to define the notion of A∞ space. (See Definition 7.1.65.) We also recall that we used the moduli space Mmain k+1 (β) to define the operations mk of our filtered A∞ algebra. In a similar way, the moduli space Nk+1 which will turn out to be homeomorphic to the moduli space Mmain k+1 (M, L, {Jρ }ρ : β; top(ρ)) when the target M is a point, is related to the definition of the notion of A∞ map. (See Definition 7.1.66.) We discuss this relationship with the Stasheff cell in Theorem 7.1.51. Let (C 1 , m), (C 2 , m ) be A∞ algebras. We recall that the condition for f∗ (∗ = 1, 2, · · · ) to be an A∞ homomorphism is given by the following formula (7.1.50)

,a

m (f∗ (x(:1) ), · · · , f∗ (x(:) )) = a a

a

±f∗ (x(3:1) , m(x(3:2) ), x(3:3) ), a a a

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 (:1) (:) where Δ−1 x = a xa ⊗ · · · ⊗ xa . In (7.1.52) in Theorem 7.1.51 below, we will describe the boundary of Nk+1 in terms of its cell decomposition each cell of  which consists of a product of Nk +1 or Mmain k +1 with k < k. We use the following notation there. Let k = {1, · · · , k} and k + = k ∪ {∗} where ∗ is a symbol. For each A ∈ (k+ ) with  < k we take a copy of N+1 and denote it by NA . We also fix copies of Mmain +1 indexed by the set A for each A and  main . Moreover for each m we fix a copy (Mmain denote it by Mmain m+1 ) of Mm+1 . A Theorem 7.1.51. For k = 1, 2, · · · , there exists a cell complex Nk+1 diffeomorphic to Dk−1 ( k − 1 dimensional disc) such that its boundary is decomposed to the union of cells described as follows: (7.1.52.1) (7.1.52.2)

Mmain (i+1,··· ,i+) × N(1,··· ,i,∗,i++1,··· ,k) . m   N(i−1 +1,··· ,i ) × (Mmain m+1 ) . Here 1 = 0 , m = k, i < i+1 − 1. i=1

When the target manifold M is a point, the moduli space Mmain k+1 (M, L, {Jρ }ρ : β; top(ρ)) is homeomorphic to the space Nk+1 above. We remark that the right hand side and the left hand side of (7.1.50) correspond to (7.1.52.1) and (7.1.52.2) respectively. Namely the moduli space Mmain (i+1,··· ,i+)  corresponds to m, Mmain corresponds to m , and N corresponds to f. k+1 m+1 The statement of Theorem 7.1.51 is equivalent to saying that Dk−1 has a decomposition into smooth cells described as in (7.1.52). Proof. We first recall the definition of the moduli space Mmain k+1 (M, L, {Jρ }ρ : β; top(ρ)) when the target space M is a point. (β = 0 as a consequence.) Let (Σ, (z0 , · · · , zk )) ∈ Mmain k+1 . Namely Σ is a genus zero bordered Riemann surface and z0 , · · · , zk are the boundary marked points respecting the order, such that (Σ, (z0 , · · · , zk )) is stable. (See Subsection 2.1.2.) Let Σ = ∪Σi be the decomposition into the irreducible components. The stability implies that there is no sphere component, (since we do not put any interior marked points.) We define a partial order < on the set of irreducible components by the one given in Definition 4.6.6. Definition 7.1.53. Let (Σ, 0, there exist X(g) (L) (g = 0, · · · , K) and multisections sd,k,β,P for (d, β) ≤ K with the following properties. (7.2.36.1) (7.2.36.2)

X(g) (L) satisfies Properties 7.2.27. Let Pi ∈ Xd(i) (L), i = 1, · · · , k. We put main,d Mmain,d k+1 (β; P1 , · · · , Pk ) = Mk+1 (β) ×Lk



Pi

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447

and define a multisection sd,k,β,P thereof. sd,k,β,P is transversal to 0. (7.2.36.3) If g = (d, β), then   sd,k,β,P ev0∗ Mmain,d k+1 (L; β; P1 , · · · , Pk ) is decomposed into elements of X(g) (L). Here and henceforth we denote sd,k,β,P := s−1 Mmain,d k+1 (L; β; P1 , · · · , Pk ) (0). d,k,β,P

(7.2.36.4) The multisections sd,k,β,P satisfy Compatibility Conditions 7.2.38 and 30.44 described below. (7.2.36.5) s−1 (0) is in a δ neighborhood of the zero of the original Kuranishi d,k,β,P map. In fact we need to state (7.2.36.3) in a way similar to (3.4.6.3). Namely the zero set s−1 (0) has a triangulation such that each of simplices determine an d,k,β,P element of X(g) (L). Hereafter we do not repeat this remark. Now we precisely state the two compatibility conditions for the multisections. First, we describe how the multisections behave near the boundary of a singular simplex Pi . Let Pi be given by a smooth map f : |Pi | = Δdi → L. We assume pi,m ∈ Δdi for each m and (7.2.37)

lim pi,m = pi,∞ .

m→∞

We assume pi,∞ ∈ ∂Δdi for some i.  We also assume pi,∞ is in the interior of the face Δdi . We can identify the face with the standard di simplex in a canonical way (using the canonical order on  the vertices of the standard simplex Δm ). We regard (Δdi , f ) as a smooth singular  simplex. It follows from (7.2.28.3) that we have g  ≤ d(i) such that (Δdi , f ) ∈  Xg (L). We put Pi = (Δdi , f ) and P  = (P1 , · · · , Pk ). (Note some of Pi may coincide with Pi .) We now describe how the multisection sd,k,β,P behaves in this limit. Note that our Kuranishi structure on Mmain,d k+1 (β) is weakly submersive (in the sense of Definition A1.13) and so satisfies the following: Let p ∈ Mmain,d k+1 (β) and let Up be its Kuranishi neighborhood. Let  (p, (p1 , · · · , pk )) ∈ Mmain,d Pi k+1 (β) ×Lk  and Ui (pi ) a neighborhood of pi in Pi . Then Up ×Lk Ui (pi ) is a Kuranishi neighborhood of (p, (p1 , · · · , pk )) in Mmain,d (β; P , · · · , P ). 1 k k+1 We suppose that (pm , (p1,m , · · · , pk,m )) lies in some Kuranishi neighborhood in the moduli space Mmain,d k+1 (β; P1 , · · · , Pk ) and limm→∞ pm = p∞ . Now our compatibility condition is given by the following: Compatibility Condition 7.2.38. Let pm , and p∞ be as above and let d be a decoration for which Pi ∈ Xd (i) (L). Then lim s (pm , (p1,m , · · · m→∞ d,k,β,P

, pk,m )) = sd ,k,β,P  (p∞ , (p1,∞ , · · · , pk,∞ )).

We next state the second compatibility condition on the multisections along main,d the boundary of Mmain,d k+1 (β). Let (Σ, w, d) ∈ Mk+1 (β). We assume that Σ is

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singular and has a double point on the boundary. Let p0 ∈ ∂Σ be one of the double points. We cut Σ at p0 into two pieces Σ1 and Σ2 . We may assume that the 0-th marked point z0 lies in Σ2 . We restrict w to Σi and will write wi . Then (Σi , wi ) ∈ Mmain ki +1 (βi ) with β1 + β2 = β, k1 + k2 − 1 = k. We write (7.2.39)

(Σ, w) = (Σ1 , w1 ) # (Σ2 , w2 ),

where p0 is the -th marked point of Σ2 . (p0 is the 0-th marked point of Σ1 .) p0 z −1

z

z1 Σ2

Σ1

z0 zk

z+k1 z+k1 −1

Figure 7.2.1 We next fix decorations of (Σ1 , w1 ), (Σ2 , w2 ). Let i ∈ {1, · · · , k1 }. Then the i-th marked point of Σ1 corresponds to some, say the j-th, marked point of Σ. We put (7.2.40.1)

d1 (i) = d(j).

Next let i ∈ {1, · · · , k2 }. If i = , then the i-th marked point of Σ2 corresponds to some, say the j-th, marked point of Σ. We define the decoration dmax by 2 (7.2.40.2) (7.2.40.3)

(i) = d(j) for i =  dmax 2 = (d1 , β1 ) for i = .

, we put (Note d1 was determined by (7.2.40.1).) Using this d1 and dmax 2 (7.2.41)

). (Σ, w, d) = (Σ1 , w1 , d1 ) # (Σ2 , w2 , dmax 2

Now (7.2.41) determines a map (7.2.42)

1 # : Mmain,d k1 +1 (β1 )

ev0

main,dmax 2

×ev Mk2 +1

(β2 ) → ∂Mmain,d k+1 (β).

Proposition 7.1.2 implies that the restriction of the Kuranishi structure of main,dmax main,d1 2 (β2 ) (by the map # ) coincides Mmain,d k+1 (β) to Mk1 +1 (β1 ) ev0 ×ev Mk2 +1 main,dmax 2

1 with the fiber product Kuranishi structure on Mmain,d k1 +1 (β1 ) ev0 ×ev Mk2 +1 Now let Pi ∈ Xd(i) (L), i = 1, · · · , k. We consider   sd ,k ,β ,P 1 1 1 1 (1) , Q = ev0∗ Mmain,d (β ; P , · · · , P ) 1  +k −1 1 k1 +1

(β2 ).

where P(1) = (P , · · · , P+k1 −1 ). The property (7.2.36.3) implies that Q is decomposed into a sum of elements of X(dmax ()) (L). For the simplicity of notation, we write Q in place of the simplex 2

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

449

which appears in the decomposition. We use this kind of abuse of notation frequently from now on. We do not repeat this remark. We set  (7.2.43)

d2 (i) =

dmax (i) if i = , 2 g if i = .

Here g is the generation of Q. We put P(2) = (P1 , · · · , P−1 , Q, P+k1 , · · · , Pk ), and consider sd2 ,k2 ,β2 ,P (2) . We remark that the obstruction bundle of the Kuranishi structure of Mmain,d k+1 (β) at its boundary can be identified with the direct sum of the obstruction bundles main,dmax 1 2 E(β1 )k1 +1 and E(β2 )k2 +1 of Mmain,d (β2 ), respectively. This k1 +1 (β1 ) and Mk2 +1 fact follows from the definition of the fiber product of Kuranishi structure. Then the second compatibility condition can be stated as follows: Compatibility Condition 7.2.44. Decompose the pull-back of sd,k,β,P by the map # into s(1) ⊕ s(2) ∈ E(β1 )k1 +1 ⊕ E(β2 )k2 +1 . Then we have (1) s(1) = sd1 ,k1 ,β1 ,P (1) . (2) At the point where s(1)= sd1 ,k1 ,β1 ,P (1) vanishes, we have s(2)= sd2 ,k2 ,β2 ,P (2). We remark that sd2 ,k2 ,β2 ,P (2) is defined only on the point where sd1 ,k1 ,β1 ,P (1) vanishes. In other words it is defined only at the point that corresponds to a main,dmax point of the Kuranishi neighborhood of Mk2 +1 2 (β2 ; P(2) ). We remark that Q is included in P(2) and that Q is the virtual fundamental chain of the zero set of sd1 ,k1 ,β1 ,P (1) . This is the reason why we state (2) as above. The value of s(2) where s(1) does not vanish is not related to the zero set of s. So this restriction does not matter at all for our construction. Hence we have finished the precise description of Proposition 7.2.35. The proof of this proposition is in order in the next subsection. 7.2.5. Proof of Proposition 7.2.35. We will construct sd,k,β,P by the induction over (d, β). In order to make sure that the induction works, we need the following lemmas. Lemma 7.2.45. β ≥ −1. β = −1 if and only if β = β0 . Proof. Obvious from definition.



Lemma 7.2.46. If d∅ is the unique function ∅ → Z≥0 , then (d∅ , β) ≥ 0. If d = d∅ , then (d, β) > max d. Proof. The second assertion is obvious from the definition (7.2.33) of (d, β) and Lemma 7.2.45. (We recall that if β = β0 then k ≥ 2 in (7.2.33) by the stability condition.) The first assertion holds by (7.2.34) since β = β0 in this case. 

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Lemma 7.2.47. Let (Σ, w, d) = (Σ1 , w1 , d1 ) # (Σ2 , w2 , d2 ) be as in (7.2.41), main,d2 1 β1 + β2 = β with (Σ1 , w1 , d1 ) ∈ Mmain,d k1 +1 (β1 ), (Σ2 , w2 , d2 ) ∈ Mk2 +1 (β2 ) and (Σ, w, d) ∈ Mmain,d k+1 (β). Let d2 be as in (7.2.43). Then for each of i = 1, 2, one of the following alternatives holds: (7.2.48.1) (7.2.48.2) (7.2.48.3)

(di , βi ) < (d, β). (di , βi ) = (d, β) and ki + βi  < k + β. (di , βi ) = (d, β), ki + βi  = k + β, and βi  < β.

, β2 ) ≥ (d2 , β2 ), it suffices to consider the case dmax = Proof. Since (dmax 2 2 d2 . We will assume this for the rest of the proof. We have k = k1 + k2 − 1, β ≥ β1  + β2  + 1.

(7.2.49.1) (7.2.49.2) Moreover it follows that (7.2.50.1) (7.2.50.2)

ki ≤ 1 ⇒ βi = β0 , k2 ≥ 1, k1 ≥ 0. z −1

z

z1 β2

β1 z+k1

z0 zk

z+k1 −1

Figure 7.2.2 We put d1 = max{d1 (i) | i = 1, · · · , k1 }, d = max{d(1), · · · , d(k)}.

d2 = max{d2 (i) | i = , i = 1, · · · , k2 },

When the set in the right hand side is empty, namely if k1 = 0 and k2 = 1 (or equivalently if k = 0), we set d1 = 0, d2 = 0 or d = 0, respectively. We then have d = max{d1 , d2 }. Now we prove (7.2.48) for i = 1. We first consider the case when β2 = β0 . Then, k2 ≥ 2 and hence we obtain (d1 , β1 ) = k1 + β1  + d1 < k + β + d = (d, β). Namely (7.2.48.1) holds. In case β2 = β0 we have β1  < β and k2 ≥ 1. Therefore we have (d1 , β1 ) = k1 + β1  + d1 < k + β + d = (d, β), which also implies (7.2.48.1). We next prove (7.2.48) for i = 2. We remark that d2 () = (d1 , β1 ) = k1 + β1  + d1 . We will consider two cases d = d1 and d = d2 separately.

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First consider the case d = d1 . Then from (7.2.49) we have (d, β) = k + β + d ≥ k1 + k2 + β1  + β2  + d1 ≥ k2 + β2  + (d1 , β1 ). We remark that d2 () = (d1 , β1 ) ≥ d1 = d ≥ d2 , where the first inequality comes from Lemma 7.2.46. Hence max{d2 (i) | i = 1, · · · , k2 } = (d1 , β1 ). Therefore (d2 , β2 ) = k2 + β2  + (d1 , β1 ) ≤ (d, β).

(7.2.51)

If strict inequality < holds in (7.2.51), we have (7.2.48.1). Now suppose the equality holds in (7.2.51). If k1 > 1, it follows k = k1 + k2 − 1 > k2 . We also have β ≥ β2 . Then (7.2.48.2) holds. If k1 = 1, we have β1 = β0 . This implies β > β2  and k = k2 . Hence (7.2.48.2) again holds. If k1 = 0, we have d1 = d = 0 ≤ (d1 , β1 ). Therefore, since the equality is assumed to hold in (7.2.51), we derive k + β = (d, β) = k2 + β2  + (d1 , β1 ) ≥ k2 + β2 . Moreover k2 = k + 1 > k. Hence (7.2.48.2) or (7.2.48.3) holds. Next we consider the case d = d2 > d1 . Again we split this into two cases (7.2.52.1) (7.2.52.2)

d2 ≤ (d1 , β1 ), d2 > (d1 , β1 ).

from For the case (7.2.52.1), we have (d1 , β1 ) = d2 , since we assume d2 = dmax 2 the beginning of this proof. So we have d2 ≤ (d1 , β1 ) = dmax () = d2 () ≤ d. 2 Then we obtain (d, β) = k + β + d ≥ k1 + k2 + β1  + β2  + d2 > k1 + k2 + β1  + β2  + d1 = k2 + β2  + (d1 , β1 ) = k2 + β2  + d2 = (d2 , β2 ). (7.2.48.1) holds. For the case (7.2.52.2), we have (d, β) = k + β + d ≥ k1 + k2 + β1  + β2  + d2 ≥ k2 + β2  + d2 = (d2 , β2 ) where we used the inequality k1 + β1  ≥ 0 which follows from the stability condition. If the equality holds, we have k1 + β1  = 0 and so β1 = β0 , k1 = 0. Therefore, k = k2 − 1, β > β2 . It follows that (7.2.48.2) or (7.2.48.3) holds. The proof of Lemma 7.2.47 is now complete.  Now we start the proof of Proposition 7.2.35.

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We fix two positive constants δ and K. We will construct a system of multisections sd,k,β,P for each (d, β) ≤ K. The construction will be carried out by an induction. To properly organize the induction steps, we introduce an order < on  ). We say (d , k , β  , P  ) < (d, k, β, P ) if and only if one of the the set of (d, k, β, P following is satisfied. (7.2.53.1) (7.2.53.2) (7.2.53.3) (7.2.53.4)  deg Pi .

(d , β  ) < (d, β). (d , β  ) = (d, β) and k + β   < k + β. (d , β  ) = (d, β), k + β   = k + β and β   < β.  (d , β  ) = (d, β), k + β   = k + β, β   = β and deg Pi >

We will prove the following two statements inductively: (7.2.54.1)

Under the assumptions that all of the multisections sd ,k ,β  ,P  are de-

 ) and that Xg (L) are defined for g < (d, β), fined for (d , k , β  , P  ) < (d, k, β, P we define sd,k,β,P satisfying the required properties. (7.2.54.2) Under the assumption that all of sd,k,β,P for g = (d, β) are defined, we define Xg (L). We start with defining X0 (L) and sd,k,β,P for (d, β) = 0. Lemma 7.2.46 implies that if (d, β) = 0 then d = d∅ and β = 0. (In particular, β = β0 .) We consider the moduli space Mmain,d 0+1 (β). Since β = 0, this moduli space cannot bubble-off and so does not have a boundary and hence the compatibility condition for Mmain,d (β) is void. Therefore we can freely take a multisection sd∅ ,0,β, 1 on 1 it. (Here 1 is the element corresponding to the unique map ∅ → S(L).) We may choose it so that it is transversal to zero. We then take all the chains of the form ev0∗ (s−1 (0)) with β = 0. We choose a smooth triangulation thereof d∅ ,0,β, 1 and consider all the singular simplices appearing in the triangulation. We use the method explained in Remark 7.2.15 (3) to identify each simplex to a singular simplex. (We do not repeat this remark any more since it is always applied in a similar situation.) If a simplex is equal to one that already appeared before, we discard it. We have thus obtained X0 (L). This is the first step of the induction. Next we describe the induction step of (7.2.54.1). Noting that we have d(i) < (d, β) by Lemma 7.2.46, putting the hypothesis Pi ∈ Xd(i) (L) makes sense. We next examine Compatibility Conditions 7.2.38 and 7.2.44. Consider the limit of a sequence of the points (pm , (p1,m , · · · , pk,m )) in the Ku ranishi neighborhood of Mmain,d k+1 (L; β)×Lk Pi . Suppose that (pm , (p1,m , · · · , pk,m )) converges to a boundary point. Then as we mentioned before, the point falls into the contexts of Compatibility Conditions 7.2.38 or 7.2.44. Then it follows from Lemma 7.2.47 that the value of sd,k,β,P at the limit is already determined by the induction hypothesis. The following lemma is essential here: Lemma 7.2.55. The boundary values of sd,k,β,P determined by Compatibility Conditions 7.2.38 and 7.2.44 are consistent at the points where the two overlap. Assuming Lemma 7.2.55 for the moment, we will complete the proof of Proposition 7.2.35.

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We can apply Theorem 3.11 in [FuOn99II] (that is Theorem A1.23 in Section A1) to extend the boundary value and obtain sd,k,β,P that is transversal to 0. The zero set s−1 (0) has a triangulation (Lemma A1.26). We can take the d,k,β,P triangulation so that it coincides with one on the boundary which is given by the earlier step of induction. Let g = (d, β). We take all the simplex of s−1 (0) d,k,β,P and regard them as a smooth singular simplices. Among them we remove singular simplices which are elements of X(g−1) (L) or are chosen in the earlier steps. All the other simplex are regarded as generation g, and as an elements of Xg (L). In this way, we can choose the subset so that the spanning property of X(g) (L) holds on s−1 (0). We have thus completed the step (7.2.54.1). d,k,β,P We remark that in case of dim P1 = dim P2 = 0, β = β0 , our moduli space M2+1 (β0 ; P ) has no boundary and hence the compatibility condition is void. Then we can freely take multisection sd,2,β0 ,P on it. The step (7.2.54.2) is easier. We consider the union of the chains of generation g obtained in step (7.2.54.1). It has all the required properties except (7.2.28.2). So we suitably add countably many singular simplices so that (7.2.28.2) is satisfied. (Clearly we can do this while keeping the properties (7.2.28.3) intact. See the last step of the proof of Lemma 7.2.12.) Finally we check if the above construction can be done in the given Kuranishi neighborhood, i.e. if the constructed multisection satisfies (7.2.36.5). This concerns the problem we mentioned in Subsection 7.2.3. Note that we have an upper bound C(K) of the number of inductive steps in the proof of the existence of sd,k,β,P for all (d, β) ≤ K. (We remark that there are only finitely many β ∈ G(L) with (d, β) ≤ K.) We start with a multisection that is C(K) δ close to the original section (that is the Kuranishi map in the definition of Kuranishi structure). (Here  is a sufficiently small number which we choose later.) We organize the above inductive steps so that the multisection also satisfies the following: (7.2.56.1) The system of multisections constructed in the I-th inductive steps is assumed to be C(K)−I δ close to the original Kuranishi map. Its zero set is also assumed to be C(K)−I δ close to zero set of the original Kuranishi map. (7.2.56.2) When we are constructing the multisection sd,k,β,P at the (I + 1)-th step of the induction, we took the fiber product of the multisections constructed in  the steps ≤ I to obtain a multisection s∂d,k,β,P on the boundary of ∂Mmain,d k+1 (β; P ). (7.2.56.3)

Then, s∂d,k,β,P is CC(K)−I δ close to the original Kuranishi map and its

zero point set is CC(K)−I δ close to zero set of the original Kuranishi map. Here C is a number which we can bound in terms of the moduli spaces we used and K. This is a consequence of the construction. (7.2.56.4) Therefore, we can choose an extended multisection sd,k,β,P , so that it is C 2 C(K)−I δ close to the Kuranishi map. And its zero set is C 2 C(K)−I δ-close to the zero of the Kuranishi map. This is again a consequence of the construction. (7.2.56.5) By choosing C 2  < 1, (7.2.56.4) implies that the induction works. Moreover by choosing δ small, we may assume that the zero set of the perturbed Kuranishi map is away from the end of the Kuranishi neighborhood. So the problem mentioned Subsection 7.2.3 does not occur.

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We note that in general we need to handle infinitely many smooth singular simplices Pi . However obstruction bundles are pull-backed from the one defined over the moduli spaces of pseudo-holomorphic discs and Kuranishi neighborhoods are defined by taking the (transversal) fiber product of the simplices Pi and Kuranishi neighborhoods of the moduli spaces. Therefore under the bound (d, β) ≤ K we need to study only finitely many moduli spaces and hence we can keep the perturbed moduli spaces away from the boundaries of Kuranishi neighborhoods. This enables us to choose the constants C and etc. independently of the singular simplices Pi . (7.2.56) implies (7.2.36.5) also. The proof of Proposition 7.2.35 is now complete.  Remark 7.2.57. The argument of (7.2.56) can be used in a more general situation to keep the perturbed multisection being close enough to the original Kuranishi map so that its zero set does not reach the end of the Kuranishi neighborhood, as long as we have only finitely many steps to construct the perturbed multisections. So we do not repeat the same argument in later subsections. Proof of Lemma 7.2.55.

z+−1

p 0

p0

z+−2

z

z −1

z1 Σ2

Σ3

Σ4

z0 zk

z++k4−2

z+k4+ k −1

z+k4+k −2

3

3

p 0

z+−1

z+−2 z z −1

z1 z0

z++k4−2

z +k4+ k3−1 z++k4++ k −2

zk

3

Figure 7.2.3 There are 3 different ways how the compatibility conditions enter in the proof. The first case requires us to apply Compatibility Condition 7.2.38 twice. The second involves both Compatibility Condition 7.2.38 and Compatibility Condition 7.2.44 once. The third involves Compatibility Condition 7.2.44 twice. We will discuss only the third case. Since the other two cases are easier, we omit their consideration. To simplify the notations, we restrict to the following case: (The general case is similar.) Σ has two singular points, say p0 and p0 at boundary.

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455

We assume that if we split Σ at p0 into the union of Σ1 and Σ2 , then Σ2 contains the 0-th (the last) marked point and p0 is contained in Σ1 . On the other hand, we assume that if we cut Σ1 at p0 into the union of Σ3 and Σ4 , p0 is contained in Σ3 . Let p0 be the  -th marked point of Σ3 and p0 the -th marked point of Σ2 . In this circumstance, we denote Σ = (Σ4 # Σ3 ) # Σ2 . Compare this with (7.2.41) and see Figure 7.2.3. To prove Lemma 7.2.55, we will compare the multisections on the boundary corresponding to two types of decompositions of Σ: one is Σ = (Σ4 # Σ3 ) # Σ2 and the other Σ = Σ4 #+ −1 (Σ3 # Σ2 ). Let Σi have ki + 1 boundary marked points. Then Σ has k + 1 boundary marked points with k = k2 + k3 + k4 − 2 and Σ1 = Σ4 # Σ3 has k1 + 1 marked points with k1 = k3 + k4 − 1. We assign βi ∈ π2 (M, L) to Σi . Then β1 = β3 + β4 and β = β2 + β3 + β4 are assigned to Σ1 and Σ, respectively. We also put a d to Σ. Then, decoration di on Σi will be determined as follows. d1 (i) = d(i +  − 1),

d4 (i) = d(i +  +  − 2).

⎧ i = 1, · · · ,  − 1. ⎪ ⎨ d(i) d2 (i) = g1 i= ⎪ ⎩ d(i + k1 − 1) i =  + 1, · · · , k2 . ⎧ i = 1, · · · ,  − 1. ⎪ ⎨ d1 (i) d3 (i) = i =  g4 ⎪ ⎩ d1 (i + k4 − 1) i =  + 1, · · · , k3 . Here g1 ≤ (d1 , β1 ) and g4 ≤ (d4 , β4 ) will be determined later. Let w : Σ → M be a map of homology class β. We restrict w to Σi and obtain a map wi : Σi → M of homology class βi . Write   (7.2.58) (Σ, w, d) = (Σ4 , w4 , d4 ) # (Σ3 , w3 , d3 ) # (Σ2 , w2 , d2 ). For Pi ∈ Xd(i) (L), we put P(4) = (P+ −1 , · · · , P+ +k4 −2 ),   sd ,k ,β ,P 4 4 4 4 (4)  . Q4 = ev0∗ Mmain,d (β ; P ) 4 (4) k4 +1 Then by the induction hypothesis, the chain Q4 is decomposed to a finite number of singular simplices with their generations ≤ (d4 , β4 ). For simplicity, we write Q4 in place of writing its simplices and assume Q4 ∈ X( (d4 ,β4 ) ) (L), by abuse of notation. We denote the generation of Q4 by g4 . We next put P(3) = (P , · · · , P+ −2 , Q4 , P+ +k4 −1 , · · · , P+k3 +k4 −2 )   3 (3) )sd3 ,k3 ,β3 ,P(3) Q3 = ev0∗ Mmain,d (β ; P 3 k3 +1

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and P(1) = (P , · · · , P+k3 +k4 −2 ),   1 (1) )sd1 ,k1 ,β1 ,P(1) . Q1 = ev0∗ Mmain,d (β ; P 1 k1 +1 (Note that  + k3 + k4 − 2 =  + k1 − 1.) See Figures 7.2.4 and 7.2.5. P(4)

P(3)

P

P

P + −1

P

P

P−1 P1

P+−2

Σ4

Σ2

Σ3

z0

P P P

P P++k4 −2  ++k4 −1

Pk− 1

P P P + k4 +k −2 3

P(3)

Q4

Q3

Figure 7.2.4 P(1)

P + −1 P+ −2

P

P

P−1 P1

P(1)

Σ3

Σ4

P

P++k

− 4 2

P ++k

4 −1

Σ2

Pk− 1

P

P P  + k4 +k

P(1) Q1

Figure 7.2.5

z0

3 −2

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Q1 is decomposed to elements of X( (d1 ,β1 ) ) (L). For simplicity, we assume Q1 ∈ X( (d1 ,β1 ) ) (L). We define g1 to be the generation of Q1 . We then put P(2) = (P1 , · · · , P−1 , Q1 , P+k3 +k4 −1 , · · · , Pk ),  P(2) = (P1 , · · · , P−1 , Q3 , P+k3 +k4 −1 , · · · , Pk ). (See Figure 7.2.5.) As in Subsection 7.2.4, we consider a sequence (pm , (p1,m , · · · , pk,m ))m=1,2,··· of points in a Kuranishi neighborhood of Mmain,d k+1 (β; P1 , · · · , Pk ). We put p∞ = (Σ, w, d) and let pi,∞ ∈ Pi such that (p∞ , (p1,∞ , · · · , pk,∞ )) = (p∞ , p∞ ) lies in the Kuranishi neighborhood. We apply Compatibility Condition 7.2.44 twice according to the splitting (7.2.58) and obtain the following. sd,k,β,P (p∞ , p∞ ) =sd4 ,k4 ,β4 ,P (4) (p(4),∞ , p(4),∞ ) ⊕ sd3 ,k3 ,β3 ,P (3) (p(3),∞ , p(3),∞ )

(7.2.59)

⊕ sd2 ,k2 ,β2 ,P (2) (p(2),∞ , p(2),∞ ). Here p(i),∞ = (Σi , wi , di ) and p(i),∞ = (wi (z1 ), · · · , wi (zki )) where zi are boundary marked points. We next decompose (Σ, w, d) in a different way. Namely we consider   (Σ, w, d) = (Σ4 , w4 , d4 ) #+ −1 (Σ3 , w3 , d3 )) # (Σ2 , w2 , d2 ) , where we put ⎧ i = 1, · · · ,  − 1. ⎪ ⎨ d(i)  i = . Here g3 ≤ (d3 , β3 ) is the generation of Q3 . d2 (i) = g3 ⎪ ⎩ d(i + k1 − 1) i =  + 1, · · · , k2 . Then we obtain sd,k,β,P (p∞ , p∞ ) =sd4 ,k4 ,β4 ,P (4) (p(4),∞ , p(4),∞ ) ⊕ sd3 ,k3 ,β3 ,P (3) (p(3),∞ , p(3),∞ )

(7.2.60)

⊕ sd ,k2 ,β2 ,P  (p(2),∞ , p(2),∞ ). 2

(2)

The difference between (7.2.60) and (7.2.59) is as follows. In (7.2.60) we applied  sd ,k2 ,β2 ,P  to get the third factor. (There are primes in d2 and in P(2) .) On the 2

(2)

other hand, we applied sd2 ,k2 ,β2 ,P (2) to obtain the third factor in (7.2.59). (There are no primes here.) The reason we do so in (7.2.59) is that we regard (Σ4 , w4 , d4 ) # (Σ3 , w3 , d3 ) 1  as an element of (a Kuranishi neighborhood of) Mmain,d k1 +1 (L; β1 ; P(1) ) in (7.2.58). Lemma 7.2.55 in our case means that the right hand side of (7.2.59) is equal to the right hand side of (7.2.60). To prove this, we recall that Q3 lies in the boundary of Q1 . Hence it is a component of the decomposition of ∂Q1 . Therefore Compatibility Condition 7.2.38, which we assumed as a part of induction hypothesis, implies the required property, that is, the multisections coincide. The proof of Lemma 7.2.55 is now complete. 

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7.2.6. Filtered An,K structures. In subsections Subsections 7.2.6-8, we deal with the problem mentioned in Subsection 7.2.3. In this subsection we generalize the notion of the AK structure to its filtered version. We will use an idea similar to the induction process adopted in Subsections 7.2.4,7.2.5 for this construction of filtered An,K structures. Let G be a monoid and πG,G : G −→ G ⊂ R≥0 × 2Z be a homomorphism to a submonoid satisfying Condition 3.1.6 such that the inverse image of each element of G is of finite order and that the inverse image −1 πG,G ({(0, 0)}) consists of one element β0 . Definition 7.2.61. For β ∈ G, we define   n 

 β = sup n  ∃βi ∈ G − {β0 }, βi = β 

+ [π1 (β)] − 1,

i=1

if β = β0 and β0  = −1. Here π1 (β) is the R≥0 component of πG,G (β) and [x] is the largest integer less than or equal to x. We remark: β1 + β2  ≥ β1  + β2  + 1.

(7.2.62)

Definition 7.2.63. We define a partial order < on (G × Z≥0 ) \ {(β0 , 0)} such that (β1 , k1 ) > (β2 , k2 ) if and only if one of the following holds: (7.2.64.1) (7.2.64.2)

β1  + k1 > β2  + k2 . β1  + k1 = β2  + k2 , β1  > β2 .

If β1  + k1 = β2  + k2 and β1  = β2 , we write (β1 , k1 ) ∼ (β2 , k2 ). We denote by (β1 , k1 )  (β2 , k2 ) if either (β1 , k1 ) > (β2 , k2 ) or (β1 , k1 ) ∼ (β2 , k2 ). In case n, n , m are non-negative integers, we use the notation (β, k) < (n, k), (m, k) < (n , k ) in a similar way. (For the case of pairs (m, k) and (n , k ), the relation ∼ is just the equality in the usual sense.) The following lemma is obvious from definition. Lemma 7.2.65. If (βi , ki ) ∈ (G × Z≥0 ) \ {(β0 , 0)}, then (β1 , k1 )  (β1 + β2 , k1 + k2 − 1). If the equality ∼ holds, then (β2 , k2 ) = (β0 , 1). Now we define the notion of G-filtered An,K structure. Let C be a free filtered ˆ Λ0,nov ∼ graded Λ0,nov module. We take a graded Q vector space C such that C ⊗ = ˆ is the completion of the algebraic tensor product.) We assume that C C. (Here ⊗ is a cochain complex and denote by δ its coboundary operator. Suppose that, for (β, k) ∈ (G × Z≥0 ) \ {(β0 , 0)}, we have an operation (7.2.66) of degree +1.

mk,β : Bk C[1] → C[1]

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Definition 7.2.67. Let n, K be non negative integers and G a monoid as above. A structure of the G-filtered An,K algebra on C is a family of operations mk,β for (β, k)  (n, K), (β, k) = (β0 , 0) with the following property: For any (β, k)  (n, K), we have (7.2.68)





(−1)deg



(1)

xi

  (1) (2) (3) = 0, mk2 ,β2 xi , mk1 ,β1 (xi ), xi

β1 +β2 =β,βi ∈G;k1 +k2 =k+1 i

where Δ2 (x) =



(1)

i

xi

(2)

⊗ xi

(3)

⊗ xi .

We remark that the left hand side of (7.2.68) is a finite sum by Condition 3.1.6 and is well defined by Lemma 7.2.65. The following lemma is easy to see from definition. Hereafter we fix G and so call a G-filtered An,K structure etc. a filtered An,K structure etc.. Lemma 7.2.69. Assume that C has a structure of filtered An,K algebra for any non-negative integers n, K such that for each (n, K) < (n , K  ) the restriction of filtered An ,K  structure coincides with filtered An,K structure. Then

T λ(β) eμ(β)/2 mk,β mk = β∈G

is a filtered A∞ structure. Here λ(β) = πG,G,1 (β) ∈ R≥0 is the first factor of πG,G (β) and μ(β) = πG,G,2 (β) ∈ 2Z is the second one. Let C, C  be filtered An,K algebras and consider a series of Q-linear homomorphisms of degree zero  fk,β : Bk C[1] → C [1]. (We assume f0,β0 = 0.) Definition 7.2.70. {fk,β } defines a filtered An,K homomorphism, if the identity





  (1) (m) mm,β  fk1 ,β1 (xi ), · · · , fkm ,βm (xi )

m,i β  +β1 +···+βm =β k1 +···+km =k

=



β1 +β2 =β,k1 +k2 =k+1

 

 (1) (1) (2) (3) (−1)deg xi fk2 ,β2 xi , mk1 ,β1 (xi ), xi i

holds for (β, k)  (n, K). We can prove a lemma similar to Lemma 7.2.69 for filtered An,K homomorphisms. We can define the notion of a model of [0, 1] × C for a filtered An,K algebra C in an obvious way. If C is a filtered An,K algebra, then P oly([0, 1], C) and C [0,1] which we introduced in Section 4.2, are both filtered An,K algebras and are models of [0, 1] × C. We can use them to define a filtered An,K homotopy between filtered An,K homomorphisms. We can show that it is an equivalence relation by using a result similar to Theorem 4.2.34. We can then define filtered An,K homotopy equivalences and an analog of Theorem 4.2.45 also holds.

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Remark 7.2.71. We remark that to prove An,K analog of Theorems 4.2.34 and 4.2.45, we use an induction over the order K, then mgeo k,β (P1 , · · · , Pk ) may not be defined. This is the reason why we need to modify mgeo k,β to mk,β . We will mention an alternative method in Remark 7.2.88. Proof. The proof of Proposition 7.2.78 follows the idea used in the construction of the canonical model carried out in Section 5.4. We first take a constant c1 such that β + k < c1

(7.2.80) for each (β, k)  (n, K) and put

gm = g0 + mc1 . We will fix another positive integer M later in the proof so that the corresponding gM satisfies gM ≤ K. We remark that C(gm ) (L; Q) is a Q-vector space generated by X(gm ) (L). We consider a chain map ΠC(g1 ) (L;Q) : C(gM ) (L; Q) → C(gM ) (L; Q) which satisfies Π2C(g

1)

(L;Q)

= ΠC(g1 ) (L;Q) ,

Image ΠC(g1 ) (L;Q) = C(g1 ) (L; Q).

Lemma 7.2.81. There exist such ΠC(g1 ) (L;Q) and a sequence of Q-linear maps k−1 k Gk : C(g (L; Q) → C(g (L; Q) M) M)

such that k (L; Q). (7.2.82.1) 1 − ΠC(g1 ) (L;Q) = −m1 ◦ Gk − Gk+1 ◦ m1 on C(g M) (7.2.82.2) Gk ◦ Gk+1 = 0. (7.2.82.3) The filtration {C(gm ) (L; Q)}m of C(gM ) (L; Q) is preserved by Gk . Namely we have: Gk (C(gm ) (L; Q)) ⊆ C(gm ) (L; Q). (7.2.82.4) Gk = 0 on C(g1 ) (L; Q).

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

463

Proof. We can find Am ⊂ C(gm ) (L; Q) such that C(gm ) (L; Q) = C(gm−1 ) (L; Q) ⊕ Am ⊕ m1 (Am ). It follows that gM 

C(gM ) (L; Q) = C(g1 ) (L; Q) ⊕

(Am ⊕ m1 (Am )).

m=g1 +1

We then put

⎧ ⎪ ⎨0 ΠC(g1 ) (L;Q) (x) = x ⎪ ⎩ 0

if x ∈ Am , if x ∈ C(g1 ) (L; Q), if x ∈ m1 (Am ),

⎧ ⎪ ⎨0 G(x) = 0 ⎪ ⎩ −(m1 |Am )−1 (x) It is easy to see that ΠC(g1 ) (L;Q) and G have and

if x ∈ Am , if x ∈ C(g1 ) (L; Q), if x ∈ m1 (Am ). the required properties.



Lemma 7.2.83. If xi ∈ C(gm ) (L; Q), i = 1, · · · , k, (β, k)  (n, K) and gm ≤ K, then mgeo k,β (x1 , · · · , xk ) ∈ C(gm+1 ) (L; Q). This is an immediate consequence of Proposition 7.2.35 and our choice of c1 , gi . Now we use the construction of canonical model in Section 5.4 using ΠC(g1 ) (L;Q) in place of ΠH . (We use Lemma 7.2.81 for Lemma 5.4.28.) Namely we define ik,β , mk,β for (β, k)  (n, K) as follows. We use the same notations as those of Subsection 5.4.4. Let Γ = (T, i, v0 , Vtad , η) ∈ G+ k+1 . We put

βη(v) . β(Γ) = 0 (T ) v∈Cint

We put G(L) = {β0 , β1 , · · · , } with E(βi ) ≤ E(βi+1 ). Now for Γ ∈ G+ k+1 with (β(Γ), k)  (n, K) we define iΓ , mΓ by induction as follows. ⎧

(1) () ⎪ m (x) = ΠC(g1 ) (L;Q) ◦ mgeo ⎪ ,βη(v1 ) (iΓ1 (xa ) ⊗ · · · ⊗ iΓ (xa )), ⎨ Γ a (7.2.84)

(1) () ⎪ ⎪ G ◦ mgeo ⎩ iΓ (x) = ,βη(v ) (iΓ1 (xa ) ⊗ · · · ⊗ iΓ (xa )). a

1

Here Γi are as in Formula (5.4.36). (Note we use the symbol f in Section 5.4. But we use the symbol i here. This is because we need to use the symbol f for another An,K homomorphism in Subsection 7.2.9.) We also put (7.2.85)

mΓk+1,η = ΠC(g1 ) (L;Q) ◦ mgeo k,βη(v) ,

iΓk+1,η = G ◦ mgeo k,βη(v) .

to initiate the induction. (Here notations are as in (5.4.34).)

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We remark that mgeo is defined only partially. So we need to show that the right hand side of (7.2.84) is well defined if x ∈ Bk C(g1 ) (L; Q). For this purpose, we will prove the following Lemma 7.2.86. For Γ = (T, i, v0 , Vtad , η) ∈ G+ k+1 we put m(Γ) = #{Γ | Γ < Γ}, where < is as in (5.4.33). Lemma 7.2.86. If x ∈ Bk C(gm ) (L; Q), gm+m(Γ) < K and  + η(v1 ) < K then (1) () mgeo ,βη(v ) (iΓ1 (xa ) ⊗ · · · ⊗ iΓ (xa )) 1

is well-defined and contained in C(gm+m(Γ) ) (L; Q). Moreover mΓ (x), iΓ (x) is well defined and iΓ (x) ∈ C(gm+m(Γ) ) (L; Q) for k + β(Γ) ≤ K. (Γ ∈ G+ k+1 .) Proof. We remark that the second half of the lemma follows from the first half of the lemma, (7.2.84) and (7.2.82.3). We start the induction from Γ = Γk+1,η . (Recall the definition of Γk+1,η from the paragraph right after (5.4.33).) In this case, the first half of the lemma follows from Lemma 7.2.83. Next assuming that the lemma has been proved for Γ with Γ < Γ, we prove the lemma for Γ. Let Γi be as in Formula (7.2.84). By Lemma 5.4.35, we have Γi < Γ. Therefore applying the induction hypothesis we derive iΓi (x(i) a ) ∈ C(gm+m(Γi ) ) (L; Q) ⊂ C(gm+m(Γ)−1 ) (L; Q). Lemma 7.2.83 now implies (1) () mgeo ,βη(v ) (iΓ1 (xa ) ⊗ · · · ⊗ iΓ (xa )) ∈ C(gm+m(Γ) ) (L; Q). 1

 Now we are ready to fix the integer M . We take " ! M = sup m(Γ) | k = 0, 1, 2, · · · , Γ ∈ G+ k+1 , (β(Γ), k)  (n, K) + 1. Then Lemma 7.2.86 implies that for x ∈ Bk (C(g1 ) (L; Q)), k = 0, 1, 2, · · · , Γ ∈ G+ k+1 , (β(Γ), k)  (n, K), mΓ (x), iΓ (x) are well-defined. So we put

mΓ (x). (7.2.87) mk,β (x) = Γ∈G+ k+1 ;β(Γ)=β

It follows from Lemma 7.2.86 that the right hand side of (7.2.87) is well-defined. By definition, we obtain mk,β (x) ∈ C(g1 ) (L; Q). We can prove that mk,β satisfies the An,K -formula in the same way as the proof of Theorem 5.4.2.

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

465

If x ∈ Bk (C(g0 ) (L; Q)), then mgeo k,βη(v) (x) ∈ C(g1 ) (L; Q). Then from (7.2.82.4) we obtain iΓk+1,η (x) = 0. Starting from this, we prove iΓ (x) = 0 by an induction. Then by definition, we obtain mk (x) = mgeo k (x). The proof of Proposition 7.2.78 is now complete.



In the course of the proof we have also proved the following Proposition 7.2.78bis. In the situation of Proposition 7.2.78, we have homomorphisms ik,β such that the identity  



(1) (m) mgeo ) m,β  ik1 ,β1 (xi ), · · · , ikm ,βm (xi m,i β  +β1 +···+βm =β k1 +···+km =k

=





β1 +β2 =β,k1 +k2 =k+1

i

(−1)deg



(1)

xi

  (1) (2) (3) ik2 ,β2 xi , mk1 ,β1 (xi ), xi

holds for (β, k)  (n, K), x ∈ Bk (C(g1 ) (L; Q)). Moreover, if (β, k) = (β0 , 1), we have ik,β = 0 on Bk (C(g0 ) (L; Q)). We remark that the domain of ik,β is described in Lemma 7.2.86. The welldefinedness of both sides of the above formula is a part of the statement. Remark 7.2.88. In this subsection, we modified the operations which are defined by Proposition 7.2.35 algebraically to obtain a filtered An,K algebra. We remark that there is an alternative way to obtain a filtered An,K algebra by modifying the proof and the statement of Proposition 7.2.35 as follows: Proposition 7.2.35 was proved by an induction with finitely many steps in which we used the finiteness of the inductive steps in (7.2.56) to control the distances of the perturbed moduli spaces and multisections from the original ones, respectively. For this purpose we need to restrict ourselves to constructing the perturbation of  ) so that not only k + β but also the generations of Pi are bounded Mk+1 (L; β; P by K. We can modify the conditions in (7.2.56) which allows us to repeat the construction of multisection on Mk+1 (L; β; P ) for infinite number of Pi ’s but for (k, β) satisfying k + β ≤ K. For this purpose, we replace the condition (7.2.56.1) by the following, which we can ensure by an inductive argument:  ) is in an K−(k+ β ) δ neighborhood The multisection sd,k,β,P of Mk+1 (L; β; P of the original Kuranishi map. The perturbed moduli space Mk+1 (L; k, β; P )sd,k,β,P is in an K−(k+ β ) δ neighborhood of the original Kuranishi map.

(∗)

Here in (∗) the number K−(k+ β ) δ is independent of the generation of Pi , while in (7.2.56.1) C(K)−I δ depends also on the generation of Pi . We leave to the readers for checking the induction works in this way as well. We can then take the union of all Xg (L), for g = 0, 1, · · · and obtain a countably generated subcomplex on which the An,K operation is everywhere defined. If we take this approach, the An,K operations mk,β coincide with mgeo k,β . Before ending this subsection, we remark that Proposition 7.2.78 immediately implies the following variant of Theorem 3.4.8.

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Corollary 7.2.89. (Unfiltered version). For β = β0 , mk,β0 defines an AK structure on C(g1 ) (L; Z) Proof. We can prove a version of Proposition 7.2.35 where we perturb the space Mmain,d k+1 (β0 ) only, in the same way. Then we obtain Xg (L). We put XL = #K g=0 Xg (L). It is easy to check (3.4.6.1), . . . ,(3.4.6.4). For this step we can work over Z-coefficients rather than Q-coefficients. This is because we are using the components Mmain,d k+1 (β0 ) only and hence we can use single-valued sections rather multisections. Then the proof of Proposition 7.2.78 works to construct AK structure over Z.  Remark 7.2.90. The proof of Corollary 7.2.89 above looks rather different from the argument explained in Subsection 7.2.1. So we explain their relationship here. We consider the moduli space Mmain,d k+1 (β0 ). This space, which consists of constant maps, is diffeomorphic to L × Grk+1 and is transversal. (In other words k+1 , it is of correct dimension.) However the evaluation map Mmain,d k+1 (β0 ) → L k+1 (which is an embedding L → L as a diagonal and is trivial with respect to the second factor Grk+1 ) is not submersive. So the obvious Kuranishi structure of Mmain,d k+1 (β0 ), where obstruction bundle E is trivial, is not the one we constructed in Propositions 7.1.1 and 7.1.2. The Kuranishi neighborhood (of the one constructed in Propositions 7.1.1 and 7.1.2) may be taken to be a neighborhood of L in Lk+1 . So, by perturbing the Kuranishi map s, we obtain s−1 (0) which perturbs the diagonal L to another manifold (chain) close to L in Lk+1 . For example in case k = 2, a −1 perturbation of L ⊂ L3 can be written as {(x, ϕ−1 1 (x), ϕ2 (x)) | x ∈ L}. Hence taking a fiber product of L with P1 × P2 we obtain m2 (P1 , P2 ) as we described in Subsection 7.2.1. 7.2.8. Perturbed moduli space running out of the Kuranishi neighborhood II. In this subsection, we complete the proofs of Proposition 3.5.2 and Theorem 3.5.11 stated in Chapter 3 using the idea explained in Subsection 7.2.3. Theorem 7.2.72 and Proposition 7.2.78 will play an essential role for this purpose. We fix an appropriate norm on the obstruction bundle to measure the distance between two multisections sd,k,β,P . We also fix Kuranishi neighborhoods of our moduli spaces: The definition of a Kuranishi structure on X a priori only assigns a germ of Kuranishi neighborhood at each point p ∈ X. Here we fix some representatives of the germs. We also assume that these fixed Kuranishi neighborhoods form a good coordinate system in the sense of Definition 6.1 [FuOn99II] or Lemma A1.11 in Section A1 of this book so that we can use the system to inductively construct the multisections that are needed for our purpose. Take two sets of positive numbers {Kλ }λ and {δλ }λ parameterized so that Kλ  ∞,

δλ  0

as λ → ∞.

From Proposition 7.2.35, we have a countable set X(g) (L, λ) (g ≤ Kλ ) which generates C(g) (L, λ) for each λ. (In order to make the notation consistent with that in Section 3.5, it might be better to denote X(g) (L, λ) by X1,(g) (L, λ). But we use the notation X(g) (L, λ) for simplicity.) We also have a family of multisections sd,k,β,P ,λ for (d, β) ≤ Kλ satisfying (7.2.36) for K = Kλ , δ = δλ .

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

467

Proposition 7.2.78 shows that we can choose this family so that g0 (λ), g1 (λ)  ∞,

K(λ)  ∞,

n(λ)  ∞

as λ → ∞

and the followings hold: (7.2.91.1) (7.2.91.2)

There exists a filtered An(λ),K(λ) structure mλk,β on C(g1 (λ)) (L, λ). on C(g0 (λ)) (L, λ). Here mλ,geo is as in (7.2.77). mλk,β = mλ,geo k,β k,β

From now on we simply denote (7.2.92.1)

C(L, λ) = C(g1 (λ)) (L, λ),

(7.2.92.2)

C0 (L, λ) = C(g0 (λ)) (L, λ).

Proposition 7.2.93. Under the situation above, there exists a sequence (n (λ), K  (λ)) with (n (λ), K  (λ))  (n(λ), K(λ)) and lim (n (λ), K  (λ)) = (∞, ∞)

λ→∞

and a constant c(λ) depending on λ such that the An(λ),K(λ) -algebra (C(L, λ), mλk ) is  An (λ),K  (λ) -homotopy equivalent to the filtered An(λ ),K(λ ) -algebra (C(L, λ ), mλk ) if λ > c(λ). Proposition 7.2.93 follows from the An,K -version of Theorems 4.6.1 and 4.6.25, which will be proved in the next two subsections. The proof of Proposition 7.2.93 will be given in Subsection 7.2.11. Assuming Proposition 7.2.93, we combine it with Theorem 7.2.72 and Lemma 7.2.69 to complete the construction of the filtered A∞ algebra in the rest of this subsection. Consider a sequence {λi }i=0,1,2,··· with λ0 < λ1 < · · · → ∞ as i → ∞, and λi+1 > c(λi ), which are inductively constructed as in Proposition 7.2.93. We will construct a filtered An (λi ),K  (λi ) structure m(λi ) on C(L, λ0 ) and a filtered An (λi ),K  (λi ) homotopy equivalence hλ0 ,λi : C(L, λ0 ) → C(L, λi ) for any λi ≥ λ0 . We use the induction on i for this purpose. As the induction hypotheses for λi , we assume that we have already constructed m(λi ) and hλ0 ,λi as above and that the filtered An (λj ),K  (λj ) -structure m(λj ) on C(L, λ0 ) coincides with the restriction of the filtered An (λi ),K  (λi ) -structure m(λi ) if j < i. Then Proposition 7.2.93 implies that there exists a filtered An (λi ),K  (λi ) homotopy equivalence hλi ,λi+1 : C(L, λi ) → C(L, λi+1 ). Since C(L, λi+1 ) has a filtered An (λi+1 ),K  (λi+1 ) structure, Theorem 7.2.72 implies that we can extend the filtered An (λi ),K  (λi ) structure on C(L, λi ) to a filtered An (λi+1 ),K  (λi+1 ) structure and the map hλi ,λi+1 can be extended to a filtered An (λi+1 ),K  (λi+1 ) homotopy equivalence. Now using the filtered An (λi ),K  (λi ) homotopy equivalence hλ0 ,λi in our induction hypothesis, we can extend m(λi ) to

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a filtered An (λi+1 ),K  (λi+1 ) structure m(λi+1 ) . Furthermore hλ0 ,λi can be also extended to a filtered An (λi+1 ),K  (λi+1 ) homotopy equivalence. We define hλ0 ,λi+1 to be the composition hλi ,λi+1 ◦ hλ0 ,λi . By the induction hypotheses put on i, we have thus obtained the desired m(λi ) and hλ0 ,λi for any i. Hence we obtain a filtered A∞ structure on C(L, λ0 ) from Lemma 7.2.69. This gives the countably generated submodule C(L; Q) in Section 3.5 which we want to construct. Therefore we have constructed a filtered A∞ algebra structure on ˆ Λ0,nov , C(L, Λ0,nov ) = C(L, Q) ⊗ assuming Proposition 7.2.93.  Remark 7.2.94. In the construction of the filtered A∞ algebra above, we started from X0 (L) and constructed a filtered A∞ algebra at once. In Theorem 3.1.10, however, we assumed that there exists an unfiltered A∞ algebra as in Theorem 3.4.8 and claimed that we can put the effect of pseudo-holomorphic discs to obtain a filtered A∞ algebra. To prove this statement we modify Proposition 7.2.35 as follows. Assume that we have X00 (L) and a system of multisections of 0 (β0 ; P1 , · · · , Pk ) for Pi ∈ X00 (L) which defines an unfiltered A∞ algeMmain,d k+1 bra. Then under this additional assumption, we can construct Xg (L) such that X0 (L) ⊃ X00 (L), satisfying all the properties stated in Proposition 7.2.35. Further0 (β0 ; P1 , · · · , Pk ) coincides with more for Pi ∈ X00 (L) the multisection on Mmain,d k+1 one we started with. We can prove this statement in the same way. We can then use this to prove Theorem 3.1.10 in the same way. 7.2.9. Construction of filtered An,K homomorphisms. In this subsection, we prove the An,K versions of Proposition 4.6.14 and of Theorem 4.6.1 (modulo sign). We first fix our notation. Recall that ψ : M → M  is a symplectic diffeomorphism with ψ(L) = L . We take sequences Kλ > 0, δλ > 0 with Kλ  ∞, δλ  0 as λ → ∞. Then for each λ (resp. λ ), we apply Proposition 7.2.35 with K = Kλ , δ = δλ (resp. Kλ , δλ ) to obtain the countable set X1,g (L, λ) for g ≤ Kλ ,

(resp. X2,g (L , λ ) for g  ≤ Kλ )

together with the multisections sL,J,d,k,β,P ,

(resp. s2,L ,J  ,d ,k,β,P  )

on



 (resp. Mmain,d (L , J  ; β  ; P  ) ) Mmain,d k+1 (L, J; β; P ), k+1 and triangulation of its zero set as in Proposition 7.2.35, for (d, β) ≤ Kλ (resp. (d , β  ) ≤ Kλ ). Later, we only consider the case λ ≤ λ . (Using the notations in Section 4.6, we denote by J an almost complex structure main,d   for Mmain,d (L , J  ; β  ; P  ). The suffix ‘2’ k+1 (L, J; β; P ) and by J one for Mk+1 in the symbol s2,L ,J  ,d ,k,β,P  indicates that the corresponding multisections are associated to X2,g (L , λ ). In Proposition 7.2.100 below, we will reserve the notation sL ,J  ,d ,k,β,P  for the multisections that are associated to another countable set X3,g (L , (λ, λ )) containing X2,g (L , λ ).)

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

We put



X1,(Kλ ) (L, λ) =

X2,(Kλ ) (L , λ ) =

X1,g (L, λ),



469

X2,g (L , λ ),

g  ≤Kλ

g≤Kλ

which generate C2,+ (L , λ ) := C2,(Kλ ) (L , λ )

C1,+ (L, λ) := C1,(Kλ ) (L, λ),



respectively. Then using Proposition 7.2.35, we define mλ,geo (resp. mλk,β,geo ), which k,β satisfy the A∞ formula as long as both sides of the formula are defined. We also choose g0 (λ), g1 (λ) (resp. g0 (λ ), g1 (λ )) as in (7.2.91), (7.2.92). We put C1 (L, λ) = C1,(g1 (λ)) (L, λ), 





C1,0 (L, λ) = C1,(g0 (λ)) (L, λ), 

C2,0 (L , λ ) = C2,(g0 (λ )) (L , λ ).

C2 (L , λ ) = C2,(g1 (λ )) (L , λ ),

Applying Proposition 7.2.78, we obtain a filtered An(λ),K(λ) (resp. An(λ ),K(λ ) )  structure mλk (resp. mλk ) on C1 (L, λ) (resp. C2 (L , λ )). Here lim n(λ) = lim K(λ) = ∞.

λ→∞

λ→∞

Our strategy of the construction of filtered A∞ homomorphisms is similar to that of the filtered A∞ algebras: We will define a countable set X3,g (L , (λ, λ )) satisfying Properties 7.2.95 below and put X3,(Kλ,λ ) (L , (λ, λ )) = This generates



X3,g (L , (λ, λ )).

g  ≤Kλ,λ

C3,+ (L , (λ, λ )) := C3,(Kλ,λ ) (L , (λ, λ )).

Here we take

  Kλ,λ Kλ,λ  ≤ Kλ ,  ≤ K λ . Then we will define a (partially defined ‘An (λ),K  (λ) ’) homomorphism 

f(λ,λ ),geo : C1,+ (L; λ) → C3,+ (L , (λ, λ )) 



together with operations m(λ,λ ),geo on C3,+ (L , (λ, λ )). Here m(λ,λ ),geo satisfies  An (λ),K  (λ) relation as long as both sides are defined. Moreover f(λ,λ ),geo together  with mλ,geo , m(λ,λ ),geo satisfies the formula defining an An (λ),K  (λ) homomorphism as long as both sides are defined. Next we use an argument similar to Subsection 7.2.7 to obtain an An (λ),K  (λ)  structure m(λ,λ ) on C3 (L , (λ, λ )) ⊂ C3,+ (L , (λ, λ )) and, by Proposition 7.2.126, an An (λ),K  (λ) homomorphism 



f(λ,λ ) : (C1 (L, λ), mλ ) → (C3 (L , (λ, λ )), m(λ,λ ) ). We then consider a sequence {λi }i with λ0 < λ1 < λ2 < · · · → ∞ and also (n , K  ) = (n (λi ), K  (λi )) → (∞, ∞) as i → ∞. Taking the limit based on Proposition 7.2.132, we will obtain the filtered A∞ structures on C1 (L, λ0 ) and

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C3 (L , (λ0 , λ0 )) and the filtered A∞ homomorphism f : C1 (L, λ0 ) → C3 (L ; (λ0 , λ0 )) as we want. Now we carry out the plan described above. We first describe the conditions we require for the set X3,g (L , (λ, λ )) of smooth singular simplices below. For a singular simplex P on L, we write as ψ(P ) its image by the symplectic diffeomorphism ψ : M → M  with ψ(L) = L . Properties 7.2.95. (7.2.96.0) X3,00 (L , (λ, λ )) is a union of X2,(Kλ ) (L , λ ) ∪ ψ(X1,(Kλ ) (L, λ)). (7.2.96.1) X3,g (L , (λ, λ )) is a countable set of smooth singular simplices of L . (7.2.96.2) The inclusion C3,(g ) (L , (λ, λ ); Q) → S(L ; Q) induces an isomorphism in cohomology. Here C3,(g ) (L , (λ, λ ); Q) is generated by X3,(g ) (L , (λ, λ )) over Q. (7.2.96.3) All the faces of each element in X3,g (L , (λ, λ )) belongs to the countable set X3,(g ) (L , (λ, λ )). The conditions (7.2.96.1)-(7.2.96.3) are the analogs to Properties 7.2.27 for the filtered An,K -homomorphisms. We remark that singular simplex of the form ψ(P ) (P ∈ X1,(Kλ ) (L, λ)) may coincides with P  ∈ X2,(Kλ ) (L , λ ). So the union X2,(Kλ ) (L , λ ) ∪ ψ(X1,(Kλ ) (L, λ)) may not be a disjoint union. If P  ∈ X3,g (L , (λ, λ )), we say that the generation of P  is g  . We note that in the point of view of (7.2.96.0), P  ∈ X2,(Kλ ) (L , λ ) has generation 0. So the generation g  of P  used as an element in X3,g (L , (λ, λ )) is different from the generation that we used to construct the filtered An,K -structure on C2 (L ; Λ0,nov ). We ignore the latter generation of the elements of X2,(Kλ ) (L , λ ) for the current construction of the filtered An,K -homomorphisms. We define the decorated moduli    space Mmain,d k+1 (L , J ; β; P ) using this newly assigned generations. On the other hand, we do not make the decoration explicit in the notation of  the moduli space Mmain k+1 (L, J; β; P ), since we take a countably generated subspace X1,(Kλ ) (L, λ) of chains on L once and for all and will not change it during our construction of An,K -homomorphisms. This being understood, we eliminate d from the notations and denote the relevant multisections by sL,J,k,β,P instead of sL,J,d,k,β,P . We next describe the moduli spaces that we will use for the construction of our filtered An,K -homomorphism. Recall we defined the moduli space of time-ordered    products Mmain k+1 (M , L , {Jρ }ρ : β; top(ρ); P ) in Definition 4.6.10. In this section, we denote this as main     Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) := Mk+1 (M , L , {Jρ }ρ : β; top(ρ); P )

to simplify the notations. Here the singular simplices P = (P1 , · · · , Pk ) are chosen from the fixed X1,(Kλ ) (L, λ). We have the evaluation map (7.2.97)

  ev0 : Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) → L .

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In Proposition 7.2.100 below, we will construct a perturbation (or a multisection) top(ρ) sβ,k,P of the Kuranishi map and define the chains   top(ρ) s geo  main   β,k, P . fk,β (P ) = ev0∗ Mk+1 ({Jρ }ρ : β; top(ρ); P ) As we discussed just after Proposition 4.6.14, codimension-one components of the  boundary of the moduli space Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) are described as a union of (4.6.16.1)-(4.6.16.5). As we examined there, the contribution of (4.6.16.1) cancels that of (4.6.16.2). We now examine other boundary components. First we consider (4.6.16.3). It can be written as (7.2.98.1)

0 Mmain k+−m+1 ({Jρ }ρ : β2 ; top(ρ); (P1 , · · · , P−1 , Q,m , Pm+1 , · · · , Pk )),

where β = β1 + β2 and

  Q0,m = ev0∗ Mmain m−+2 (L, J; β1 ; (P , · · · , Pm )) .

(7.2.98.2)

We remark that (7.2.98.2) is the moduli space we used to define a singular chain mJ,s m−+1 (P , · · · , Pm ). Here we include J, s in the notation to clarify its dependence. We also remark that we already fixed the perturbation of the moduli space (7.2.98.2) when we defined mJ,s m−+1 (P , · · · , Pm ). (Note again that Pi ∈ X1,(Kλ ) (L, λ) and we do not change X1,(Kλ ) (L, λ).) We next consider (4.6.16.4). It can be written as   0 0 Mmain m+1 (L , J ; β2 ; R1 ,2 , · · · , Rm−1 ,m )

(7.2.99.1) where β = β2 +

m−1 i=1

β1,i , 1 = 1 ≤ 2 ≤ · · · ≤ m = k + 1 and

  (7.2.99.2) R0i ,i+1 = ev0∗ Mmain i+1 −i +1 ({Jρ }ρ : β1,i ; top(ρ); (Pi , · · · , Pi+1 −1 )) . (7.2.99.1) is a moduli space we will use to define a filtered An,K structure on C3 (L , Λ0,nov ) and (7.2.99.2) is the moduli space we will use to define a filtered An,K homomorphism. Contrary to the case of (7.2.98), we have not fixed yet the perturbation of either of the moduli spaces (7.2.99.1) and (7.2.99.2). Now we state the main technical result in the proof of Proposition 4.6.14. Proposition 7.2.100. We take sequences Kλ > 0, δλ > 0 with Kλ  ∞, δλ  0 as λ → ∞. For such sequences, we have X1,g (L, λ), X2,g (L , λ ) and the multisections sL,J,k,β,P , s2,L ,J  ,d,k,β,P  etc., as we explained in the beginning of this subsec  tion. We assume that λ ≤ λ . Keeping the notations above, let δλ,λ  > 0, Kλ,λ > 0 be sequences such that  Kλ,λ  ≤ Kλ ,

 Kλ,λ  ≤ K λ ,

 δλ,λ  ≥ δλ ,

 δλ,λ  ≥ δλ

  and Kλ,λ   ∞, δλ,λ  0 as λ → ∞. Then, if λ and λ are large enough, there exist multisections sL ,J  ,d,k,β,P  for top(ρ)

  (d, β) ≤ Kλ,λ  , and piecewise smooth multisections s for β + k ≤ Kλ,λ , β,k,P  and a countable set X3,g (L , (λ, λ )) for g  ≤ Kλ,λ  with the following properties.

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(7.2.101.1) X3,g (L , (λ, λ )) has Properties 7.2.95.    (7.2.101.2) sL ,J  ,d,k,β,P  is a multisection of Mmain,d k+1 (L , J ; β; P ) transversal to zero where P  = (P1 , · · · , Pk ) with Pi ∈ X3,g (L , (λ, λ )). Furthermore, if Pi ∈ X2,g (L , λ ) (g  ≤ Kλ ), sL ,J  ,d,k,β,P  coincides with the multisection s2,L ,J  ,d,k,β,P  which we have already defined for the construction of filtered An(λ),K(λ) - structure on L . top(ρ)  sβ,k,P is a multisection of Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) for Pi ∈ X1,g (L, λ) (g ≤ Kλ ) with P = (P1 , · · · , Pk ) and is transversal to zero. (7.2.101.3) If (d, β) = g  , then       sL ,J  ,d,k,β,P  ev0∗ Mmain,d k+1 (L , J ; β; P ) is decomposed into elements of X3,(g ) (L , (λ, λ )). (7.2.101.4) If β + k = g  , then   top(ρ) s main   β,k, P ev0∗ Mk+1 ({Jρ }ρ : β; top(ρ); P ) is decomposed into elements of X3,(g ) (L , (λ, λ )). top(ρ)

(7.2.101.5) The multisections sL,J,k,β,P and sβ,k,P are compatible to each other in the sense of Compatibility Condition 7.2.102 below. top(ρ) The multisections sL ,J  ,d,k,β,P  and sβ,k,P are compatible to each other in the sense of Compatibility Condition 7.2.104 below. top(ρ)  (7.2.101.6) The zero sets (sL,J,k,β,P )−1 (0) and (sβ,k,P )−1 (0) are in δλ,λ  neighborhoods of the zero sets of the original Kuranishi maps. top(ρ)  The zero sets (sL ,J  ,d,k,β,P  )−1 (0) and (sβ,k,P )−1 (0) are in δλ,λ  neighborhoods of the zero sets of the original Kuranishi maps. top(ρ)

See Remark 7.2.122 for the reason why sβ,k,P is piecewise smooth instead of smooth. Note that in (7.2.101.2), we have X2,g (L , λ ) ⊂ X3,00 (L , (λ, λ )) ⊂ X3,0 (L , (λ, λ )) ⊂ X3,g (L , (λ, λ )). Since Proposition 7.2.100 holds for each pair (λ, λ ) with λ, λ large enough, we may fix λ and λ large enough to prove the proposition. Therefore hereafter (until the end of the proof of Proposition 7.2.100), we drop λ and λ from the notations and put X1 (L) = X1,(Kλ ) (L, λ), X2 (L ) = X2,(Kλ ) (L , λ ), X3 (L ) = X3,(Kλ,λ ) (L , (λ, λ )) to simplify the notations. We now explain the compatibility of the multisections stated in (7.2.101.5). First the compatibility among sL ,J  ,d,k,β,P  are exactly the same as the one in (7.2.36.4). So we do not repeat it. top(ρ) Let us describe the compatibility of sβ,k,P with other multisections at the top(ρ)

 boundary. sβ,k,P is a multisection on Mmain k+1 ({Jρ }ρ : β; top(ρ); P ), whose boundary

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

473

is described either by (7.2.98), (7.2.99) or is Mmain k+1 ({Jρ }ρ : β; top(ρ); (P1 , · · · , ∂Pi , · · · , Pk )) for some i. In the last case, the compatibility condition is the same as Compatibility Condition 7.2.38. We consider the boundary corresponding to (7.2.98). The perturbation (multisection) sL,J,k,β1 ,(P ,··· ,Pm ) of Mmain m−+2 (L, J; β1 ; (P , · · · , Pm )) is already fixed when s,J we defined m . We then obtain   sL,J,k,β1 ,(P ,··· ,Pm ) . Q,m = ev0∗ Mmain m−+2 (L, J; β1 ; (P , · · · , Pm )) We know that Q,m ∈ X1 (L). Then Proposition 7.2.100 asserts that we have a top(ρ) multisection sβ,k,(P1 ,··· ,P−1 ,Q,m ,Pm+1 ,··· ,Pk ) of Mmain k+−m+1 ({Jρ }ρ : β2 ; top(ρ); (P1 , · · · , P−1 , Q,m , Pm+1 , · · · , Pk )). Compatibility Condition 7.2.102. The multisection top(ρ)

sβ2 ,k−m+,(P1 ,··· ,P−1 ,Q,m ,Pm+1 ,··· ,Pk ) ⊕ sL,J,m−+1,β1 ,(P ,··· ,Pm ) top(ρ)

coincides with the pull-back of sβ,k,P . See Figure 7.2.6. P

P

P −1

β1

P

β2

P1

z0 Pk

P

Pm

Pm + 1

Q , m

Figure 7.2.6 Here the pull-back is taken similarly as in the case of Compatibility Condition 7.2.44. We remark that we actually need to state Compatibility Condition 7.2.102 more carefully as stated in Compatibility Condition 7.2.44, by the same reason. Since it can be done in the same way as Compatibility Condition 7.2.44, we state Compatibility Condition 7.2.102 simply as above. We will not repeat this kind of remarks later when similar circumstance occur. We next describe the compatibility condition corresponding to (7.2.99). Propotop(ρ) sition 7.2.100 asserts that we have a multisection sβ1 ,m−+1,(P ,··· ,Pm ) of (7.2.103.1)

Mmain m− ({Jρ }ρ : β2 ; top(ρ); (P , · · · , Pm )).

(Actually we need Lemma 7.2.107 proven later to see that β1  + m −  + 1 ≤ K.) We define

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(7.2.103.2)

 Ri ,i+1 = ev0∗ Mmain i+1 −i +1 ({Jρ }ρ : β1,i ; top(ρ) top(ρ) 1,i ,i+1 −i ,(Pi ,··· ,Pi+1 −1 )

sβ

; (Pi , · · · , Pi+1 −1 ))

 .

We put

d max (i) = β1,i  + i+1 − i . Then (7.2.101.4) asserts that the chain Ri ,i+1 is decomposed into elements of X3,(d max (i)) (L ). To avoid heavy notations we pretend as if Ri ,i+1 ∈ X3,gi (L ) for  = (R , , · · · , R some gi ≤ d max (i). We put R ) and define the decoration 1 2 m−1 ,m d : {1, · · · , m} → {0, 1, · · · , K};

d (i) = gi .

Again Proposition 7.2.100 asserts that we have a multisection sL ,J  ,d ,m+1,β2 ,R of     the moduli space Mmain,d m+1 (L , J ; β2 ; R). (Here we need to apply Lemma 7.2.107 below, though.) Compatibility Condition 7.2.104. The multisection sL ,J  ,d ,m,β2 ,R ⊕

m−1 

top(ρ)

sβ1,i ,i+1 −i ,(P i=1

i

,··· ,Pi+1 −1 )

top(ρ)

coincides with the pull-back of sβ,k,P . See Figure 7.2.7.

P

R2 ,3

P

R1 ,2

P2 −1 P

P2

P1 =P1

z0

P

P3−1

P3

P

P

R3 ,4

P4−1

R3 ,4 P

Figure 7.2.7

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475

We have thus stated Proposition 7.2.100 precisely. Once the precise statement thereof is made, its proof is rather a direct analog of that of Proposition 7.2.35. top(ρ) Namely we construct sL ,J  ,d,k,β,P  , sβ,k,P , together with X3,g (L ), inductively so that it satisfies Compatibility Conditions 7.2.102 and 7.2.104. We can use Theorem A1.23 in Section A1 (that is Lemma 3.11 of [FuOn99II]) to extend the multisection defined on the boundary by the induction hypothesis. We can prove a lemma corresponding to Lemma 7.2.55 in the same way as the proof of Lemma 7.2.55. So we only need to describe the relevant order used to define the multisections and check if the induction works. The order of the construction of the multisection is as follows: Definition 7.2.105. (7.2.106.1)

sL ,J  ,d ,k ,β1 ,P  < sL ,J  ,d ,k ,β2 ,P  if one of the following holds: 1

1

2

(1)

2

(2)

(7.2.106.1.1) (d1 , β1 ) < (d2 , β2 ). (7.2.106.1.2) (d1 , β1 ) = (d2 , β2 ) and k1 + β1  < k2 + β2 . (7.2.106.1.3) (d1 , β1 ) = (d2 , β2 ), k1 + β1  = k2 + β2  and β1  < β2 .      (d1 , β1 ) = (d2 , β2 ), k1 + β1  = k2 + β2 , β1  = β2  and  (7.2.106.1.4)  deg P(1),i > deg P(2),i . (7.2.106.2)

top(ρ) 1 ,k1 ,P(1)

top(ρ) 2 ,k2 ,P(2)

< sβ

sβ

if one of the following holds:

(7.2.106.2.1) k1 + β1  < k2 + β2 . (7.2.106.2.2) k1 + β1  = k2 + β2  and β1  < β2 .  (7.2.106.2.3) k1 +β1  = k2 +β2 , β1  = β2  and deg P(1),i > deg P(2),i . (7.2.106.3)

1

(7.2.106.3.1) (7.2.106.3.2) (7.2.106.3.3) (7.2.106.4)

top(ρ) 2 ,k2 ,P(2)

sL ,J  ,d ,k ,β1 ,P  < sβ (d1 , β1 ) (d1 , β1 ) (d1 , β1 )

top(ρ) 1 ,k1 ,P(1)

sβ

(7.2.106.4.1) (7.2.106.4.2) never occurs.) (7.2.106.4.3)

1

(1)

if one of the following holds:

< k2 + β2 . = k2 + β2  and k1 + β1  < k2 + β2 . = k2 + β2 , k1 + β1  = k2 + β2  and β1  < β2 .

< sL ,J  ,d ,k ,β2 ,P  if one of the following holds: 2

2

(2)

k1 + β1  < (d2 , β2 ). k1 + β1  = (d2 , β2 ) and k1 + β1  < k2 + β2 . (Actually this

k1 + β1  = (d2 , β2 ), k1 + β1  = k2 + β2  and β1  < β2 .

Now we prove Lemmas 7.2.107 and 7.2.120 which show that we can use an induction with respect to the order < for the proof of Proposition 7.2.100. Lemma 7.2.107. Let Ri ,i+1 , d max , β, β2 , β1,i etc. be as in (7.2.103). Then we have: top(ρ)

(7.2.108.1)

sL ,J  ,d ,m,β2 ,R < sβ,k,P .

(7.2.108.2)

sβ1,i ,i+1 −i ,(P

top(ρ)

top(ρ)

i

,··· ,Pi+1 −1 )

< sβ,k,P , for each i.

Proof. We first prove (7.2.108.1). We may assume d max = d without loss of generality, because sL ,J  ,d ,m,β2 ,R ≤ sL ,J  ,d max ,m,β2 ,R .

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We have d (j) = β1,j  + j+1 − j

(7.2.109.1) (7.2.109.2)

m

(j+1 − j ) = k

j=1

(7.2.109.3)

β2 +

m

β1,j = β.

j=1

 2 − 1

β1,1

β2

β1,2

z0

β1,m

Figure 7.2.8 

We choose j0 so that d (j0 ) = max d . Then we have (7.2.110)

(d , β2 ) = β2  + d (j0 ) + m = β2  + β1,j0  + (j0 +1 − j0 ) + m.

(7.2.49.2) and (7.2.109.3) imply (7.2.111)

β ≥ β2  +

m

β1,j  + m.

j=1

By (7.2.110) and (7.2.111) we have: (7.2.112)

(d , β2 ) ≤ β −



β1,j  + (j0 +1 − j0 ).

j=j0

By stability, we have β1,j  + (j+1 − j ) ≥ 0, where the equality holds only if β1,j  = 0, j+1 = j , β1,j = β0 . Therefore, by (7.2.109.2) and (7.2.112), we have either (7.2.113.1)

(d , β2 ) < β + k,

or (7.2.113.2)

(d , β2 ) = β + k,

β1,j  = 0 and j+1 = j for all j = j0 .

Moreover in case (7.2.113.2) the equality holds in (7.2.111). In case (7.2.113.1), we have (7.2.106.3.1) and hence we are done.

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477

Let us assume (7.2.113.2). Then using (7.2.109.2) and (7.2.112), we have (7.2.114)

(d , β2 ) = β + (j0 +1 − j0 ) = β + k.

Moreover m + β2  = (d , β2 ) − d (j0 ). Therefore we have either (d , β2 ) = β + k,

(7.2.115.1)

m + β2  < k + β,

or d (j0 ) = 0.

(7.2.115.2)

If (7.2.115.1) holds, then we have (7.2.106.3.2) and hence we are done. Let us assume (7.2.115.2). Then by (7.2.109.1) we have (7.2.116)

β1,j0  = 0 and

j0 +1 − j0 = 0.

(7.2.116), (7.2.113.2) and (7.2.109.2) imply k = 0. Therefore using (7.2.114) we have (d , β2 ) = β. Since the equality holds in (7.2.111), it follows from (7.2.116), (7.2.113.2) that β = β2 +m. Note m > 0. Hence we have (7.2.106.3.3). The proof of (7.2.108.1) is complete. We next prove (7.2.108.2). We use (7.2.109.2) together with β1,i  ≤ β, j+1 − j ≥ 0 to obtain i+1 − i + β1,i  ≤ β + k.

(7.2.117)

If the strict inequality holds in (7.2.117), then we have (7.2.106.2.1) and we are done. Let us assume that the equality holds in (7.2.117). Then j+1 − j = 0

(7.2.118)

for all j = i,

and (7.2.119)

β 2 = β0

and

β1,j = β0

for all j = i.

(7.2.118) and (7.2.119) imply m = 1, by stability. (Namely there is no j = i.) By stability and m = 1, we have β2 = β0 . This contradicts to (7.2.119). Hence we have proved (7.2.108.2).  Lemma 7.2.120. In the situation of Compatibility Condition 7.2.102 we have top(ρ)

top(ρ)

sβ2 ,k−m+,(P1 ,··· ,P−1 ,Q,m ,Pm+1 ,··· ,Pk ) < sβ,k,P . Proof. We have β1 + β2 = β. If β1 = β0 , then  < m. Hence we obtain β2  + k − m +  < k + β, as required. If β1 = β0 , then β2  + 1 ≤ β, k − m +  ≤ k + 1. Therefore either β2  + k − m +  < β + k, or β2  < β. β2  + k − m +  = β + k, Therefore either (7.2.106.2.1) or (7.2.106.2.2) holds.



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   We need a similar lemma to study ∂Mmain,d k+1 (L , J ; β; P ). But it is exactly the same as Lemma 7.2.47.

Now we are ready to complete the proof of Proposition 7.2.100. We first define X3,0 (L ), the set of the generation 0 elements. We take a countable set  X3,00 (L ) = X2 (L ) ∪ ψ(X1 (L)). If k + β = 0 for Mmain k+1 ({Jρ }ρ : β; top(ρ); P ), then by Lemma 7.2.45 we have either (i) k = 0, β = β0 , β = 0, or (ii) k = 1, β = β0 . The stability implies the case (ii) does not happen. For the case (i), the  moduli space Mmain 0+1 ({Jρ }ρ : β; top(ρ); 1) has no boundary. We fix a multisection top(ρ) sβ,0, 1 transversal to zero. We take a simplicial decomposition of   top(ρ) s  β,0, 1 ev0∗ Mmain , ({J } : β; top(ρ); 1) ρ ρ 0+1 regard it as a smooth singular chain, and then add the singular simplices appearing in the above decomposition to X3,00 (L ) if not already there. We thus obtain X3,0 (L ).    We remark that when (d, β) = 0, if the moduli space Mmain,d k+1 (L , J ; β;P ) is nonempty, then we have k = 0 and β = β0 β = 0. Therefore the perturbation of    the moduli space Mmain,d 0+1 (L , J ; β; 1) was already fixed in the last subsection. The inductive step of the proof of Proposition 7.2.100 is the same as that of Proposition 7.2.35. We need a lemma similar to Lemma 7.2.55. It can be proved in the same way as Lemma 7.2.55. Therefore the proof of Proposition 7.2.100 is now complete.  We now explain the reason why we obtain only a piecewise smooth multisection top(ρ) sβ,k,P not smooth one in Proposition 7.2.100. We first recall that in our construction of multisection, we start with a multisection given on each of the components of the boundary of our space with Kuranishi structure. (They are compatible at the corners.) And we extend it to a neighborhood of the boundary. (We then apply Theorem A1.23.) We actually use the following lemma in the step of extending the multisection to a neighborhood of the boundary. We recall that a smooth function on a subset of Rn is, by definition, a restriction of a smooth function defined on its neighborhood. Lemma 7.2.121. Let V = {(x1 , · · · , xn ) ∈ Rn | xi ≥ 0, i = 1, · · · , n} and Vi = V ∩ {(x1 , · · · , xn ) ∈ Rn | xi = 0}. Let fi : Vi → R be smooth functions such that fi = fj on Vi ∩ Vj . Then there exists a smooth function f : V → R such that f = fi on Vi . Proof. By adding constants, we may assume fi (0) = 0 for all i = 1, · · · , n. For x = (x1 , · · · , xn ) ∈ Vi , we consider the new functions fi defined by

fi (0, · · · , 0, xj , 0 · · · , 0). fi (x) = fi (x) − j=i

j

The fi ’s still satisfy the assumption fi = fj on Vi ∩ Vj , and fi (0, · · · , 0, xj , 0 · · · , 0) = 0 holds for j = i in addition. Since

fi (0, · · · , 0, xj , 0 · · · , 0) + fk (0, · · · , 0, xi , 0, · · · , 0) j=i

j

i

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

479

is a smooth function on V for k = i, we may assume without loss of generality that the equality fi (0, · · · , 0, xj , 0 · · · , 0) = 0 holds for j = i. By a similar process this time considering pairs of indices j = k, we can also reduce the case to one for which fi (0, · · · , 0, xj , 0, · · · , 0, xk , 0, · · · , 0) = 0 for all j, k = i. Continuing this inductively, we can reduce to the case where fi = 0 for all i, which is then trivial to prove.  Remark 7.2.122. On the other hand, we can not generalize Lemma 7.2.121 to the polygons that enters in the proof of Proposition 7.2.100: Let P ⊂ Rn be a convex polygon and decompose ∂P to ∪Pa : Suppose we have smooth functions fa on Pa such that fa = fb on Pa ∩ Pb : In this situation it is not true in general that there exists a smooth function f : P → R such that f = fa on each Pa . Now let us consider the situation of Proposition 7.2.100 and examine the contop(ρ)  struction of the multisection sβ,k,P on Mmain k+1 ({Jρ }ρ : β; top(ρ); P ). The boundary  of Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) is decomposed into a fiber product of the moduli  spaces, say, {Mα }α . This decomposition {Mα }α of ∂Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) is induced by the decomposition (7.1.52). As we mentioned in Remark 7.1.69, the decomposition (7.1.52) is not the standard decomposition of (the Kuranishi  neighborhood of) Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) into its boundaries and corners with various dimensions. So the decomposition {Mα }α is not a decomposition to the boundary and corners. In order to extend the multisections which are given on each of Mα (and which are compatible at their intersections), we can not use Lemma top(ρ) 7.2.121. This is the reason why we do not claim that our multisection sβ,k,P is smooth. top(ρ) On the other hand, we can find a piecewise smooth multisection sβ,k,P in the

following way. We can decompose (Kuranishi neighborhoods of) Mmain k+1 ({Jρ }ρ :  ) to {Nγ }γ where Nγ is a manifold with corner. Moreover each of Mα β; top(ρ); P is a component of the standard decomposition of Nγ as a manifold with corner. (We may take simplicial decomposition for example.)  Now we can extend a given multiseciton on ∂Mmain k+1 ({Jρ }ρ : β; top(ρ); P ) which main is smooth on each of Mα to a multisection of the neighborhood of ∂Mk+1 ({Jρ }ρ :  ) so that it is smooth on each Nγ . We can prove this using Lemma β; top(ρ); P 7.2.121. Therefore we can obtain a piecewise smooth multisection. We remark that we can use piecewise smooth multisections in the same way as smooth ones for our purpose of defining virtual fundamental chains. In fact we triangulate the zero set of our multisection and use it to obtain a virtual fundamental chain. The zero set of a piecewise smooth multisection has a triangulation if it is transversal. Using Proposition 7.2.100, we can define  X3,(K  ) (L , (λ, λ )) = λ,λ

X3,g (L , (λ, λ )),

g  ≤Kλ,λ

which generates

C3,+ (L , (λ, λ )) := C3,(K

λ,λ

) (L



, (λ, λ )).

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Then we define an operator m(λ,λ ),geo on C3,+ (L , (λ, λ )) by    (λ,λ ),geo       )sL ,J  ,d,k,β,P  (7.2.123) mk,β (P ) = ev0∗ Mmain,d (L , J ; β; P k+1 and a (partially defined filtered ‘An (λ),K  (λ) ’) homomorphism 

f(λ,λ ),geo : C1,+ (L, λ) → C3,+ (L , (λ, λ )) by (7.2.124.1)

(λ,λ ),geo  fk,β (P )

(7.2.124.2)

f1,β0

(λ,λ ),geo

  top(ρ) s main   β,k, P , = ev0∗ Mk+1 ({Jρ }ρ : β; top(ρ); P )

(P ) = ψ(P ).

 Here n (λ), K  (λ) are determined by Proposition 7.2.78 for K = Kλ,λ  , where is determined in Proposition 7.2.100.  m(λ,λ ),geo satisfies An (λ),K  (λ) relation whenever both sides of relation are de  fined. f(λ,λ ),geo , m(λ,λ ),geo , m(λ),geo satisfy defining formula for filtered An (λ),K  (λ) homomorphisms as long as both sides are defined. We modify them in a similar way as Subsection 7.2.7 as follows. We first use Proposition 7.2.78 to obtain g1 (λ, λ ), g0 (λ, λ ) such that if we put  C3 (L , (λ, λ )) = C3,(g1 (λ,λ )) (L , (λ, λ )),  Kλ,λ 

C3,0 (L , (λ, λ )) = C3,(g0 (λ,λ )) (L , (λ, λ )), (λ,λ )

then there exists a filtered An (λ),K  (λ) structure mk,β (λ,λ )

(7.2.125)

mk,β

(λ,λ ),geo

on C3 (L , (λ, λ )) such that

on C3,0 (L , (λ, λ )).

= mk,β

(λ,λ )

(λ,λ ),geo

We recall that mk,β is constructed by using mk,β C3,+ (L , (λ, λ )). By construction we have (λ,λ ),geo

mk,β



= mλk,β,geo

and the operator Gk on

on C2 (L , λ ) ⊆ C3,+ (L , (λ, λ )).

(See (7.2.101.2).) Therefore we may choose Gk so that (λ,λ )

mk,β



on C2 (L , λ ).

= mλk,β

Now we prove the following: Proposition 7.2.126. If λ > C(λ) then there exists a filtered An (λ),K  (λ) homomorphism 



f(λ,λ ) : (C1 (L, λ), mλ ) → (C3 (L , (λ, λ )), m(λ,λ ) ) 

such that the image of C1,0 (L, λ) by f(λ,λ ),geo lies in C3 (L , (λ, λ )) and (7.2.127)





f(λ,λ ) = f(λ,λ ),geo

on C1,0 (L, λ).

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481



Proof. Let us put f(λ,λ ) = f(λ,λ ),geo ◦ i where i is as in (7.2.84) and the composition is defined in the same way as the case of filtered A∞ homomorphisms.  By choosing c(λ) large enough we may assume that f(λ,λ ) is well defined on C1 (L, λ).  Moreover, we may assume that the image of f(λ,λ ) is contained in C3,0 (L , (λ, λ )), geo where m coincides with m. Then, using Proposition 7.2.78bis and (7.2.125), we  can easily prove that f(λ,λ ) is a filtered An (λ),K  (λ) homomorphism. (7.2.127) is a consequence of Proposition 7.2.78 and the definition.  In Propositions 7.2.78 and 7.2.126, we use the technique of Section 5.4 to modify mgeo , fgeo to m, f. We need to use the same technique several times again in later (sub)sections. However, since the method to do so is the same as in Propositions 7.2.78 and 7.2.126, we will not repeat the detail. Remark 7.2.128. In case (M, L) = (M  , L ), ψ = id, λ = λ , s = s , J = J  , X1,s (L, λ) = X2,s (L , λ ), and the multisections to define operators m are also the same, we can take the multisections in Proposition 7.2.100 so that C3,+ (L , (λ, λ )) = C2 (L , λ ) and 



f(λ,λ ),geo = f(λ,λ ) = identity. The bimodule version of the above equality is used at the end of Subsection 6.1.4 to complete the proof of Theorem 5.3.14. See Remark 7.2.172 (2). We now extend a filtered An (λ),K  (λ) structure and a filtered An (λ),K  (λ) homomorphism in Proposition 7.2.126 to a filtered A∞ structure and a filtered A∞ homomorphism. We use the following general lemma for this purpose. We use the notation of Subsection 7.2.6. Lemma 7.2.129. Let (n, K) < (n , K  ) and C1 , C2 , C1 , C2 be filtered An ,K  algebras. Let h : C1 → C2 , h : C1 → C2 be filtered An ,K  homotopy equivalences. Let g(1) : C1 → C1 be a filtered An,K homomorphism and g(2) : C2 → C2 a filtered An ,K  homomorphism. We assume that g(2) ◦ h is An,K homotopic to h ◦ g(1) .  Then there exists a filtered An ,K  homomorphism g+ (1) : C1 → C1 such that g+ (1) coincides to g(1) as a filtered An,K homomorphism and that g(2) ◦ h is filtered An ,K  homotopic to h ◦ g+ (1) . Proof. The proof is similar to the proof of Theorem 7.2.72 and based on obstruction theory. We remark that it suffices to consider the cases (7.2.130.1) (7.2.130.2)

(n, K) < (n , K  ) = (n + 1, K − 1), (n, K) = (n, 0) < (n , K  ) = (0, n + 1).

In each case we can find an obstruction class 

[oK  ,β (g(1) )] ∈ H(Hom(BK  C 1 [1], C 1 [1])) to extend the filtered An,K homomorphism g(1) to a filtered An ,K  homomorphism  g+ (1) . (See (7.2.76.2). Here β ∈ G with β = n , (Definition 7.2.61).) The naturality of the obstruction class and the presence of g(2) imply that the obstruction vanishes and hence we have g+ (1) . To extend the filtered An,K homotopy between g(2) ◦ h and h ◦ g(1) to a filtered An ,K  homotopy between g(2) ◦ h and h ◦ g+ (1) , there is obstruction classes

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in H(Hom(BK  C 1 [1], C2 [1])) similar to ones studied in Section 4.5. (Here C2 is a  model of [0, 1] × C 2 . See Subsection 4.2.1.) Then using the obstruction theory to extend the homotopy, we find that we can eliminate the obstruction to extend the homotopy by modifying the choice of g+ (1) appropriately. (See Remark 7.2.131 below for the argument. Compare the argument at the end of the proof of Subsection 7.2.6.) Lemma 7.2.129 follows.  Remark 7.2.131. In this remark, we describe the argument of the obstruction theory to extend the homotopy. Assume that (n, K) < (n , K  ) as in (7.2.130). Let g(0) , g(1) : C → C  be filtered An ,K  homomorphisms and F : C → C a filtered An,K homotopy between g(0) and g(1) . We want to extend F to a filtered An ,K  homotopy. (In the situation of the proof of Lemma 7.2.129, we take C = C1 , C  = C2 , C = C2 and g(0) = g(2) ◦ h, g(1) = h ◦ g(1) .) Let β = n . Since C is a model of [0, 1] × C  ,  we can find FIK  ,β : BK  (C[1]) → C such that (r )

Evalr=r0 ◦ FIK  ,β = gK0 ,β ,

r0 = 0, 1.



Denote by oK  ,β (F) ∈ Hom(BK  (C[1]), C [1]) the obstruction cycle to extend F to a filtered An ,K  homomorphism. We put      K = Ker Evalr=0 ⊕ Evalr=1 : C → C r=0 ⊕ C r=1 . It is easy to see that oK  ,β (F) + δ1 (FIK  ,β ) ∈ Hom(BK  (C[1]), K[1]). (Here δ1 is induced by m1 .) Since Evalr=0 is a homotopy equivalence, the exactness of 

0 → Hom(BK  (C[1]), K[1]) → Hom(BK  (C[1]), C [1]) 



→ Hom(BK  (C[1]), C r=0 [1] ⊕ C r=1 [1]) → 0 implies that the connecting homomorphism 

δ1 : H p−1 (Hom(BK  (C[1]), C r=1 [1])) → H p (Hom(BK  (C[1]), K[1])) 

is an isomorphism. Therefore there exists Cor1K  ,β ∈ Hom(BK  (C[1]), C r=1 [1]) of degree 0 such that δ1 (Cor1) = 0,

δ1 ([Cor1]) = [oK  ,β (F) + δ1 (FIK  ,β )]. 

Then there exists Cor2K  ,β ∈ Hom(BK  (C[1]), C [1]) such that Evalr=0 ◦ Cor2K  ,β = 0,

Evalr=1 ◦ Cor2K  ,β = Cor1K  ,β .

Moreover there exists Cor3K  ,β ∈ Hom(BK  (C[1]), K[1]) such that oK  ,β (F) + δ1 (FIK  ,β ) = δ1 (Cor2K  ,β ) + δ1 (Cor3K  ,β ). We put FK  ,β  = FIK  ,β − Cor2K  ,β − Cor3K  ,β ,

(1)

(1)

gK  ,β = gK  ,β − Cor1K  ,β .

Then g(1) is a filtered An ,K  homomorphism and F is a filtered An ,K  homotopy  between g(0) and g(1) .

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

483

Now we are ready to wrap-up the proof of Theorem 4.6.1 (modulo sign). We have the filtered An (λ),K  (λ) homomorphism f(λ,λ) : C1 (L, λ) → C3 (L , (λ, λ)). Here we are taking the case λ = λ and can regard C1 (L, λ) as a filtered An (λ),K  (λ) algebra. (We recall that in the beginning, C1 (L, λ) is a filtered An(λ),K(λ) algebra where n(λ), K(λ) are determined from Propositions 7.2.78 and 7.2.35 depending on Kλ , while n (λ), K  (λ) are determined from Propositions 7.2.126 and 7.2.100   depending on Kλ,λ which satisfies Kλ,λ ≤ Kλ . Thus we have (n (λ), K  (λ)) ≤ (n(λ), K(λ)).) Let gλ : C1 (L, λ) → C2 (L , λ) be the filtered An (λ),K  (λ) homomorphism obtained as the composition of the filtered An (λ),K  (λ) homomorphism f(λ,λ) : C1 (L, λ) → C3 (L , (λ, λ)) and the homotopy inverse to the inclusion C2 (L , λ) → C3 (L , (λ, λ)). Under this situation, it suffices to extend gλ0 to a filtered A∞ homomorphism. Recall that n (λ)  ∞ and K  (λ)  ∞ as λ → ∞. Proposition 7.2.132. There exist n (λ), K  (λ) with n (λ)  ∞, K  (λ)  ∞ as λ → ∞ such that the following holds. If λ < λ , then there exist filtered An (λ),K  (λ) homotopy equivalences 

h1,(λ,λ ) : C1 (L, λ) → C1 (L, λ ), 





h2,(λ,λ ) : C2 (L , λ) → C2 (L , λ ), 

such that gλ ◦ h1,(λ,λ ) is An (λ),K  (λ) homotopic to h2,(λ,λ ) ◦ gλ . Proposition 7.2.132 follows from the result of the next subsection and is proved in Subsection 7.2.11. Suppose that λ < λ . Then Theorem 7.2.72 implies that we can extend the filtered An (λ),K  (λ) structures on C1 (L, λ), C2 (L , λ) to filtered An (λ ),K  (λ )   structures and the filtered An (λ),K  (λ) homomorphisms h1,(λ,λ ) , h2,(λ,λ ) to filtered An (λ ),K  (λ ) homomorphisms. We denote them by the same symbol. Moreover, Lemma 7.2.129 and Proposition 7.2.132 imply that the An (λ),K  (λ) homomorphism gλ extends to a filtered An (λ ),K  (λ ) homomorphism (denoted by    the same symbol) such that gλ ◦ h1,(λ,λ ) is An (λ),K  (λ) homotopic to h2,(λ,λ ) ◦ gλ . Now using the sequence λ0 < λ1 < · · · → ∞ in the same way as the last part of Subsection 7.2.8, we obtain filtered A∞ structures on C1 (L, λ0 ) and C2 (L , λ0 ), and a filtered A∞ homomorphism g : C1 (L, λ0 ) → C2 (L , λ0 ). By composing the A∞ homotopy equivalence C2 (L , λ0 ) → C3 (L , (λ0 , λ0 )), we obtain the filtered A∞ homomorphism f : C1 (L, λ0 ) → C3 (L , (λ0 , λ0 )) as we want. The proof of Theorem 4.6.1 is now complete.  7.2.10. Construction of filtered An,K homotopies. The purpose of this subsection is to prove the An,K versions of Proposition 4.6.37 and of Theorem 4.6.25. The proof of Theorem 4.6.25 (the A∞ version) will be completed in Subsection 7.2.13. We use the same notations as those of Subsection 4.6.2. We start with fixing the notation we use in this subsection. We fix λ and write C1 (L, Λ0,nov ) = C1 (L, λ) as in the proof of Proposition 7.2.100. We also write C2 (L; Λ0,nov ) = C2 (L, λ ). The countable sets of chains X1 (L) = X1 (L, λ), X2 (L ) = X2 (L , λ ) generate them, respectively. From now on we will not concern the generations of elements of X1 (L), X2 (L ).

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We use the two parameter family {Jρ,s }ρ,s of almost complex structures in (4.6.26). We put {Jρs=0 }ρ = {Jρ,0 }ρ ,

{Jρs=1 }ρ = {Jρ,1 }ρ .

Let ψs : (M, L) → (M  , L ) be a family of symplectic diffeomorphisms as in Subsection 4.6.2, such that ψ0 = ψ, ψ1 = ψ  . We use {Jρ }ρ and ψ = ψ0 to perform the construction of Subsection 7.2.9. We s=0 then obtain X3,g (L ) (for g ≤ K) and ss=0,d,k,β,P  ,

top(ρ)

ss=0,k,β,P .

(We also omit λ, λ in the notation.) It induces a structure of filtered An,K algebra   s=0 (L )) and a filtered An,K mJ ,ss=0 on C3s=0 (L , Λ0,nov ) (which is generated by X3,(g) homomorphism top(ρ)

ψ0 ,{Jρs=0 }ρ ,ss=0

(7.2.133) fs=0

J  ,s s=0

s=0 : (C1 (L; Λ0,nov ), mJ,s (L ; Λ0,nov ), mk k ) → (C3

).

(Here (n, K) = (n(λ), K(λ)), which goes to infinity as λ goes to infinity. Strictly s=0 speaking, we should write C3s=0 (L ; Λ0,nov ) as C3,(g) (L ; Λ0,nov ) but we omit (g) for short, as we omit λ. We also remark that we will first construct fgeo and modify it as we have done for the construction of mgeo . We do not discuss the detail on this.) We next use {Jρs=1 }ρ and ψ  = ψ1 to perform the construction of Subsection s=1 (L ) (for g ≤ K = Kλ ) and 7.2.9. We then obtain X3,g ss=1,d,k,β,P  , 

top(ρ)

ss=1,k,β,P .



It defines a filtered An,K structure mJ ,ss=1 on C3s=1 (L ; Λ0,nov ) (which is generated s=1 by X3,(g) (L )) and a filtered An,K homomorphism top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

(7.2.134) fs=1

top(ρ)

ψ0 ,{J s=0 }ρ ,s

J  ,s s=1

s=1 : (C1 (L; Λ0,nov ), mJ,s (L ; Λ0,nov ), mk k ) → (C3

).

top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

s=0 and gs=1 gs=0 ρ homotopy inverses of

are compositions of (7.2.133), (7.2.134) with

C2 (L ; Λ0,nov ) → C3s=0 (L ; Λ0,nov ),

C2 (L ; Λ0,nov ) → C3s=1 (L ; Λ0,nov ),

respectively. We will find an An,K homotopy between top(ρ)

ψ0 ,{Jρs=0 }ρ ,ss=0

gs=0

top(ρ)

ψ1 ,{Jρs=1 }ρ ,ss=1

and gs=1

.

Since the target of (7.2.133) is different from the target of (7.2.134), we also need to construct a filtered An,K algebra (C([0, 1] × L ) ⊗ Λ0,nov , m) interpolating them. Hereafter we just write X3s=0 (L ), X3s=1 (L ) dropping the subscript g encoding the generation given to the elements thereof. (In this subsection, we will not increase the number of the chains chosen from L or in L but fix the set of chains.) So we omit the symbol d in the notation ss=0,d,k,β,P  , ss=1,d,k,β,P  and write ss=0,k,β,P  , ss=1,k,β,P  . (The symbol d is also omitted from the notation of the moduli spaces on which those multisections are defined.)

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485

We recall that the following condition for singular simplex on [0, 1] × L was introduced in Section 4.6. Definition 4.6.30. Let P = (|P|, f ) (f : |P| → [0, 1] × L ) be a smooth singular simplex. We say that P is an adapted simplex on [0, 1] × L if following conditions are satisfied for s0 = 0, 1. (4.6.31.1) f −1 ({s0 } × L ) ∩ |P| is either |P|, empty or consists of a single face (of arbitrary codimension) of a simplex |P|. (4.6.31.2) If f −1 ({0} × L ) ∩ |P| is nonempty then |P| ⊂ [0, 1/3) × L . If f −1 ({1} × L ) ∩ |P| is nonempty then |P| ⊂ (2/3, 1] × L . m m (4.6.31.3) Let Δm a be a face of |P| and Δb another face containing Δa . Suppose  m −1  m −1 that x ∈ Δa ⊂ f ({s0 } × L ) and Δb is not contained in f ({s0 } × L ). Then,  m for any vector N ∈ Tx Δm b \ Tx (∂Δb ), the [0, 1]-component of f∗ (N ) is nonzero. A singular chain of dimension m is regarded as a cochain of degree d = 1 + d dim L − m. We write S+ ([0, 1] × L ) the abelian group of cochains of degree d.  Let S+ ([0, 1] × L ) be the free abelian group generated by adapted singular simplex on [0, 1] × L . It is easy to see that this is a cochain complex. We have a canonical inclusion S({0, 1} × L ) ⊂ S+ ([0, 1] × L ) where S({0, 1} × L ) is the smooth singular chain complex. S({0, 1} × L ) is a subcomplex of S+ ([0, 1] × L ). We define S([0, 1] × L ) the quotient S([0, 1] × L ) =

S+ ([0, 1] × L ) . S({0, 1} × L )

It follows that S([0, 1] × L ) is free over the basis of adapted singular simplices not contained in {0, 1} × L . We next define the evaluation map Evals=s0 : S([0, 1] × L ) → S(L ) by defining its value to be Evals=s0 (P) = 0 −1



if f ({s0 } × L ) ∩ |P| is not of codimension one in |P|. If f −1 ({s0 } × L ) ∩ |P| is of codimension one in |P|, we set Evals=s0 (P) = (−1)s0 +c+1 (Δm−1 , fc ◦ jc ), where Δm−1 = (v0 , . . . , vc , . . . , vm ) is the face f −1 ({s0 } × L ) ∩ |P|, fc = f |Δm−1 c c and jc : Δm−1 → Δm−1 is the simplicial map given by c  vi if i < c jc (vi ) = vi+1 if i > c. We remark that Evals=s0 has degree zero since we use the cohomology notation for which the degree is given by the codimension of the chain. We will use the decomposition of [0, 1] into [0, 1/4], [1/4, 3/4] and [3/4, 1]. The homomorphism Incl1,β0 : S(L ) → S([0, 1] × L ) is defined by using the prism

486

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decomposition. Let [s, s ] be one of the intervals [0, 1/4], [1/4, 3/4] and [3/4, 1]. For Δm = (v0 , . . . , vm ), we consider the standard prism simplicial decomposition m $

[s, s ] × Δm =

(v0 , . . . , vi , vi , . . . , vm ),

i=0

where vi = (s, vi ) and vi = (s , vi ). Define the simplicial map ji : Δm+1 = (v0 , . . . , vm+1 ) → (v0 , . . . , vi , vi , . . . , vm ) by ji (vj ) = vj if j ≤ i and ji (vj ) = vj−1 if j > i. Then, for a singular simplex σ : Δm → L , we define Incl1,β0 (σ) =

m

(−1)i (id[0,1/4] × σ) ◦ ji

i=0

(7.2.135)

+

m

(−1)i (id[1/4,3/4] × σ) ◦ ji i=0

m

+ (−1)i (id[3/4,1] × σ) ◦ ji . i=0

By the choice of the decomposition of [0, 1], in particular, by the choice of 1/4 < 1/3, the output of the map Incl1,β0 of each simplex in the prism decomposition is an adapted simplex. Recall that m1,β0 on C(L ; Λ0,nov ) is defined to be (−1)n ∂, n = dim L . We set m1,β0 on S([0, 1] × L ; Λ0,nov ) to be (−1)n+1 ∂ mod S({0, 1} × L; Λ0,nov ). Note that n + 1 = dim[0, 1] × L . Then we easily see that m1,β0 ◦ Incl1,β0 = Incl1,β0 ◦ m1,β0 . See Subsection 8.9.2 for the compatibility of signs in the filtered A∞ algebra structure. Lemma (7.2.137.1) (7.2.137.2) (7.2.137.3)

7.2.136. Evals=s0 is a chain map with respect to m1,β0 . Evals=0 ⊕ Evals=1 : S([0, 1] × L ) → S(L ) ⊕ S(L ) is surjective. Evals=s0 and Incl1,β0 are chain homotopy equivalences.

Proof. We first prove (7.2.137.1). Let (P, f ) be an adapted singular simplex of dimension m not contained in {0, 1} × L . We consider 4 cases (7.2.138.1) (7.2.138.2) (7.2.138.3) (7.2.138.4)

dim f −1 ({0} × L ) ∩ |P| = m, dim f −1 ({0} × L ) ∩ |P| < m − 2, dim f −1 ({0} × L ) ∩ |P| = m − 1, dim f −1 ({0} × L ) ∩ |P| = m − 2,

separately and prove ∂(Evals=s0 (P)) = −Evals=s0 (∂P), which is equivalent to m1,β0 ◦ (Evals=s0 (P)) = Evals=s0 ◦ m1,β0 (P).

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487

For the cases (7.2.138.1) and (7.2.138.2), the left and right hand sides are both 0. For the case (7.2.138.3), we have f −1 ({0} × L ) = Δm−1 by the condition (4.6.31.1) c where Δm−1 is a codimension one face of |P|. Therefore we have c Evals=s0 (P) = (−1)s0 +c+1 (Δm−1 , fc ◦ jc ),

fc := f |Δm−1 c

for some c = 0, · · · , m. By the condition for adapted singular simplices, all other faces (Δm−1 , f |Δm−1 ) for b = c are not contained in {0} × L . We denote ∂b P = b b  (Δm−1 , f |Δm−1 ◦ jb ) and then we have ∂P = (−1)b ∂b P and hence b %m &

Evals=s0 (∂P) = Evals=s0 (−1)b ∂b P b=0

⎞ ⎛ m

= Evals=s0 (−1)c ∂c P + Evals=s0 ⎝ (−1)b ∂b P ⎠ b=c

=

m

(−1)b Evals=s0 (∂b P),

b=c

because ∂c P ⊂ {s0 } × L by definition of S+ ([0, 1] × L ). On the other hand, we have Evals=s0 (∂P) 



= (−1)s0 +1 (−1)b+c (∂c ∂b Δm , f ) + (−1)b+c−1 (∂c−1 ∂b Δm , f ) bc

(−1)b+c (∂b−1 ∂c Δm , f ) +

bc

= (−1)

=− (−1)b ∂b Evals=s0 (P) = −∂(Evals=s0 (P)).

Thus we obtain ∂(Evals=s0 (P)) = −Evals=s0 (∂P). , Δm−1 such For the case (7.2.138.4), we have two codimension one faces Δm−1 i j that f −1 ({0} × L ) ∩ ∂Δm−1 = −f −1 ({0} × L ) ∩ ∂Δm−1 = f −1 ({0} × L ) ∩ |P|. i j Then the left and right hand sides are both 0 again. The proof of (7.2.137.1) is complete. The proof of (7.2.137.2) is obvious and so omitted. We now prove (7.2.137.3). Since Evals=s0 ◦ Incl1,β0 is the identity it follows that Evals=s0 induces a surjection in homology. It remains to prove the injectivity in homology. Let

C= ai Pi ∈ S d ([0, 1] × L ) i∈I

be a cycle in S d ([0, 1] × L ) such that Evals=0 (C) = ∂D for some chain D ∈ S(L ). d We represent Pi by an adapted singular simplex element in S+ ([0, 1] × L ) and

488

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d regard D ∈ S(L ) as an element of S+ ([0, 1] × L ) via the obvious inclusion S(L ) →  d  d ([0, 1] × L ) such that S({0} × L ) ⊂ S+ ([0, 1] × L ). Then C − D is a chain in S+

∂(C − D) ∈ S({1} × L ). Since the relative homology group H([0, 1] × L ; {1} × L ) is zero we can find a d−1 ([0, 1] × L ). To singular chain E that bounds C − D. However E may not lie in S+ d−1  find a chain E contained in S+ ([0, 1] × L ), we proceed as follows. Let I0 ⊂ I be the subset of I such that Pi intersect with {0}×L for i ∈ I0 . We put Pi = (|Pi |, fi ). By the adaptedness of Pi or more specifically by (4.6.31.2), there exists  > 0 such that fi (|Pi |) ⊂ [0, 1/3 − ) × L . We put fi = (fi,s , fi,L ). We define fˆi by   t + fi,s (x), fi,L (x) . (7.2.139) fˆi (x, t) = 2 Using this map and the prism decomposition of |Pi | × [0, 1], we obtain a singular chain Ψ(Pi ) ∈ S+ ([0, 1] × L ). It is easy to see that the support of ∂Ψ(Pi ) −  Ψ(∂Pi ) − Pi does  not intersect with {0} × L . We define Ψ(D) in a similar way and put E1 = i∈I0 ai Ψ(Pi ) − Ψ(D). Then C − D − ∂E1 is a chain which is disjoint from {0} × L and satisfies ∂(C − D − ∂E1 ) = ∂(C − D) ∈ S({1} × L ). Now it is easy to find E2 such that C − D − ∂E1 − ∂E2 ∈ S({1} × L ) d−1 so that the chain E = E1 + E2 lies in S+ ([0, 1] × L ). The proof of (7.2.137.3) is complete. 

We now introduce a family of countable subsets Xg ([0, 1] × L ) of singular simplices on [0, 1] × L for g = 1, 2, · · · whose defining conditions will be given in Condition 7.2.141 below. To state these conditions, we need one more definition. Definition 7.2.140. For P ∈ X1 (L) we define a chain {ψs }s∗ (P ) on [0, 1] × L as follows: Let P = (|P |, f ), where |P | is a simplex which is regarded as a manifold with corners and f : |P | → L is smooth. We define {ψs }s ◦f : [0, 1]×|P | → [0, 1]×L by ({ψs }s ◦ f )(s, x) = (s, ψs (f (x))). We take the prism decomposition of [0, 1] × |P |. (We decompose [0, 1] to [0, 1/4], [1/4, 3/4] and [3/4, 1] as before.) We then obtain a singular chain on [0, 1] × L . We denote it by {ψs }s∗ (P ). We identify an element of X3s=s0 (L ) with that of S({s0 } × L ) in the next condition. Then we may regard X3s=s0 (L ) as a subset of S+ ([0, 1] × L ). Condition 7.2.141. We write X(g0 ) ([0, 1] × L ) =



Xg ([0, 1] × L ).

g≤g0

We denote by C(g0 ) ([0, 1] × L ; Q) the Q vector space spanned by X(g0 ) ([0, 1] × L ).

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

489

(7.2.142.0) X00 ([0, 1]×L ) is the union of the singular simplices which appear in the simplicial decompositions of the elements of {[0, 1]×P  | P  ∈ X2 (L )}∪{{ψs }s∗ (P ) | P ∈ X1 (L)} and which is not contained in {0, 1} × L . Note we have already fixed simplicial decomposition, when we defined Incl1,β0 and in Definition 7.2.140. (7.2.142.1) Xg ([0, 1]×L ) is a countable set of smooth singular simplex of [0, 1]×L . (7.2.142.2) The inclusion C(g) ([0, 1] × L ; Q) → S([0, 1] × L ; Q) induces an isomorphism in cohomology. (7.2.142.3) Every face of each element of Xg ([0, 1] × L ) is an element of X(g) ([0, 1] × L ) ∪ X3s=0 (L ) ∪ X3s=1 (L ). (7.2.142.4) (7.2.142.5)

Elements of Xg ([0, 1] × L ) satisfy Condition 4.6.30. Evals=s0 of elements of Xg ([0, 1] × L ) is contained in X3s=s0 (L ).

We remark that (7.2.142.1)–(7.2.142.3) are the analogs to Properties 7.2.27. We say that elements of Xg ([0, 1] × L ) have generation g. Hereafter we sometimes write P|s=s0 for Evals=s0 (P). We next review the two kinds of the moduli spaces we will use. One is    Mmain k+1 (M , L , {Jρ,s }ρ,s : β; top(ρ), twp(s); P )

which is defined in (4.6.29). Here P stands for the k-tuple (P1 , · · · , Pk ) and Pi ∈ X1 (L). For the simplicity of notations, we just write it as (7.2.143)

 Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ).

We have the evaluation map   (7.2.144) ev0+ = (evs , ev0 ) : Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) → [0, 1] × L .

Note that we can choose the Kuranishi structure of our space Mk+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s)) so that the evaluation map ev : Mk+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s)) → Lk × ([0, 1] × L ) is weakly submersive. In fact the weak-submersivity to the factor Lk × L can be proved in the same way as in Section 7.1. We can extend the Kuranishi neighborhood by adding [0, 1] directions and add one dimensional obstruction bundle, so that [0, 1] factor of the evaluation map is also weakly submersive. It follows that (7.2.144) is weakly submersive. See Remark A1.44. (This point was also mentioned by Akaho and Joyce [AkJo08].) The other moduli space we use is (7.2.145)

   Mmain,d k+1 (M , L , {J1,s }s : β; twp(s); P)

 = (P1 , · · · , Pk ) and Pi ∈ Xg ([0, 1]×L ). which is defined just after (4.6.35). Here P i We put d(i) = gi and simplify the notation for (7.2.145) as (7.2.146)

 Mmain,d k+1 ({J1,s }s : β; twp(s); P).

(Here we omit M  and L to simplify the notation.) There is an evaluation map (7.2.147)

  ev0+ = (evs , ev0 ) : Mmain,d k+1 ({J1,s }s : β; twp(s); P) → [0, 1] × L .

See (4.6.35). We may assume that (7.2.147) is weakly submersive in the same way as the weak submersivity of (7.2.144).

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We next describe how those moduli spaces behave at s = 0 or s = 1. Lemma 7.2.148. For s0 = 0 or 1, there exists a map main   Ics0 : Mmain k+1 ({Jρ,s0 }ρ : β; top(ρ); P ) → Mk+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P )

such that the following diagram commutes: ev

 Mmain k+1 ({Jρ,s0 }ρ : β; top(ρ); P ) ⏐ ⏐ Ics0 (

L ⏐ ⏐ x→(s0 ,x)(

0 −−−− →

+

ev0   Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) −−−−→ [0, 1] × L

Diagram 7.2.1  Here Mmain k+1 ({Jρ,s0 }ρ : β; top(ρ); P ), (s0 = top(ρ) top(ρ) ψ0 ,{Jρs=0 }ρ ,ss=0 ψ1 ,{Jρs=1 }ρ ,ss=1 and fs=1 . define fs=0

0, 1) are the moduli spaces used to

The map Ics0 lifts to a fiberwise injective homomorphism of the obstruction bundles and the lift is compatible with the Kuranishi maps. Lemma 7.2.148 is actually obvious from the construction. Remark 7.2.149. The last part of the statement of Lemma 7.2.148 may be formulated as Ics0 being a morphism between the spaces with Kuranishi structure. We have not defined such a notion yet partly because it is not needed so far and also because its optimal definition is still not clear to us.)  = (P1 ,· · ·, Pk ), Next consider the case of the moduli space (7.2.146). Let P  Pi ∈ Xgi ([0, 1] × L ). We put  s=s = (P1 |s=s , · · · , Pk |s=s ) , P| 0 0 0 

where Pi |s=s0 ∈

X3s=0 X3s=1

for s = 0 for s = 1.

Lemma 7.2.150. For s0 = 0 or 1, there exists a map main,d main     Icρ=1 s0 : Mk+1 (L , Js0 : β, P|s=s0 ) → Mk+1 ({J1,s }s : β; twp(s); P)

such that the following diagram commutes.    Mmain k+1 (L , Js0 : β, P|s=s0 ) ⏐ ⏐ Icρ=1 s0 (

ev

L ⏐ ⏐ x→(s0 ,x)(

0 −−−− →

+

ev0   Mmain,d k+1 ({J1,s }s : β; twp(s); P) −−−−→ [0, 1] × L

Diagram 7.2.2 The map lifts to a fiberwise injective homomorphism of the obstruction bundles and the lift is compatible with the Kuranishi maps. Icρ=1 s0

The proof is again immediate from the construction. We next describe boundaries of the moduli spaces we use. We start with the  = (P1 , · · · , Pk ). Let the singular simplex Pi be realized by case of (7.2.146). Let P (7.2.151)

fi : |Pi | → [0, 1] × L .

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

491

  We put fi = (fis , fiL ). An element p of Mmain,d k+1 ({J1,s }s : β; twp(s); P) is written as   (7.2.152.1) p = s, (Σ, (z0 , · · · , zk )), u, (x1 , · · · , xk ) ,

where Σ is a genus zero bordered Riemann surface, (z0 , · · · , zk ) are its marked points on the boundary respecting the cyclic order. It also satisfies the following: (7.2.152.2) (7.2.152.3) (7.2.152.4) (7.2.152.5)

u : (Σ, ∂Σ) → (M  , L ) is a J1,s holomorphic map. xi ∈ |Pi |. fis (xi ) = s for i = 1, · · · , k.  fiL (xi ) = u(zi ) for i = 1, · · · , k.

We recall that the evaluation map is defined by ev0+ (p) = (s, u(z0 )).

(7.2.152.6)

  Now let pα = sα , (Σα , (z0,α , · · · , zk,α )), uα , (x1,α , · · · , xk,α ) be a sequence of el ements of Mmain,d k+1 ({J1,s }s : β; twp(s); P) converging to the boundary. We then have three possibilities: (7.2.153.1) (7.2.153.2) (7.2.153.3)

limα→∞ xi,α ∈ ∂|Pi |. The boundary bubble occurs for (Σα , (z0,α , · · · , zk,α )) as α → ∞. sα → 0 or 1, as α → ∞.

The case (7.2.153.2) can be described as follows. (The following description corresponds to the case when z,α , · · · , zm,α lie on the bubble.) Let , m ∈ {1, · · · , k} with  − 1 ≤ m. (The case  − 1 = m is included.) We put   + (7.2.154.1) Q0,m = ev0∗ Mmain m−+2 ({J1,s }s : β1 ; twp(s); (P , · · · , Pm )) . Then the bubble corresponding to (7.2.153.2) can be described as   0 (7.2.154.2) Mmain k+−m+1 {J1,s }s : β2 ; twp(s); (P1 , · · · , P−1 , Q,m Pm+1 , · · · , Pk ) , where β = β1 + β2 . See Figure 7.2.9. (7.2.153.3) gives an element which is zero in S([0, 1] × L ) since it is contained in the boundary. P

P

P−1

β2

P1

β1

z0 Pk

Pm

Pm+1

Q0,m

Figure 7.2.9 We next describe the boundary of the moduli space (7.2.143).

492

7.

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Let P = (P1 , · · · , Pk ). Suppose that Pi is realized by fi : |Pi | → L. An element  ˜ = (s, p, (x1 , · · · , xk )) ˜ of Mmain p k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) is written as p such that: (7.2.155.1) s ∈ [0, 1].   (7.2.155.2) p ∈ Mmain k+1 (M , L , {Jρ,s }ρ : β; top(ρ)). Here the right hand side is the moduli space defined in Definition 4.6.8. We remark that here we consider the one parameter family {Jρ,s }ρ of almost complex structures for each s ∈ [0, 1]. (7.2.155.3) xi ∈ |Pi | and fi (xi ) = evi (p). Here evi is the evaluation map defined just before Definition 4.6.10. We remark that the evaluation map ev0+ is given by (7.2.156)

ev0+ (s, p, (x1 , · · · , xk )) = (s, ev0 (p)) ∈ [0, 1] × L .

Now we suppose that  (sα , pα , (x1,α , · · · , xk,α )) ∈ Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) is a sequence converging to a point in the boundary. We put (i) pα = ((Σ(i) zα(i) ), (u(i) α , α ), (ρα )).

(See Definitions 4.6.8 and (4.6.7) for this notation.) We have the following possibilities to occur in the limit: (7.2.157.1) (7.2.157.2) (7.2.157.3) (7.2.157.4) (7.2.157.5) (7.2.157.6)

(4.6.9.1) occurs for pα . (4.6.9.2) occurs for pα . (4.6.9.3) occurs for pα . (4.6.9.4) occurs for pα . limα→∞ xi,α = ∂|Pi | for some i. limα→∞ sα = 0 or 1.

As in the case of (4.6.9), the boundary component corresponding to (7.2.157.1) cancels with (7.2.157.2). The boundary corresponding to (7.2.157.6) is again zero in S([0, 1] × L ). We now describe the boundary components corresponding to (7.2.157.3) and (7.2.157.4) below. (i) We first consider the case of (7.2.157.3). Suppose limα→∞ ρα = 0. Let () (m) (i) zα , · · · , zα be the marked points on Σα . (We may assume that , m are independent of α by taking a subsequence if necessary.) Here  ≤ m + 1. (The case (i)  − 1 = m is included. In that case there is no marked point on Σα .) Then it is easy to see that such a component is described by (7.2.158.1)

  0 {J Mmain } : β ; top(ρ), twp(s); (P , · · · , P , Q , P , · · · , P ) ρ,s ρ,s 2 1 −1 m+1 k k−m++1 ,m

where β = β1 + β2 and (7.2.158.2)

  Q0,m = ev0∗ Mmain m−+2 (L, J; β1 ; (P , · · · , Pm )) .

We next consider the boundary component corresponding to (7.2.157.4). Sup(i) (i) pose limα→∞ ρα = 1. Then the component Σα must contain the 0-th marked point  by Condition (4.6.7.2) in the definition of Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ).

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

493

(i)

Let w1 , · · · , wj be the marked or singular points on ∂Σα . Suppose that wj = zj (namely wj is the j th marked point). Then we define a chain R0j ,j+1 in [0, 1] × L by (7.2.159.1)

R0j ,j+1 = {ψs }s∗ (Pj ),

where {ψs }s∗ is as in Definition 7.2.140. We also put β1,j = β0 = 0. Suppose that wj is a singular point. We consider the union of discs and spheres which lie on the tree of discs rooted at wj . Let zj , · · · , zj+1 −1 be all the marked points on this union of discs. Let β1,j be the sum of the homology classes of the discs contained in this tree. We define (7.2.159.2)   + Mmain R0j ,j+1 = ev0∗ ({J } : β ; top(ρ), twp(s); (P , · · · , P )) . ρ,s ρ,s 1,j j j+1 −1 j+1 −j +1 Now it is clear that the boundary component corresponding to (7.2.157.4) in our case is   0 0 (7.2.159.3) Mmain j+1 −j +1 {J1,s }s : β2 ; twp(s); (R1 ,2 , · · · , Rm−1 ,m ) . We are now ready to state the following Proposition 7.2.160 which is an analog of Propositions 7.2.35 and 7.2.100. Proposition 7.2.160 is used to prove Proposition 4.6.37. Proposition 7.2.160. Let K be a given positive integer. Then there exists a twp(s) top(ρ),twp(s) system of multisections sd,k,β,P (for (d, β) ≤ K) and sk,β,P (for k + β ≤  K). There also exists Xg ([0, 1] × L ) for g ≤ K. They have the following properties. (7.2.161.1) (7.2.161.2)

Xg ([0, 1] × L ) satisfies Condition 4.6.30. twp(s) s is a piecewise smooth multisection of d,k,β,P  Mmain,d k+1 ({J1,s }s : β; twp(s); P)

and is transversal to zero. top(ρ),twp(s)  s is a multisection of Mmain k+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) and is k,β,P transversal to zero. (7.2.161.3) If (d, β) = g, then   twp(s) s +   d,k,β,P ev0∗ Mmain,d ({J } : β; twp(s); P) 1,s s k+1 is decomposed into elements of X(g) ([0, 1] × L ). (7.2.161.4) If β + k = g, then   top(ρ),twp(s) s + main   k,β, P ev0∗ Mk+1 ({Jρ,s }ρ,s : β; top(ρ), twp(s); P ) is decomposed into elements of X(g) ([0, 1] × L ). twp(s) top(ρ),twp(s) are compatible in the sense (7.2.161.5) The multisections sd,k,β,P and sk,β,P described below.

494

7.

TRANSVERSALITY

twp(s)

top(ρ),twp(s)

(7.2.161.6) The zero sets (sd,k,β,P )−1 (0) and (sk,β,P )−1 (0) are in the δ neighborhood of the zero set of the original Kuranishi map. Here δ is a positive number such that it converges to zero as λ → ∞. (7.2.161.7) The multisection ss=0,k,β,P| (resp. ss=1,k,β,P| ) is compatible with s=0

s=1

twp(s)

(resp. Icρ=1 ) in Lemma 7.2.148. sd,k,β,P by the map Icρ=1 0 1 (7.2.161.8) map

Icρ=1 0

top(ρ)

top(ρ)

top(ρ),twp(s)

ss=0,k,β,P (resp. ss=1,k,β,P ) is compatible with sk,β,P (resp.

Icρ=1 ) 1

by the

in Lemma 7.2.150.

We need to describe the compatibility condition required in (7.2.161.5) more explicitly according to the various cases discussed in (7.2.153), (7.2.157). For the case (7.2.153.1), the compatibility is one described as in Compatibility Condition 7.2.38. For the case (7.2.153.2), it is Compatibility Condition 7.2.44, in view of (7.2.154.1), (7.2.154.2). For the case (7.2.157.3), it is Compatibility Condition 7.2.102, in view of (7.2.158.1), (7.2.158.2). For the case (7.2.157.4), it is Compatibility Condition 30.104, in view of (7.2.159.1), (7.2.159.2), (7.2.159.3). For the case (7.2.157.5), is Compatibility Condition 7.2.38. Proof. Once we make the statements precise, the scheme of the proof is not very different from that of Proposition 7.2.100 or Proposition 7.2.35. We first define required multisections and triangulations inductively over a partial order j0 . Since dim C = dim Cj , C ⊇ Cj , it follows that C \ Cj is nowhere dense (with # respect to the complex analytic topology). Hence*by Baire’s category theorem, j>j0 (C \ Cj ) is nowhere dense in C. In particular j≥j0 Cj is * * * non-empty. Since j≥j0 Cj ⊂ j Zj by definition, it follows that j Zj is nonempty * and hence we can find h1 ∈ j Zj . Now suppose the sublemma for I. For j > I, we put WI,j = {h ∈ V (n(j), K(j)) | Res(n(j),K(j)),(n(I),K(I)) (h) = hI }. By the same reasoning as above we show that it is nonempty for each j. Then since Res(n(j),K(j)),(n(I+1),K(I+1)) )(WI,j ) is a non-increasing sequence of constructible sets, we can show by the same way that its intersection is nonempty. Hence we can find hI+1 .  

Therefore the proof of Lemma 7.2.177 is now complete.

Now we are ready to complete the proof of Theorem 4.6.25 for the case where R is C or a finite field. Let us recall what we have proved already. The filtered A∞ structures on C1 (L, λ0 ) and C2 (L , λ0 ) are already constructed (in Subsection 7.2.8. Lemma 7.2.93 which was deferred at Subsection 7.2.8 was proved in this top(ρ)

ψ0 ,{J s=0 }ρ ,s

top(ρ)

ψ1 ,{J s=1 }ρ ,s

s=0 s=1 subsection). The filtered A∞ homomorphisms gs=0 ρ , gs=1 ρ between them are also constructed (in Subsection 7.2.9. Proposition 7.2.132 which was deferred at Subsection 7.2.9 is proved in this subsection.). It is proved that they are An,K homotopic to one another for any n, K, in Subsection 7.2.10. We recall that C1 (L, λ0 ) and C2 (L , λ0 ) are filtered A∞ homotopy equivalent to the filtered A∞ algebras defined on H(L, Λ0,nov ) and H(L , Λ0,nov ), respectively, by Theorem 5.4.1. We remark that the latter algebras are finitely generated. top(ρ)

ψ0 ,{Jρs=0 }ρ ,ss=0

Thus we can apply Lemma 7.2.177 to show that gs=0 top(ρ)

ψ1 ,{J s=1 }ρ ,ss=1 gs=1 ρ

is filtered

A∞ homotopic to . The proof of R = C or a finite field version of Theorem 4.6.25 is now complete. 

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

505

Remark 7.2.181. Unfortunately the proof of Lemma 7.2.177 given above does not work for a general coefficient ring R. The projective limit of geometric points of nonempty algebraic variety over, say Q, could be empty in general. For example, consider a sequence of sets Xn = {(x1 , · · · , xn ) ∈ Qn | xi = 1 + x2i+1 , i = 1, · · · , n − 1}. Then we have the projection map Xn+1 → Xn by forgetting xn+1 . If (x1 , · · · , xn ) ∈ Xn , an induction argument proves that x1 ≥ n − 1. Hence lim Xn = ∅. The authors ←− do not know whether Lemma 7.2.177 is true for general R or not. This difficulty is related to the fact that the functor lim is not an exact functor ←−

in various categories. This point had appeared in algebraic topology in [Mil62]. 7.2.12. Constructions of filtered A∞ homotopies II: the algebraic framework on homotopy of homotopies. The discussions of this subsection and the next involve a use of heavy notations. However these are natural continuations of the study of general A∞ or An,K structures given in the previous subsections and hence may be unavoidable in the proof of Theorem 4.6.25 for a general coefficient ring R. Since the result in the last subsection dealing with the coefficient Z2 or C is general enough to cover all the applications we know of at the time of writing of this book, the rest of this section can be skipped for those whose main interest lies in the applications of the Floer theory. We advise them to directly proceed to Section 7.3 skipping this and the next subsections. Subsection 7.2.14 is used only in Section 7.4. The materials we present in this subsection and the next provide a systematic study of homotopy of homotopies. We use them in our derivation of a filtered A∞ homotopy out of given filtered An,K homotopies for a general coefficient ring R. In this subsection and the next, we fix a monoid G and all filtered A∞ homomorphisms etc. are assumed to be G-gapped. Also we simply say a filtered An,K structure and etc. for a G-filtered An,K structure and etc.. This subsection sets up the algebraic framework for the above purpose. We first recall that our construction of filtered A∞ homomorphisms in the previous subsection involves the sequences n(λi ) and k(λi ). We wanted them to satisfy λi → ∞,

n(λi ) → ∞,

k(λi ) → ∞

λi

n(λi )

k(λi ) → ∞.

→ ∞,

→ ∞,

We also assume n(λi ) ≤ n(λi ),

k(λi ) ≤ k(λi ).

Definition 7.2.182. (1) Let λi → ∞ be given and let (n, k) be a sequence (n(λi ), k(λi )) given as above. (7.2.182.1) Suppose Ci is a sequence of filtered An(λi ),k(λi ) algebras. We call a sequence {Ii : Ci → Ci+1 } of An(λi ),k(λi ) homotopy equivalences a directed system of equivalences of A(n,k) algebras. Hereafter in this book we call a directed system of A(n,k) algebras in place of a directed system of equivalences of A(n,k) algebras, for simplicity. (We do not use the case when Ii is not a homotopy equivalence in this book.)

506

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 (7.2.182.2) Let {Ii : Ci → Ci+1 } and {Ii : Ci → Ci+1 } be two directed systems of A(n,k) algebras. We define a morphism, denoted by {fi : Ci → Ci }, from {Ii : Ci →  Ci+1 } to {Ii : Ci → Ci+1 } to be a sequence of filtered An(λi ),k(λi ) homomorphisms  fi : Ci → Ci satisfying

(7.2.183)

Ii ◦ fi ∼ fi+1 ◦ Ii , I

C1 −−−1−→ ⏐ ⏐f (1 I

( ∼ means filtered An(λi ),k(λi ) homotopic). I

Ik−1

Ck −−−k−→ · · · ⏐ ⏐f ··· (k

I

 Ik−1

I

C2 −−−2−→ · · · −−−−→ ⏐ ⏐f ··· (2

I

C1 −−−1−→ C2 −−−2−→ · · · −−−−→ Ck −−−k−→ · · · Diagram 7.2.3 (2) Let (C, {mk,β }(β,k)(n,K) ) be a G-gapped filtered An,K algebra. Its pro˜ k,β }) such that m ˜ k,β = mk,β for (β, k)  motion to a filtered A∞ algebra is (C, {m (n, K). A promotion to a filtered An ,K  algebra (where (n, k)  (n , K  )) is defined in the same way. (3) Let {fk,β }(β,k)(n,K) be a G-gapped filtered An,K homomorphism : C → C  between filtered A∞ algebras. Its promotion to a filtered A∞ homomorphism is a filtered A∞ homomorphism {˜fk,β } such that ˜fk,β = fk,β for (β, k)  (n, K). A promotion to a filtered An ,K  homomorphism (where (n, k)  (n , K  )) is defined in the same way. (4) A promotion of a filtered An,K homotopy to a filtered A∞ homotopy is defined in the same way. The argument at the ends of Subsections 7.2.8 and 7.2.9 implies the following.  Lemma 7.2.184. Let {Ii : Ci → Ci+1 }and {Ii : Ci → Ci+1 }be directed systems  of A(n,k) algebras and {fi : Ci → Ci } a morphism between them. Then we have:

(7.2.185.1) We can promote Ci to filtered A∞ algebras and Ii to filtered A∞ homomorphisms between them. We denote them by the same system. We call a system of such A∞ algebras and homotopy equivalences a promotion  of the directed system {Ii : Ci → Ci+1 }. The same holds for {Ii : Ci → Ci+1 }. (7.2.185.2) For any such promotion as in (7.2.185.1), the homomorphism fi : Ci → Ci promotes to a filtered A∞ homomorphism so that Diagram 7.2.3 commutes up to a filtered A∞ homotopy. We write them by ˜fi . We call a system {˜fi } of such fi a promotion of the morphism {fi }. (Hereafter we will frequently write fi in place of ˜fi , if no confusion can occur.) The lemma is proved by the induction argument similar to those used in Subsections 7.2.8 and 7.2.9. See also Lemma 7.2.208. By an inductive construction of the following diagram, we can prove the promotion of {Ii : Ci → Ci+1 } in (7.2.185.1) is unique up to a filtered A∞ homotopy equivalence. I

C1 −−−1−→ ⏐ ⏐i (1 I

I

Ik−1

Ck −−−k−→ · · · ⏐ ⏐i ··· (k

I

Ik−1

I

C2 −−−2−→ · · · −−−−→ ⏐ ⏐i ··· (2

I

C1 −−−1−→ C2 −−−2−→ · · · −−−−→ Ck −−−k−→ · · · Diagram 7.2.4

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

507

Namely we assume that there are two different ways of promoting the structures as in (7.2.185.1). We put them in the first and second lines of the Diagram 7.2.4 respectively. We then construct ij (j ≤ i) as filtered An(λi ),k(λi ) homomorphisms such that the first i columns of Diagram 7.2.4 commute up to An(λi ),k(λi ) homotopy. We can do it inductively starting from i1 = id (as filtered An(λ1 ),k(λ1 ) homomorphism) and using Lemma 7.2.129. We fix a promotion constructed in (7.2.185.1) from now on. Our concern here is whether the filtered A∞ homomorphism fi in (7.2.185.2) is unique up to filtered A∞ homotopy. In fact if we try to prove this by imitating the proof of the uniqueness of the construction (7.2.185.1) given above, then we will be in trouble. The main source of trouble lies in the fact that the Diagram 7.2.3 is only homotopy commutative. On the other hand, it is clear that, for any fixed n, K, the filtered An,K homotopy type of fi is independent of the promotion {fi }. We thus meet the problem pointed out at the end of the last subsection. To resolve this trouble we introduce the following refinement of Definition 7.2.182. Definition 7.2.182bis. Let {fi : Ci → Ci } be a morphism from a directed (r)  system {Ii : Ci → Ci+1 } to another {Ii : Ci → Ci+1 }. Suppose Ci is a filtered An(λi ),k(λi ) algebra which is a model of [0, 1] × Ci . Consider a sequence of filtered An(λi ),k(λi ) homomorphisms (r)

Fi : Ci → Ci+1 such that (7.2.186)

Evalr=0 ◦Fi = Ii ◦ fi ,

Evalr=1 ◦Fi = fi+1 ◦ Ii .

(r)

We call such a system {Fi : Ci → Ci+1 } a commuting error homotopy of the morphism {fi : Ci → Ci }. (Note we use the symbol r instead of s as the parameter of the interval, since it is more consistent with other notations which will come later.) Of course existence of such Fi is nothing but the condition (7.2.183). The upshot here is that we make a specific choice of homotopy Fi . We can now generalize Lemma 7.2.184 in the following way. (r)

Lemma 7.2.184bis. Suppose {Fi : Ci → Ci+1 } is the commuting error homotopy of the morphism {fi : Ci → Ci } of the directed systems {Ii : Ci → Ci+1 } and  {Ii : Ci → Ci+1 }. Then we have: (7.2.185.3) We can promote {Fi } to a filtered A∞ homotopy of the promotions  of {Ii : Ci → Ci+1 } and {Ii : Ci → Ci+1 }. Namely Fi can be promoted to a filtered A∞ homomorphism that satisfies (7.2.186) as an equality of filtered A∞ homomorphisms. More precisely we have Evalr=0 ◦F = Ii ◦ fi ,

Evalr=1 ◦F = fi+1 ◦ Ii

where f = {fi } and F = {Fi } are the promotions of {fi } and {Fi } respectively. We call this promotion F a commuting error homotopy of the promotion.

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In fact Lemma 7.2.184bis was already proved in Subsection 7.2.9. (See the proof of Lemma 7.2.129.) See also Lemma 7.2.208. Now we have:  Proposition 7.2.187. Let {Ii : Ci → Ci+1 } and {Ii : Ci → Ci+1 } be two (r) directed systems and {fi } a morphism between them. Suppose {Fi : Ci → Ci+1 } is a commuting error homotopy of {fi }. Let A∞ promotions {Ii : Ci → Ci+1 }  and {Ii : Ci → Ci+1 } be given, and assume f0 = {f0i } and f1 = {f1i } are two A∞ promotions of {fi } on the given promotions. Suppose that A∞ promotions (r) {Fi : Ci → Ci+1 } of the commuting error homotopy of both f0 and f1 are given respectively. (We do not assume however that the filtered A∞ promotions of Fi are the same.) Then f0 is filtered A∞ homotopic to f1 .

We postpone the proof of Proposition 7.2.187 until after we state Theorem 7.2.212 and Corollary 7.2.219 below, in which we weaken the condition that the An,k -commuting error homotopy {Fi } is common to both f0 , f1 by a more natural and flexible condition. To make the precise statement of Theorem 7.2.212 below, we need to systematically use the notion of homotopy of homotopies. We use the same notation Eval, Incl for the various models we use whose meaning should be clear from the context. Definition 7.2.188. Let Cr0 ,s0 (r0 = 0, 1, s0 = 0, 1) be 4 filtered A∞ algebras containing a common subalgebra C0 (that is, the structure mk of C0 is the restriction of that of Cr0 ,s0 ), such that each inclusion C0 ⊂ Cr0 ,s0 is a homotopy equivalence. We denote the quintuple by C = (C0 , {Cr0 ,s0 }r0 ,s0 ).  is called a model of [0, 1]2 × C if the following conditions A filtered A∞ algebra C hold: r=r of [0, 1] × (C0 , {Cr ,0 , Cr ,1 }) (r0 = 0, 1) (7.2.189.1) There are models, C 0 0 0  and Cs=s0 of [0, 1] × (C0 , {C0,s0 , C1,s0 }) (s0 = 0, 1) respectively. (See Definition r=r by Evals=s (s0 = 0, 1) and Eval of C  s=s by 7.2.174.) We denote Eval of C 0 0 0 Evalr=r0 (r0 = 0, 1). Cr= 0 ,s=1

s

s= 1 C

r= 0 C

 C

Cr= 0 ,s=0

s= 0 C

Cr= 1 ,s=1

r= 1 C

Cr= 1 ,s=0 r

Figure 7.2.10 (7.2.189.2)

There exist filtered A∞ homomorphisms →C r=r , Evalr=r0 : C 0

→C s=s . Evals=s0 : C 0

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

509

(7.2.189.3) (Evalr=r0 )k and (Evals=s0 )k are zero for k = 1. (Evalr=r0 )1,β and (Evals=s0 )1,β are zero for β = β0 = 0. (7.2.189.4) Diagram 7.2.5 commutes. Evalr=r0  −− r=r C −−−→ C 0 ⏐ ⏐ ⏐Eval ⏐Eval s=s0 s=s0 ( ( Evalr=r0 s=s −− −−−→ Cr0 ,s0 C 0 Diagram 7.2.5

(7.2.189.5)

Diagram 7.2.6 is exact. A  −→ C



s=s ⊕ C 0

s0 =0,1





B r=r −→ C 0

r0 =0,1

Cr1 ,s1 −→0.

r1 =0,1,s1 =0,1

Diagram 7.2.6  Here A = s0 =0,1 (Evals=s0 )1 ⊕ r0 =0,1 (Evalr=r0 )1 and B are defined as follows: 0 0 } ⊕ {Brr=r }. Then we define Denote B componentwise as B = {Brs=s 1 ,s1 1 ,s1 

0 Brs=s = δs0 s1 (Evalr=r1 )1 , 1 ,s1

0 Brr=r = −δr0 r1 (Evals=s1 )1 1 ,s1

where δij is the Kronecker-delta.  (7.2.189.6) There exists a strict and filtered A∞ homomorphism Incl : C0 → C such that the following conditions hold: (7.2.189.6.1) (7.2.189.6.2)

s=s . Evals=s0 ◦ Incl = Inclr . Here both sides map C0 to C 0 r=r . Evalr=r0 ◦ Incl = Incls . Here both sides map C0 to C 0

(7.2.189.7) Both Incl and Eval induce chain homotopy equivalence after reduction of the coefficient ring to R. The filtered An,K version and unfiltered version are defined in the same way. We will prove existence of the model of [0, 1]2 × C in the next proposition. Proposition 7.2.190. Let Cr0 ,s0 (r0 = 0, 1, s0 = 0, 1) be four filtered A∞ algebras containing a common subalgebra C0 (that is the structure mk of C0 is the restriction of one of Cr0 ,s0 ), such that each inclusion C0 ⊂ Cr0 ,s0 is a homotopy equiv r=r and C s=s be alence. We denote the quintuple by C = (C0 , {Cr0 ,s0 }r0 ,s0 ). Let C 0 0 models of [0, 1] × (C0 , {Cr0 ,0 , Cr0 ,1 }) (r0 = 0, 1) and of [0, 1] × (C0 , {C0,s0 , C1,s0 }) (s0 = 0, 1), respectively. We assume that everything is G-gapped.  which, together with C r=r Then there exists a G-gapped filtered A∞ algebra C 0 2 s=s , forms a model of [0, 1] × C. and C 0 Proof. Let B:

 s0 =0,1

s=s ⊕ C 0



r=r → C 0

r0 =0,1

be as in Diagram 7.2.6. It induces   s=s ⊕ r=r → B: C C 0 0 s0 =0,1

r0 =0,1



Cr1 ,s1

r1 =0,1,s1 =0,1

 r1 =0,1,s1 =0,1

C r1 ,s1 .

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Namely s=s [e, e−1 ] = C 0



Cs=s0 + s=s Λ0,nov C 0

etc.. We put 



 = C  ⊗R Λ0,nov . C

 = KerB, C

(7.2.191)

 is well-defined as a chain complex. (Note B = B ⊗R Λ0,nov by (7.2.189.1).) C We define Evals=s0 , Evalr=r0 so that (7.2.189.5) is satisfied in an obvious way. Moreover we have an exact sequence (30.192) 

%

 −→ 0 −→ C A

&



s=s C 0

% ⊕

s0 =0,1

&



r=r C 0



B

−→

r0 =0,1

C r1 ,s1 −→0

r1 =0,1,s1 =0,1

of chain complexes, and a similar exact sequence with A, B in place of A and B. The statements (7.2.189.2), (7.2.189.3) can be easily derived, except the statement  . about the filtered A∞ structure of C   so that it is compatible with A i.e., satisfies We define Incl1,β : C 0 → C 0

A ◦ Incl1,β0 =



0 Incls=s 1,β0 ⊕



0 Inclr=r 1,β0 .

s=s and Inclr=r0 : C0 → C r=r are filtered Here and hereafter Incls=s0 : C0 → C 0 0 A∞ homomorphism given as in Definition 7.2.174.   is not necessarily a chain homotopy equivaWe remark that Incl1,β0 : C 0 → C  which lence. But by the standard technique of homological algebra, we can find C  as a chain complex and such that Incl1,β : C 0 → C  is a chain homotopy contains C 0   so that it is a free R module having C  as a equivalence. Moreover we may take C direct summand. Namely 



=C  ⊕C  , C

=C  ⊗R Λ0,nov C

as an R module or a Λ0,nov module, (but not as a chain complex). Using this  so that A decomposition we can extend A (that is, Evals=s0 and Evalr=r0 ) to C becomes a chain map. Then A will not be injective. But Diagram 7.2.6 remains to be exact.  and a filtered A∞ structure {mk,β } We now define Incl = {Inclk,β }k,β : C0 → C  on C so that (7.2.189) is satisfied. We will define Inclk,β , mk,β , by induction with respect to the order < defined on the set of (n, k) = (β, k). Here < is defined in Definition 7.2.63.  s=s , on C  r=r , and on C0 , by ms=s0 , We write the filtered A∞ structure on C 0 0 ∗ r=r0 C0 m∗ , m∗ , respectively. Suppose that Inclk ,β  , mk ,β  is defined for (β  , k ) < (n, k). Lemma 7.2.193. Let (β, k) = (n, k). Then there exist the maps  Inclk,β : BC 0 [1] → C,

 mk,β : Bk Im(Incl1,β0 ) → C

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

511

such that the A∞ relation holds on Bk Im(Incl1,β0 ) and that Inclk,β together with Inclk ,β  with for (β  , k ) < (n, k) defines a filtered An,k homomorphism. We remark that the second half of the statement of Lemma 7.2.193 makes sense even though mk,β is defined only on Bk Im(Incl1,β0 ). In fact the defining formula for a filtered An,k homomorphism on Bk C 0 involves mk,β only on Bk Im(Incl1,β0 ). (See (7.2.196).) s=s and C  r=r are models of [0, 1] × (C0 , {Cr,s }r=0,1 ) and Proof. Since C 0 0 0 [0, 1] × (C0 , {Cr0 ,s }s=0,1 ), respectively (see Definition 7.2.174), we have (7.2.194)

r=r0 0 Evalr=r0 ◦ Incls=s k,β = Evals=s0 ◦ Inclk,β .

Therefore by the exactness of (7.2.192), there exists Inclk,β such that (7.2.195)

0 Evalr=r0 ◦ Inclk,β = Inclr=r k,β ,

0 Evals=s0 ◦ Inclk,β = Incls=s k,β .

The condition that Incl is a filtered An,K homomorphism is written as

(7.2.196)

0 C (mk,β ◦ Incl⊗k 1,β0 )(x) + (m1,β0 ◦ Inclk,β )(x) − (Inclk,β ◦ m 1,β0 )(x)

  : =− m,β  Inclk1 ,β1 (x:1 a ), · · · , Inclk ,β (xa )

0 C + Inclk1 ,β1 ◦ m k2 ,β2 (x).

Here the sum of the first term of the right hand side is taken over a, , β , ki , βi such  that k1 + · · · + k = k, β + βi = β with (, β  ) = (k, β) and (ki , βi ) = (k, β). The sum of the second term is taken over k1 , k2 , β1 , β2 such that k1 + k2 = k + 1, β1 + β2 = β and (ki , βi ) = (k, β). We put

: x:1 Δ−1 (x) = a ⊗ · · · ⊗ xa . a

(7.2.196) determines mk,β on Bk Im(Incl1,β0 ) uniquely. We now prove the An,k formula on Bk Im(Incl1,β0 ) i.e., prove

 1,β0 + m1,β0 ◦ mk,β = −  k2 ,β2 . mk1 ,β1 ◦ m (7.2.197) mk,β ◦ m Here the sum is taken over all k1 , k2 , β1 , β2 such that k1 +k2 = k+1, β1 +β2 = β and (ki , βi ) = (k, β). To prove (7.2.197) we use (7.2.196). We obtain the the following equality for x ∈ Bk C 0 [1]:



  :   ,β  ◦ Inclk1 ,β1 (x:1 m ,β  ◦ m (7.2.198) 0 = a ), · · · , Inclk ,β (xa ) ,  ,β   ,β  ki ,βi     where the sum is taken over all  , β ,  , β , (k1 , β1 ), · · · , (k , β ) and a such that    +  =  + 1, ki = k. Consider the sum over all  , β  ,  , β  , (k1 , β1 ), · · · , (k , β ), and a such that (ki , βi ) = (1, β0 ) for some i. We take one such choice of (k1 , β1 ), · · · , (k , β ), a and fix  it. We then take the sum over all  ,  , β  , β  with  +  =  + 1, β  + β  = β − βi . By the induction hypothesis the sum is zero. Thus (7.2.198) implies that the sum of the right hand side for (ki , βi ) = (1, β0 ) is zero. This implies (7.2.197) on Bk Im(Incl1,β0 ). Lemma 7.2.193 is proved. 

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Lemma 7.2.199. There exists  C)  mIk,β ∈ Hom(Bk C, such that (7.2.200.1) mIk,β extends the operator mk,β , which was defined by Lemma 7.2.193 on Bk Im(Incl1,β0 ). ⊗k

s=s0 0 Evals=s0 ◦ mIk,β = ms=s k,β ◦ Evals=s0 , where mk,β is the filtered An,K s=s . structure on C 0 ⊗k r=r0 0 (7.2.200.3) Evalr=r0 ◦ mIk,β = mr=r k,β ◦ Evalr=r0 , where mk,β is the filtered An,K r=r . structure on C

(7.2.200.2)

0

0 ,r=r0 be the filtered An,K structure on Cs=s0 ,r=r0 . We have Proof. Let ms=s k,β

⊗k

⊗k

⊗k

s=s0 ,r=r0 0 Evalr=r0 ◦ ms=s ◦ Evalr=r0 ◦ Evals=s0 k,β ◦ Evals=s0 = mk,β ⊗k

0 = Evals=s0 ◦ mr=r k,β ◦ Evalr=r0 .

Lemma 7.2.199 then follows from the exactness of Diagram 7.2.6 and split injectivity of Incl1,β0 .  We remark that the operators mIk,β may not satisfy the An,K relation. We next  We put modify them and define the operator mk,β on B C. K=

) r0 =0,1

Ker(Evalr=r0 ) ∩

)

 Ker(Evals=s0 ) ⊂ C.

s0 =0,1

 ∈ Hom(Bk C,  We define ok,β (C)  C)  by It is a subcomplex of C.



 =  Ik2 ,β2 . mIk1 ,β1 ◦ m ok,β (C) k1 +k2 =k β1 +β2 =β

We have (7.2.201.1) (7.2.201.2) (7.2.201.3)

 = 0, δ1 (ok,β (C))  ∈ Hom(Bk C,  K), ok,β (C)  ◦ Inclk⊗ = 0. ok,β (C) 1,β0

 C)  → Hom(Bk C,  C)  is induced by m1 and m ˆ 1. Here δ1 : Hom(Bk C, (7.2.201.1) is a consequence of induction hypothesis. (7.2.201.2) is a consequence of (7.2.200.2) and (7.2.200.3). (7.2.201.3) is a consequence of (7.2.200.1) and the fact that mk,β satisfies A∞ relation on Bk Im(Incl1,β0 ). Since the map  K) → Hom(Bk Im(Incl1,β ), K) Hom(Bk C, 0 induces isomorphism on δ1 cohomology, it follows from (7.2.201) that there exists  K) such that Cor = 0 on Bk Im(Incl1,β ) and that Cor ∈ Hom(Bk C, 0

 + δ1 Cor = 0. ok,β (C)

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

513

It is easy to see that mk,β = mIk,β + Cor has the required properties. The proof of Proposition 7.2.190 is complete.  ∼ H 2 ([0, 1]2 ; ∂[0, 1]2 )⊗H d−2 (C).  Hence Remark 7.2.202. We can show H d (K) = 0 −2  ∈ H (Hom(Bk C,  K)) ∼  C)).  This is consistent with the [ok,β (C)] = H (Hom(Bk C, geometric interpretation of our algebraic obstruction argument. We can take the filtered An,K version of Proposition 7.2.190 to obtain a model of [0, 1]2 × C of filtered An,K algebra C. Remark 7.2.203. We can also construct a model of [0, 1]2 × C by iterating the construction of the model [0, 1] × C. Namely we may take P oly([0, 1], P oly([0, 1], C)) = P oly([0, 1]2 , C) (see Subsection 4.6.5) or (C [0,1] )[0,1] = C ⊕ C[−1] ⊕ C[−1] ⊕ C[−2]. (Compare Lemma 4.2.25.) We omit the detail of this construction. Those models are the case when Cr0 ,s0 = C0 = C for all r0 , s0 . In such a case it is natural to assume that there exist maps  s=s → C,  Inclr : C 0

r=r → C,  Incls : C 0

such that (7.2.189bis1) β = β0 . (7.2.189bis2)

(Inclr )k = (Incls )k = 0 for k = 1. (Inclr )1,β = (Incls )1,β = 0 if The compositions

 r=r Incl −→r C C 0

Evalr=r0

−→

r=r , C 0

 s=s Incl  C −→s C 0

Evals=s0

−→

s=s C 0

are the identity maps. r=r by Incls (r0 = (7.2.189bis3) Diagram 7.2.7 commutes: (We denote Incl of C 0  0, 1). We denote Incl of Cs=s0 by Inclr (s0 = 0, 1).) Incl r=r C −−−−s→ C 0 ⏐ ⏐ ⏐Incl ⏐Incl ( r ( r Incls s=s −− −−→ C 0

(7.2.189bis4)

 C

Diagram 7.2.7 The following diagrams commute: Inclr r=r −− C −−→ 0 ⏐ ⏐Eval s=s0 (

C

 C ⏐ ⏐Eval s=s0 (

Incls s=s −− −−→ C 0 ⏐ ⏐Eval r=r0 (

Incl s=s −−−−r→ C C 0 Diagram 7.2.8

Incl  r=r −−−−s→ C 0

We remark (7.2.189bis) implies (7.2.189.6) by putting Incl = Inclr ◦ Incls .

 C ⏐ ⏐Eval r=r0 (

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We, however, do not impose (7.2.189bis) as a part of the definition of models of [0, 1]2 ×C, since in the next subsection we use the case where Cr0 ,s0 vary for different (r0 , s0 ). To state the next theorem, we need to introduce a parametric form of Definition 7.2.182. We parameterize [0, 1] by r or by s according to the situation and denote it by [0, 1]r or [0, 1]s when we want to specify the parameter. To distinguish the two, we also denote the models of [0, 1]r × C and [0, 1]s × C by Inclr , Incls respectively. We say (C, {mk,β }) is of An,K (resp. A∞ ) class when it is a filtered An,K (resp. A∞ ) algebra. The notion of An,K and A∞ class for the homomorphism, the homotopy, the model of [0, 1] × C and etc. are defined in a similar way.  Definition 7.2.204. Let {Ii : Ci → Ci+1 } and {Ii : Ci → Ci+1 } be directed s=s0  systems of A(n,k) algebras, and {fi : Ci → Ci } two morphisms between them for s0 = 0, 1.  of [0, 1]2 × C  of A Suppose that a model C i i n(λi ),k(λi ) class is given. (Here we denote the quintuple by Ci = (Ci , {Ci }r0 ,s0 ). Namely in our case the five An(λi ),k(λi ) algebras appearing in Definition 7.2.188 are all Ci .) Assume that a commuting error homotopy

 Fis=s0 : Ci → C i+1,s=s0 0 of the directed system {fs=s } is given. By definition, they satisfy (7.2.186) for each i s0 = 0, 1. 0 Suppose that we are given a homotopy between two morphisms {fs=s } of dii rected systems, i.e., are given An(λi ),k(λi ) homomorphisms

 Hi : Ci → C i,r=1 satisfying 0 . Evals=s0 ◦Hi = fs=s i

(7.2.205)

A commuting error homotopy of the homotopy {Hi } (between two morphisms 0 {fs=s }) is defined to be a pair of sequences i    ({Ii : C i,r=1 → Ci+1,r=0 }, {Hi : Ci → Ci+1 })   such that Ii : C i,r=1 → Ci+1,r=0 is a filtered An(λi ),k(λi ) homomorphism satisfying (7.2.206)

   Evals=s0 ◦ Ii = Ii ◦ Evals=s0 : C i,r=1 → Ci+1,r=0,s=s0 = Ci+1 ,

 and Hi : Ci → C i+1 is a filtered An(λi ),k(λi ) homomorphism with the following properties: (7.2.207.1) (7.2.207.2) (7.2.207.3)

 Evalr=0 ◦ Hi = Ii ◦ Hi : Ci → C i+1,r=0 ,  Evalr=1 ◦ Hi = Hi+1 ◦ Ii : Ci → C i+1,r=1 , s=s0   Evals=s0 ◦ Hi = Fi : Ci → Ci+1,s=s0 .

See Figure 7.2.11.

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515

s =1

fi+1

Hi+1 ˆ C i+1,s=1

ˆ C i+1 s =0

fi+1

r

Ci+1

Ii



Ii

s

ˆ C i+1,r=0

s =0 Fi

Ii

Ii

Hi

Ci

s =1

fi Ci

ˆ C i,r=1

Hi s =0

fi

Ci

Figure 7.2.11 We remark that the heart of Definition 7.2.204 is (7.2.207) which states that Hi is a version of homotopy of homotopies. Lemma 7.2.208, Proposition 7.2.211 and Theorem 7.2.212 below state that the directed system of A(n,k) algebras appearing in Definition 7.2.204 will be promoted to A∞ class.  } We first promote the directed systems {Ii : Ci → Ci+1 } and {Ii : Ci → Ci+1 to A∞ class using Lemma 7.2.184 (7.2.185.1). Next we will promote the morphisms {fi } and its commuting error homotopy {Fi }. For completeness’ sake, we restate and give its proof. s=s0  Lemma 7.2.208. We can promote {C } and {Fis=s0 } to filtered i,s=s0 }, {fi A∞ algebra and homomorphisms so that

(7.2.209) {Fis=s0 } is promoted to an A∞ commuting error homotopy of the pro0 motions of {fs=s } respectively for s0 = 0, 1 and i   (7.2.210) Inclr of C i,s=s0 is promoted to a filtered A∞ homomorphism and Ci,s=s0  promoted to a model of [0, 1]r × Ci of A∞ class. (Here a promotion of a model of [0, 1] × C is its promotion as a filtered An,K algebra together with promotion of Eval and Incl so that they form a model of A∞ class.) s=s0  and Fis=s0 . HereThere are various choices of such promotions of C i,s=s0 , fi after we will fix any one choice of their promotions.

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  of [0, 1]2 × C  , which Proposition 7.2.211. We can promote the models C i i are of An(λi ),k(λi ) class, to models of A∞ class, except the assertion on Incl. We   homomorphisms I : C → C can also promote the filtered A i

n(λi ),k(λi )

i,r=1

i+1,r=0

to filtered A∞ homomorphisms so that the equality (7.2.206) holds as filtered A∞ homomorphisms. We will discuss the promotion of Incl in Proposition 7.2.217 later. Postponing the proof of Lemma 7.2.208 and Proposition 7.2.211 for a while, we state the main result of this subsection. Theorem 7.2.212. For any given homotopy {Hi } between two morphisms 0 {fs=s } of directed system of A(n,k) algebras, we can promote it to a filtered A∞ i homomorphism satisfying (7.2.205) for the promotions fs=s0 . Similarly any com0 muting error homotopy {Hi } of {Hi } between two such morphisms {fs=s } in the i sense of Definition 7.2.204 can be promoted so that the conditions in (7.2.207) hold as filtered A∞ homomorphisms. Remark 7.2.213. We remark that we did not assume any compatibility con  ditions between Incls : Ci → C i,r=r0 and Ii in Definition 7.2.204. Because of this,   Incls : Ci → C i,r=r0 , Incl : Ci → Ci may not extend to filtered A∞ homomorphisms under the assumption in Theorem 7.2.212. In fact, during the proof of Theorem 7.2.212, we will never use Inclr , Incls , Incl. In particular, Theorem 7.2.212 holds without assuming the condition (7.2.189.6) for  . C i On the other hand, Theorem 7.2.212 alone does not imply that fs=0 is filtered i s=1   A∞ homotopic to fi . This is because Ci,r=1 may not be a model of [0, 1]s × Ci , as a filtered A∞ algebra. (It is a model of [0, 1]s × Ci as a filtered An(λi ),k(λi ) algebra by Definition 7.2.204.) We need to assume appropriate compatibility conditions between Incls and Ii to prove that fs=0 is filtered A∞ homotopic to fs=1 . We will i i next describe this compatibility condition as Condition 7.2.215. Before doing it, we give an example to highlight the role of Incl in the homotopy theory of filtered A∞ algebras. Example 7.2.214. Let C be a filtered A∞ algebra and C a model of [0, 1]r ×C. We assume that h : C → C is an isomorphism which is not homotopic to identity. We define Eval : C → C by Evalr=0 = Evalr=0 ,

Evalr=1 = h ◦ Evalr=1 .

(C; Evalr=0 , Evalr=1 ) has the properties of the model of [0, 1]r × C except it does not have Incl. Let F = Incl : C → C. Then Evalr=0 ◦ F = id,

Evalr=1 ◦ F = h.

We assumed that h is not homotopic to identity. This example shows that we do need to use Incl to have correct definition of homotopy. To state Proposition 7.2.217, we introduce another definition describing the above mentioned compatibility conditions. Let    ({Ii : C i,r=1 → Ci+1,r=0 }, {Hi : Ci → Ci+1 })

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0 be a homotopy of homotopies of two morphisms {fs=s } (s0 = 0, 1) of directed i    systems. Let Incls : Ci → C be a filtered A i,r=r0 n(λi ),k(λi ) homomorphism such that Evals=s0 ◦ Incls = id

as filtered An(λi ),k(λi ) homomorphisms. Existence of such Incls is a part of Defini  tion 30.204, which is contained in the definition C i,r=r0 being a model of [0, 1]s × Ci of An(λi ),k(λi ) class. Condition 7.2.215. For each i, there exist a filtered An(λi+1 ),k(λi+1 ) algebra   Ci+1 and a filtered An(λi ),k(λi ) homomorphism i : C → C  H i i+1 that satisfy the following properties: (7.2.216.1)

 is a model of [0, 1]2 × C  as a filtered A C i+1 i+1 n(λi+1 ),k(λi+1 ) algebra.      Ci,r=r0 = Ci,r=0 as models of [0, 1]s × Ci of An(λi ),k(λi ) class. (Here

(7.2.216.2) r0 = 0 or 1.)  i = I ◦ Incls as filtered A (7.2.216.3) Evalr=0 ◦ H i n(λi ),k(λi ) homomorphisms.   i = Incls ◦ I as filtered A (7.2.216.4) Evalr=1 ◦ H i n(λi ),k(λi ) homomorphisms.   i = Inclr ◦ I as filtered A homomorphisms. (7.2.216.5) Evals=s ◦ H 0

i

n(λi ),k(λi )

Roughly speaking Condition 7.2.215 means that Ii ◦ Incls ∼ Incls ◦ Ii where ∼ is ‘homotopy relative to the boundary’. Proposition 7.2.217. Consider the commuting error homotopy    ({Ii : C i,r=1 → Ci+1,r=0 }, {Hi : Ci → Ci+1 }) of a homotopy {Hi } of two morphisms of directed system. If the commuting error    homotopy satisfies Condition 7.2.215, then Incls : Ci → C i,r=r0 and Incl : Ci → Ci can be promoted to a filtered A∞ homomorphism between the promoted A∞ algebras associated to the two directed systems: With respect to the promotions of Incls and Incl given above, the promotions of 2   {C i,r=r0 }, {Ci } become models of [0, 1]s × Ci and of [0, 1] × Ci of A∞ class where  }. Ci denote promoted A∞ algebras of the directed system {Ii : Ci → Ci+1 The proof of Proposition 7.2.217 will be given at the end of this subsection. Remark 7.2.218. In the situation of Proposition 7.2.217, the filtered A∞  are already given. (Namely we have structure on the promotions of Ci and of C i fixed them during the proof of Theorem 7.2.212.) So we are not allowed to change them. The proposition concerns only promotions of the homomorphisms Incls and Incl.    Corollary 7.2.219. If ({Ii : C i,r=1 → Ci+1,r=0 }, {Hi : Ci → Ci+1 }) satisfies s=0 s=1 is filtered A∞ homotopic to fi . Condition 7.2.215, then fi Corollary 7.2.219 immediately follows from Theorem 7.2.212 and Proposition 7.2.217.

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Proof of Corollary 7.2.219 ⇒ Proposition 7.2.187. We will use the notations of Proposition 7.2.187. We put fs=0 = fs=1 = fi and will embed the i i given hypotheses of Proposition 7.2.187 into those of Definition 7.2.204 together and verify Condition 7.2.215. Then we can apply Corollary 7.2.219 to derive Proposition 7.2.187. (r)  so that C  Take C that appears in Definition 7.2.182bis. To i i,s=s0 become Ci be more explicit, we use Lemma 4.2.25 and take the model  = (C(r) )[0,1]s . C i i (r)

(The corresponding maps Evalr=r0 and Inclr thereof are induced by those of Ci .) (r) We remark that (7.2.189bis) hold for this Ci . (r)  = (C )[0,1]s , we have Since we take C i i  [0,1]s   . C i,r=0 = Ci,r=1 = (Ci ) (r)   By assumption, we are given Fi : Ci → Ci+1 = C i+1,s=1 = Ci+1,s=0 . We put s=1 = Fi = Fi . Then the property of commuting error homotopy of {Fis=s0 } follows from (7.2.186). Since fi = fs=0 = fs=1 as a map from Ci to Ci , we may take Hi = Incls ◦fi : i i  Ci → Ci . Then homotopy relation of Hi is obvious.   )[0,1]s is induced by Ii : Ci → Ci+1 by Theorem The map Ii : (Ci )[0,1]s → (Ci+1 4.2.34. It is easy to see that they satisfy (7.2.206). Finally we put  . Hi = Incls ◦Fi : Ci → C i+1 It is straightforward to check (7.2.207). We have thus verified all the conditions spelled out in Definition 7.2.204. We next check Condition 7.2.215. Since Ii is induced from Ii by Theorem 4.2.34 it follows that

Fis=0

Ii ◦ Incls = Incls ◦ Ii .

(7.2.220)

  = [0, 1]2 × C  = (C )[0,1]r and We put C i+1 i+1 i+1  i = Inclr ◦ Incls ◦ I  . H i Since (7.2.189bis) holds in our case, (7.2.220) implies (7.2.216). Then for any promotions f0 and f1 of {f0i } and {f1i }, Corollary 7.2.219 implies that f0 is filtered A∞ homotopic to f1 . The proof of Proposition 7.2.187 is complete.  Proof of Lemma 7.2.208. Let C be filtered A∞ algebras and {Ii : Ci → a directed system of An,k algebras. Then {Ci } are promoted to filtered A∞ algebras by Theorem 7.2.72 and Lemma 7.2.69. We fix one i. We first prove:

 Ci+1 }

 Lemma 7.2.221. Let g : C → Ci+1 be a filtered An(λi+ ),k(λi+ ) homomorphism  and f : C → Ci a filtered An(λi+−1 ),k(λi+−1 ) homomorphism. (r)

 Let Ci+1 be a model of [0, 1]r × Ci+1 as a filtered An(λi+ ),k(λi+ ) algebra and (r)

F : C → Ci+1 a filtered An(λi+−1 ),k(λi+−1 ) homomorphism.

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519

Suppose F satisfies the following identities of filtered An(λi+−1 ),k(λi+−1 ) homomorphisms: (7.2.222.1)

Evalr=0 ◦ F = g,

(7.2.222.2)

Evalr=1 ◦ F = I ◦ f

with I = Ii .

Then F and f extend to filtered An(λi+ ),k(λi+ ) homomorphisms so that (7.2.222) holds as filtered An(λi+ ),k(λi+ ) homomorphisms. Proof. We may assume that ((n, K), (n , K  )) = ((n(λi+−1 ), k(λi+−1 )), (n(λi+ ), k(λi+ )) is one of the types (7.2.130.1) or (7.2.130.2). (The general case will follow from this by induction.) A conclusion which we will prove below is that (7.2.222.1) holds as filtered An ,K  homomorphisms. This can be written as Evalr=0 ◦ FK  ,β = gK  ,β

(7.2.223)

for β = n by definition. Since g is a filtered An ,K  homomorphism and Evalr=0 I is an An ,K  homotopy equivalence, we can find a map FK  ,β satisfying (7.2.223) but not necessary define a filtered An ,K  homomorphism. For β = n , let oK  ,β (F) ∈ (r)

Hom(BK  (C[1]), Ci+1 [1]) be the obstruction cycle to extend {FK,β  | (β  , K) < (n , K  )} to a filtered An ,K  homomorphism. We put   (r)  K = Ker Evalr=0 : Ci+1 [1] → C i+1 [1] . Then I oK  ,β (F) + δ1 (FK  ,β ) ∈ Hom(BK  (C[1]), K[1]).  1 .) Note K is acyclic, because (Here δ1 (A) = δ1 (A) = m1 ◦ A − (−1)deg A A ◦ m Evalr=0 is a homotopy equivalence. Therefore there is an element Cor1K  ,β ∈ Hom(BK  (C[1]), K[1]) such that II I FK  ,β := FK  ,β + Cor1K  ,β

together with {FK,β  | (β  , K) < (n , K  )} defines a filtered An ,K  homomorphism (r) form C to Ci+1 satisfying (7.2.223). Then it follows from Lemma 7.2.129 that there exists fIK  ,β such that together with fK,β  for (β  , K) < (n , K  ), it defines a filtered An ,K  homomorphism, which we denote by fI . Since (7.2.224)



II I Evalr=1 ◦ FK  ,β − (I ◦ f )K  ,β ∈ Hom(BK  (C[1]), C i+1 [1])

is a δ1 cycle, it follows that we may change fIK  ,β so that (7.2.224) is a δ1 boundary. Namely II I Evalr=1 ◦ FK  ,β − (I ◦ f )K  ,β = δ1 (Cor2K  ,β ) for some Cor2K  ,β . By the surjectivity (4.2.2.4), we can find (r)

Cor3K  ,β ∈ Hom(BK  (C[1]), Ci+1 [1]) such that Evalr=0 ◦ Cor3K  ,β = 0,

Evalr=1 ◦ Cor3K  ,β = Cor2K  ,β .

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II We now put FK  ,β = FK  ,β − δ1 (Cor3K  ,β ). It is easy to see that this has the required properties. Hence the proof of Lemma 7.2.221.    We remark that C i,s=s0 is given as a model of [0, 1]r × Ci in An(λi ),k(λi ) class. We can promote this model to a model in A∞ class. This is the one-parameter version of (the induction step of the proof of) Proposition 7.2.190. Since the proof of Proposition 7.2.190 (which is the two-parameter version) is similar to and more non-trivial than the one-parameter version, we omit the detail of this promotion process for the one-parameter version. We remark that the situation of Lemma 7.2.208 is different from the situation  of Proposition 7.2.217. In Proposition 7.2.217, a promotion of C i,r=r0 to a filtered A∞ algebra is already given, and we want to promote Incls to a filtered A∞ ho momorphism with respect to this given structure on the promotion of C i,r=r0 . (See the proof given at the end of this subsection.) On the other hand, in the proof  of Lemma 7.2.208, we will promote C i,s=s0 to a A∞ structure and Inclr to a A∞ homomorphism simultaneously. In fact, in the proof of Lemma 7.2.193, we constructed mk,β and Inclk,β simultaneously. The proof of Lemma 7.2.193 cannot be applied to construct Inclk,β in the situation where mk,β is given in advance. 0 We are now ready to complete the proof of Lemma 7.2.208. We promote fs=s i s=s0 and Fi to filtered An(λi+ ),k(λi+ ) homomorphisms, by a double induction on over (, i) with respect to the order 0 such that fa (|Pa |) ⊂ [0, 1/3 − )2 × L1 for all a. We put fa = (fa,r , fa,s , fa,L ) : |Pa | → [0, 1]2 × L1 . We define fˆa : |Pa | × [0, 1] → [0, 1]2 × L1 by   t (7.2.294) fˆa (x, t) = fi,r (x), + fa,s (x), fi,L1 (x) . 2 Sublemma 7.2.295. We can take a prism decomposition of |Pa | × [0, 1] such that fˆa restricted to the top dimensional simplices of |Pa | × [0, 1] are adapted. In particular, the decomposition defines an element of S+ ([0, 1]2 × L1 ). Assuming Sublemma 7.2.295, we now complete the proof of (7.2.292.3). Let Ψ0,0 (Pa ) be the singular chain in S+ ([0, 1]2 × L1 ) obtained by Sublemma 7.2.295. We can easily prove that ∂Ψ0,0 (Pa ) − Ψ0,0 (∂Pa ) − Pa does not intersect with {(0, 0)} × L1 . Now we define

C = C − ca ∂Ψ0,0 (Pa ). a∈A

 Then C does not intersect with {(0, 0)} × L1 . (In fact, a∈A ca ∂Pa does not intersect with {(0, 0)} × L1 . This is because ∂C does not intersect with {(0, 0)} × L1 .)

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We repeat the same construction at {(1, 0)} × L1 , {(0, 1)} × L1 , {(1, 1)} × L1 . Then we may (and will) assume that our cycle C does not intersect with the corner of [0, 1]2 × L1 . Let {Pa }a∈A be the set of all simplices in C which intersects with [0, 1]r × {0} × L1 . We have

ca Pa . C= a∈A

We put Pa = (|Pa |, fa ). Let fˆa be as in (7.2.294). Since Pa does not intersect the corner, (|Pa | × [0, 1], fˆa ) is decomposed, by the prism decomposition, to adapted simplices. We denote it by Ψ0 (Pa ). Then

C− ca ∂Ψ0 (Pa ) a∈A

does not intersect [0, 1]r × {0} × L1 . We repeat the construction at s = 1. Then we may (and will) assume that C does not intersect with [0, 1]r × ∂[0, 1]s × L1 . Now we repeat the proof of Lemma 7.2.136. Namely we regard [0, 1]2 × L1 = [0, 1]r × ([0, 1]s × L1 ). We regard [0, 1]r here as [0, 1] in proof of Lemma 7.2.136. We conclude that C is a boundary, as required. This finishes the proof of (7.2.292.3). Proof of Sublemma 7.2.295. We will prove only (7.2.289.3) since the other conditions are easy to check. Let fa : |Pa | → [0, 1/3 − )2 × L1 be the map as above. If fa (|Pa |) ⊂ (∂([0, 1/3 − )2 × L1 )), then (7.2.289.3) is easy to check. So we may assume that fa (|Pa |) intersects both {0} × (0, 1/3 − )s × L1 and (0, 1/3 − )r × {0} × L1 . We first remark that there cannot be any vertex v of Pa such that fa (v) ∈ (0, 1/3 − )2 × L1 : Otherwise we could join it with v0 with fa (v0 ) ∈ {(0, 0)} × L1 and then this one dimensional face of Pa would violate (7.2.289.3). Consider the triangle Δ2 = {(σ, ρ) | 0 ≤ σ, ρ, σ + ρ ≤ 1} and define an affine map ψ : |Pa | → Δ2 such that (1) If a vertex v ∈ |Pa | is on {(0, 0)} × L1 then ψ(v) = (0, 0). (2) If a vertex v ∈ |Pa | is on {0} × (0, 1/3)s × L1 then ψ(v) = (0, 1). (3) If a vertex v ∈ |Pa | is on (0, 1/3)r × {0} × L1 then ψ(v) = (1, 0). See Figure 7.2.15. We first consider the case where dimension of Pa is 2. In this case we may identify |Pa | ∼ = Δ2 . We can take a prism decomposition such that the vertex (0, 0, 0) is not joined with (1, 0, 1). We remark that fˆa−1 ((0, 1/3)2 × L1 ) contains only one vertex (1, 0, 1) and fˆa−1 ({(0, 0)} × L1 ) contains only one vertex (0, 0, 0). Moreover the prism decomposition does not change the set of vertices. Hence we can easily check (7.2.289.3) in this case. In the general case, we consider the affine map ψ ×id : |Pa |×[0, 1] → Δ2 ×[0, 1]. We pull back the above simplicial decomposition of Δ2 × [0, 1] to |Pa | × [0, 1] and obtain a decomposition on it. (It may not be a simplicial decomposition yet.) We can take our prism decomposition so that it is a subdivision of this decomposition. (7.2.289.3) can be checked easily for this decomposition. The proof of Sublemma 7.2.295 is complete. 

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 Figure 7.2.15 We next prove (7.2.292.4). The surjectivity of B is a consequence of (7.2.137.2) which can be easily proved. B ◦ A = 0 is a consequence of (7.2.292.2). Let



0 0 0 0 cr=r Pr=r , Cs=s0 = cs=s Ps=s , Cr=r0 = a a a a a∈Ir=r0

a∈Is=s0

such that B(Cs=0 , Cs=1 , Cr=0 , Cr=1 ) = 0. We will show that the element (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) lies in the image of A. We consider the set of all a ∈ Ir=0 such that Pr=0 intersects {(0, 0)} × L1 . We a denote it by Ir=0,(0,0) . We define Is=0,(0,0) in the same way. We put



r=0 s=0 cr=0 Cs=0,(0,0) = cs=0 Cr=0,(0,0) = a Pa , a Pa . a∈Ir=0,(0,0)

a∈Is=0,(0,0)

By the assumption, we have Evals=0 (Cr=0,(0,0) ) = Evalr=0 (Cs=0,(0,0) ). We denote this common chain by Y. 2  Let a ∈ Ir=0,(0,0) . We define fa : |Pr=0 a | × [0, 1] → [0, 1] × L1 by fa (x, r) = (r/4, fa (x)) where fa (x) ∈ {0} × [0, 1]s × L1 ∼ = [0, 1]s × L1 . By taking an appropriate prism r=0 2  decomposition of (|Pa | × [0, 1], fa ) we obtain φ(Pr=0 a ) ∈ S+ ([0, 1] × L1 ) such that φ satisfies the following: r=0 Evalr=0 (φ(Pr=0 a )) = Pa ,

r=0 Evals=0 (φ(Pr=0 a )) = φ(Evals=0 (Pa )),

Evals=1 (φ(Pr=0 a )) = 0,

Evalr=1 (φ(Pr=0 a )) = 0.

We put E1 =



r=0 2  cr=0 a (φ(Pa )) ∈ S+ ([0, 1] × L1 ).

a∈Ir=0,(0,0)

Then we have A(E1 ) = (φ(Y), 0, Cr=0,(0,0) , 0). Therefore replacing (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) by (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) − A(E1 ), it suffices to consider the case Cr=0,(0,0) = 0. In the last case, we have Y = 0, by (7.2.292.2). We apply the same argument as above with s and r exchanged and obtain E2 . Then replacing (Cs=0 ,Cs=1 ,Cr=0 ,Cr=1 ) by (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) − A(E2 ), we find that Cr=0,(0,0) = Cs=0,(0,0) = 0. In

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547

other words, we may assume that (Cs=0 , Cs=1 , Cr=0 , Cr=1 )−A(E2 ) does not intersect {(0, 0)} × L1 . We denote E(0,0) = E2 ∈ S+ ([0, 1]2 × L1 ). Repeating the same argument at each corner, we obtain E(1,0) , E(0,1) and E(1,1) so that all 4 components of (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) − A(E(0,0) + E(1,0) + E(0,1) + E(1,1) ) =: (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) do not intersect the corners. We put Cr=0 =



ca Pr=0 a .

a∈A

Since Pr=0 does not intersect with corners, we have a 

  A ca φ(Pr=0 a ) = (0, 0, Cr=0 , 0). Repeating the same construction for r = 1 and s = 0, 1, we find that the quadruple (Cs=0 , Cs=1 , Cr=0 , Cr=1 ) is in the image of A. The proof of Lemma 7.2.291 is now completed.  The rest of the proof of Lemma 7.2.285 is similar to the argument of Subsection 7.2.9. We will define a set X ([0, 1]2 × L1 ) of singular simplices as follows. Elements of this set are singular simplices P ∈ S([0, 1]2 × L1 ) that satisfy Evals=s0 (P) ∈ C s=s0 ([0, 1]r × L1 , (λ0 , λ1 )), Evalr=r0 (P) ∈ C r=r0 ([0, 1]s × L1 , (λ0 , λ1 )). We consider the singular simplices which appear in the decomposition of (7.2.296.1)

Incl1,β0 (P ) ∈ S([0, 1]2 × L1 )

where P ∈ C(L1 , λ1 ). We remark C(L1 , λ1 ) plays a role of C0 in Definition 7.2.188. We define (7.2.296.2)

{Φs,t }s,t∗ (P ) ∈ S([0, 1]2 × L1 )

in the same way as Definition 7.2.140. Now we start from the subcomplex of S([0, 1]2 × L1 ) generated by (7.2.296.1) and (7.2.296.2). We next use (7.2.292.4) to add singular chains so that (7.2.189.5) is satisfied. Then we add the singular simplices that appear in the triangulation of fiber product of the twp(s), twp(r) moduli spaces of pseudo-holomorphic discs with the singular simplices obtained in the earlier steps. Note that we can find a triangulation of the moduli space involved so that it gives an element of S+ ([0, 1]2 × L1 ) as follows: By reparameterizing the isotopies ψr,s to become constant near ∂[0, 1]2 and then choosing the perturbations (multisections) so that they become constant along the normal direction from the corner (that is (s, r) = (s0 , r0 )) in the neighborhood of the corner. So we may regard the moduli space (that is, the fiber product) near the simplex in the corner as the direct product of simplex (which is contained in the corner) with [0, ] × [0, ]. Thus we obtain a simplex satisfying Condition 7.2.288 in the same way as in the proof of Sublemma 7.2.295. See Figure 7.2.16.

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Figure 7.2.16 We remark that there may be two chains P1 , P2 in S([0, 1]2 ×L1 ) whose (naive) intersection does not satisfy (7.2.289.3). For example if we consider two 2 simplices P1 , P2 in [0, 1]2 × S 1 such that the projection of P1 , P2 is as in Figure 30.13 (1). Then it is easy to find an example such that the projection of the intersection P1 ∩ P2 is a line joining (0, 0) and (, ). We may also take that the intersection of P1 and P2 is transversal (and dimension 1 = 2 + 2 − 3.) Therefore the naive intersection P1 ∩ P2 does not satisfy (7.2.289.3). We however remark that in such an example the restriction of P1 to the corner is not transversal to the restriction of P2 to the corner. (For the above example, the expected dimension of intersection at the corner is 0 + 0 − 1 = −1 and is not 0.) So we have already perturbed the intersection m2 in such a way that such phenomenon does not occur. Once we have taken care of the above mentioned transversality problem near the corner, the construction is completely parallel to the case of Subsection 7.2.10. Thus we have constructed the maps (7.2.286.1) and (7.2.286.3) which satisfy (7.2.287.1), (7.2.287.2). To construct Inclk,β (that is the map (7.2.286.2) satisfying (7.2.287.1), we start with Incl1,β0 and define Inclk,β by induction. The way to do so is the same as the proof of Proposition 7.2.162. The proof of Lemma 7.2.285 is now complete.  Remark 7.2.297. We can define (Incls )1,β0 : S([0, 1]r × L1 ) → S([0, 1]2 × L1 ) (Inclr )1,β0 : S([0, 1]s × L1 ) → S([0, 1]2 × L1 ) in the same way as (7.2.135). We may choose the prism decomposition so that the image of (Incls )1,β0 ◦ (Inclr )1,β0 : S(L1 ) → S([0, 1]2 × L1 ) and that of (Inclr )1,β0 ◦ (Incls )1,β0 : S(L1 ) → S([0, 1]2 × L1 ) both satisfy (7.2.289.3). Therefore we may take either (Incls )1,β0 ◦ (Inclr )1,β0 or (Inclr )1,β0 ◦ (Incls )1,β0 as the definition of (Incl)1,β0 . We however remark that the formula (7.2.189bis3)

(Inclr )1,β0 ◦ (Incls )1,β0 = (Incls )1,β0 ◦ (Inclr )1,β0

does not hold: Consider one simplex P = [0, 1] in C(L ). Let us consider the part of {(s, r) | s + r = 1/4} × L of (Incls ◦ Inclr )(P ) and (Inclr ◦ Incls )(P ). Both of them are rectangles. But their simplicial decompositions are different. (See Figure 7.2.17.)

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Figure 7.2.17 We have now completed the study of (7.2.280.2). This is in fact the main geometric part of the proof of Proposition 7.2.275. The other parts, which will follow, concern the difference between f and g. This difference occurs since we need to modify the chain complex several times during our construction. Unfortunately, this part is rather technical and involves heavy notations. Discussion on (7.2.280.1): We first decompose homotopy (7.2.280.1) into a composition of two homotopies. Hereafter we denote by j the homotopy inverse of various inclusions. (For example j : C(L0 , (λ0 , λ0 )) → C(L0 , λ0 ) is a homotopy inverse to the inclusion C(L0 , λ0 ) → C(L0 , (λ0 , λ0 )). Hereafter we always assume that j = id on C(L0 , λ0 ) etc. We remark that (7.2.280.1) is a composition of 2 homotopies 





(7.2.298.1)

gφ ◦ gψ0,s0 = j ◦ fφ ◦ j ◦ fψ0,s0 ∼ j ◦ fφext,s=s0 ◦ fψ0,s0 ,

(7.2.298.2)

j ◦ fφext,s=s0 ◦ fψ0,s0 ∼ j ◦ fφ ◦ψ0,s0 = gφ ◦ψ0,s0 .









We will define fφext,s=s0 later. We first discuss the homotopy (7.2.298.1) and its s-parameterized version. Lemma 7.2.299. There exist a subcomplex Cas=s0 (L1 , ((λ0 , λ0 ), λ1 )) of smooth singular chain complex of L1 , a filtered An ,K  structure on it, and a filtered An,K homomorphism 

fφext,s=s0 : C(L0 , (λ0 , λ0 )) → Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ) such that the inclusion C(L1 , (λ1 , λ1 )) → Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ) is a homotopy equivalence and that Diagram 7.2.17 commutes: C(L1 , λ1 ) ⏐ ⏐ ( Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ) ←−−−− C(L1 , (λ0 , λ1 )) , , ⏐ φ ⏐ φ ⏐fext,s=s0 ⏐f ψ0,s 0

f

C(L0 , λ0 ) −−−−→

C s=s0 (L0 , (λ0 , λ0 ))

←−−−−

C(L0 , λ0 )

Diagram 7.2.17 The proof is the same as the argument of Subsection 7.2.9. (See also the end of Subsection 4.6.3.)

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We remark that Diagram 7.2.17 strictly commutes, not just up to homotopy. This will be important for our argument since we are required to establish several strict equalities. However after inverting the direction of the arrows (that is after replacing inclusions by j), the diagram commutes only up to filtered An,K homotopy. This homotopy is (7.2.298.1). (We will come back to this point later.) Now we will construct an s-parameterized version of Diagram 7.2.17. Lemma 7.2.300. • We can choose subcomplexes C([0, 1]s × L1 , λ1 ),

C([0, 1]s × L1 , (λ0 , λ1 )),

Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 ))

of S([0, 1]s × L1 ), filtered An ,K  structures on them, and Eval’s and Incl’s so that C([0, 1]s × L1 , λ1 ),

C([0, 1]s × L1 , (λ0 , λ1 )),

Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 ))

become models of [0, 1]s × C(L1 , λ1 ), [0, 1]s × C(L1 , (λ0 , λ1 )), [0, 1]s × (C(L1 , λ1 ), {Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ))}s0 ). Here the model means one in the sense of Definition 7.2.174 and of An ,K  class. • We can choose subcomplexes C([0, 1]s × L0 , λ0 ),

C ([0, 1]s × L0 , (λ0 , λ0 ))

of S([0, 1]s × L0 ), filtered An,K structures on them, and Eval’s and Incl’s so that C([0, 1]s × L0 , λ0 ),

C ([0, 1]s × L0 , (λ0 , λ0 )))

become models of [0, 1]s × C(L0 , λ0 ),

[0, 1]s × (C(L0 , λ0 ), {C s=s0 (L0 , (λ, λ0 ))}s0 ).

Here the model means one in the sense of Definition 7.2.174 and of An,K class. • There are filtered An,K homomorphisms 

fid×φ : C([0, 1]s × L0 , λ0 ) → C([0, 1]s × L1 , (λ0 , λ1 )) 

: C([0, 1]s × L0 , (λ0 , λ0 )) → Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 )). fid×φ ext • They satisfy the following properties: (7.2.301.1) (7.2.301.2) (7.2.301.3)





Evals=s0 ◦fid×φ = fφ ◦ Evals=s0 .   Evals=s0 ◦fid×φ = fφext,s=s0 ◦ Evals=s0 . ext

C([0, 1]s × L1 , λ1 ) ⊂ C([0, 1]s × L1 , (λ0 , λ1 )) ⊂ Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 )). (7.2.301.4) The above inclusions preserve An ,K  structures. They commute with Eval’s and Incl’s. (7.2.301.5) We may take C ([0, 1]s × L0 , (λ0 , λ0 )) so that C([0, 1]s × L0 , λ0 ) ⊂ C ([0, 1]s × L0 , (λ0 , λ0 )) and that the inclusion preserves An,K structures and the inclusion commutes with Eval’s and Incl’s. (7.2.301.6) Diagram 7.2.18 commutes:

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C([0, 1]s × L1 , λ1 ) ⏐ ⏐ ( Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 ))) ←− C([0, 1]s × L1 , (λ0 , λ1 )) , , ⏐ id×φ ⏐ id×φ ⏐fext ⏐f f{ψ0,s }s

C(L0 , λ0 ) −−−−−→ C ([0, 1]s × L0 , (λ0 , λ0 ))

←−

C([0, 1]s × L0 , λ0 )

Diagram 7.2.18 Proof. The proof is a parameterized version of the proof of Lemma 7.2.299 and is similar to the discussion of Subsection 7.2.10. We first explain the construc tion of fid×φ . We define C([0, 1]s × L0 , λ0 ) and C([0, 1]s × L1 , λ1 ) similarly as in Subsection 7.2.9 - 7.2.10. We next use the moduli space (7.2.302)

 [0, 1]s × Mmain k+1 ((L1 , {Jρ }ρ ) : β; top(ρ)).

 Here Mmain k+1 ((L1 , {Jρ }ρ ) : β; top(ρ)) is the moduli space that we used to define φ  f . We denote (7.2.302) by Mmain k+1 (L1 : β; top(ρ); [0, 1]s ). There exists an obvious evaluation map   k+1 ev = (ev1 , · · · , evk , ev0 ) : Mmain . k+1 (L1 : β; top(ρ); [0, 1]s ) → ([0, 1] × L1 )

 = Let Pi be smooth singular chains contained in C([0, 1]s × L0 , λ0 ). We put P (P1 , · · · , Pk ) and define   Mmain k+1 (L1 : β; top(ρ); [0, 1]s ; P)  = Mmain k+1 (L1 : β; top(ρ); [0, 1]s ) ×id×φ (P1 × · · · × Pk ).

Here the fiber product is taken over ([0, 1] × L1 )k using ev1 , · · · , evk and id × φ . We now use multisections to define its virtual fundamental chain and regard it as a smooth singular chain on [0, 1]s × L1 by using ev0 . Then in the same way as in Subsections 7.2.9 - 7.2.10 we obtain C([0, 1]s × L1 , (λ0 , λ1 )), a filtered An ,K   structure on it and a filtered An,K homomorphism fid×φ such that (7.2.301.1) is satisfied. We finally extend An ,K  structure and An,K homomorhpism to C ([0, 1]s ×   L0 , (λ0 , λ0 )) and obtain Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 )), fid×φ ext , which satisfy the required properties.  Remark 7.2.303. (1) We have proved that the filtered An,K -homomorphisms  fid×φ and others are compatible with Eval in that the strict identities (7.2.301.1), (7.2.301.2) hold, but we do not claim they are compatible with Incl. Their compatibility with Eval is based on the Compatibility Conditions required for the mul tisections of the moduli spaces that we use to construct fid×φ and others. On the id×φ so that the strict other hand it seems difficult to define the homomorphism f equality   fid×φ ◦ Incls = Incls ×fφ

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holds because (Incls )k,β = 0 for (k, β) = 0 when the moduli space (7.2.302) is used  for the definition of fid×φ . This point is related to Proposition 7.2.217 in Subsection 7.2.12 and Proposition 7.2.277 which we prove later. (2) Actually we can use Theorem 7.2.176 to define 

id × fφ : C([0, 1]s × L0 , λ0 ) → C([0, 1]s × L0 , (λ0 , λ1 )) 



that satisfies (7.2.301). It moreover satisfies (id × fφ ) ◦ Incl = Incl ◦ fφ . In other  words, this map id × fφ can be defined by a purely algebraic method and can play id×φ as far as the proof of Lemma 7.2.300 is concerned. the role of f However, in the proof of Lemma 7.2.327 we will use our geometric construction   of fid×φ . Especially we use the fact that fid×φ is defined by correspondence associated to the virtual fundamental chain of the moduli space (7.2.302). See Remark  7.2.329. By this reason we do not use this map id × fφ . Lemma 7.2.304. There exist filtered An,K homomorphism ea : C ([0, 1]s × L0 , (λ0 , λ0 )) → C([0, 1]s × L0 , λ0 ) and filtered An ,K  homomorphisms Ea : Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 ))) → Ca ([0, 1]s × L1 , (λ0 , λ1 )) j[0,1]s : Ca ([0, 1]s × L1 , (λ0 , λ1 )) → C([0, 1]s × L1 , λ1 ) such that (7.2.305.1) ea , Ea and j[0,1]s are identity on C([0, 1]s × L0 , λ0 ), Ca ([0, 1]s ×    L1 , (λ0 , λ1 )), C([0, 1] × L1 , λ1 ), respectively. Evals=s0 ◦ea = js=s0 ◦ Evals=s0 . Evals=s0 ◦Ea = js=s0 ◦ Evals=s0 . Evals=s0 ◦j[0,1]s = j ◦ Evals=s0 .

(7.2.305.2) (7.2.305.3) (7.2.305.4)

We can construct such ea , Ea , j[0,1]s by an obstruction theory similar to one in the proof of Proposition 7.2.211. We will work out a similar argument later in a more difficult case of homotopy of homotopies. So we omit the proof of the proposition here. See the proof of Lemma 7.2.310 and Proposition 7.2.312. (7.2.305) implies that the following Diagram 7.2.19 commutes up to homotopy. C([0, 1]s × L1 , λ1 ) , ⏐j ⏐ [0,1]s Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 )) −−−a−→Ca ([0, 1]s × L1 , (λ0 , λ1 )) , , ⏐ id×φ ⏐ id×φ ⏐fext ⏐f E

f{ψ1,s }s

C(L0 , λ0 ) −−−−−→ C ([0, 1]s × L0 , (λ0 , λ0 )) −−−a−→ e

C([0, 1]s × L0 , λ0 )

Diagram 7.2.19 We will specify the homotopy between two compositions in the square of Diagram 7.2.19 in Lemma 7.2.309 below. We first start its un-s-parameterized version.

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553

Let Ca,r (L1 , λ1 ) be a model of [0, 1]r × C(L1 , λ1 ). More precisely we take the model C(L1 , λ1 )[0,1]r in Subsection 4.2.1. Then there exists Fas=s0 : C s=s0 (L0 , (λ0 , λ0 )) → Ca,r (L1 , λ1 ) such that (7.2.306.1) (7.2.306.2) (7.2.306.3)



Evalr=0 ◦Fas=s0 = j ◦ fφ ◦ js=s0 ,  Evalr=1 ◦Fas=s0 = j ◦ js=s0 ◦ fφext,s=s0 ,  Fas=s0 = Inclr ◦ j ◦ fφ on C(L0 , λ0 ).

Existence of the map Fas=s0 follows from the homotopy commutativity of the diagram obtained by inverting the directions of three inclusions in Diagram 7.2.17. (7.2.306.3) follows from the fact that the diagram strictly commutes on C(L0 , λ0 ). (7.2.306.1) and (7.2.306.2) imply that Fas=s0 ◦ fψ0,s0 : C(L0 , λ0 ) → Ca,r (L1 , λ1 ) defines a homotopy (7.2.298.1). a (L , λ ) be a model of [0, 1]2 × C(L , λ ) such that Now let C 1 1 1 1 a (L , λ ))s=s = Ca,r (L , λ ), (C 1 1 1 1 0

a (L , λ ))r=r = C([0, 1] × L , λ ). (C 1 1 1 1 0

More precisely we take (7.2.307)

a (L , λ ) = C([0, 1]s × L , λ )[0,1]r . C 1 1 1 1

 a (L , λ ) such that In particular, there exists Inclr : C([0, 1]s × L1 , λ1 ) → C 1 1 (7.2.308)

Evals=s0 ◦ Inclr = Inclr ◦ Evals=s0 : C([0, 1]s × L1 , λ1 ) → Ca,r (L1 , λ1 ).

Remark 7.2.309. We remark that in the context of the general model of [0, 1]2 × C, existence of Inclr , Incls that satisfy (7.2.308) is not assumed. (See  a (L , λ ) purely Remark 7.2.203.) Here (7.2.308) is satisfied because we defined C 1 1   algebraically from C([0, 1]s × L1 , λ1 ) by (7.2.307). Lemma 7.2.310. For any Fas=s0 satisfying (7.2.306), there exists a filtered An,K homomorphism a (L , λ ) Ha : C ([0, 1]s × L0 , (λ0 , λ0 )) → C 1 1 such that 

(7.2.311.1)

Evalr=0 ◦Ha = j[0,1]s ◦ fid×φ ◦ ea ,

(7.2.311.2)

Evalr=1 ◦Ha = j[0,1]s ◦ Ea ◦ fid×φ ext ,

(7.2.311.3)

Evals=s0 ◦Ha = Fas=s0 ◦ Evals=s0 ,

(7.2.311.4)



Ha = Inclr ◦ j[0,1]s ◦ fid×φ



on C([0, 1]s × L0 , λ0 ).

The proof of Lemma 7.2.310 will be based on an obstruction theory which we summarize in the Proposition 7.2.312 below. Let C0 , C  be filtered A∞ algebras and h : C0 → C  be a strict filtered An ,K  homomorphism.

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 C  s=s , C r=r , C = (C0 , {Cr ,s }r ,s ) be as in Definition 7.2.188. Namely Let C, 0 0 0 0 0 0  is a model of [0, 1]2 × C. We assume everything is G-gapped. C  and Hr=r0 : C  → C r=r , Hs=s0 : Proposition 7.2.312. Let H0 : C0 → C 0  s=s be G-gapped filtered An ,K  homomorphisms. Suppose that h is a hoC → C 0 motopy equivalence and h1,β0 is split injective, and assume (7.2.313.1) (7.2.313.2) (7.2.313.3)

Evals=s0 ◦ Hr=r0 = Evalr=r0 ◦ Hs=s0 , Hr=r0 ◦ h = Evalr=r0 ◦ H0 , Hs=s0 ◦ h = Evals=s0 ◦ H0 .

 such Then there exists a G-gapped filtered An ,K  homomorphism H : C  → C that (7.2.314.1) (7.2.314.2) (7.2.314.3)

Evals=s0 ◦ H = Hs=s0 , Evalr=r0 ◦ H = Hr=r0 , H ◦ h = H0 .

 in the proof we will give below. We remark that we will never use Incl of C Proof. The proof is similar to that of Proposition 7.2.190. We will construct the map HK  ,β inductively over the order < which was defined on the set of (n , K  ) = (β, K  ) in Definition 7.2.63. The first step of the induction, which is the construction of H1,β0 , is similar to and easier than the main induction step and omitted. Suppose that we have defined HK,β for β, K with (β, K) < (n , K  ). We will define HK  ,β , for β = n . The condition that (7.2.314.3) is satisfied as an identity of filtered An ,K  homomorphism is

  (7.2.315) HK  ,β ◦ h⊗K H,β  ◦ (hk1 ,β1 ⊗ · · · ⊗ hk ,β ) ◦ Δ−1 , 1,β0 = (H0 )K ,β − wherethe sum in the  right hand side is taken over all , β, (k1 , β1 ),· · ·, (k , β ) such that ki = K  , β  + βi = β and (, β  ) = (K  , β).  (7.2.315) determine HK  ,β uniquely on the image of h⊗K 1,β0 . Lemma 7.2.316. There exists   HIK  ,β : BK  C [1] → C

such that: (7.2.317.1) (7.2.317.2) (7.2.317.3)



HIK  ,β = HK  ,β satisfies (7.2.315) on the image of h⊗K 1,β0 . 0 Evals=s0 ◦ HIK  ,β = Hs=s . K  ,β 0 Evalr=r0 ◦ HIK  ,β = Hr=r K  ,β .

Proof. We define (7.2.318)

K = Ker

% 

s=s ⊕ C 0



r=r −→ C 0 B

1 1  

& C s=s0 ,r=r0

r0 =0 s0 =0

By (7.2.313.1) we have 

s=1 r=0 r=1 (Hs=0 K  ,β , HK  ,β , HK  ,β , HK  ,β ) ∈ Hom(BK  C [1], K).

.

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555





 We note that the image of h⊗K 1,β0 is a direct summand of BK C by the split injectiveness assumption on h. Moreover the evaluation map induces an surjection  → K. Lemma 7.2.316 follows. C 



 by We define the obstruction class oK  ,β (H) ∈ Hom(BK  C [1], C) oK  ,β (H) + δ1 (HIK  ,β ) = − +



m,β  ◦ (Hk1 ,β1 ⊗ · · · ⊗ Hk ,β ) ◦ Δ−1  k2 ,β2 , Hk1 ,β1 ◦ m

where the sum of  the first term is taken over all , β  , (k1 , β1 ), · · · , (k , β ) such that    ki = K , β + βi = β, (, β  ) = (1, β0 ). The sum of the second term is taken over all k1 , k2 , β1 , β2 such that k1 + k2 = K  + 1, β1 + β2 = β and (k2 , β2 ) = (1, β0 ). We have the following: Lemma 7.2.319. 

(7.2.320.1)

oK  ,β (H) ◦ h⊗K 1,β0 = 0.

(7.2.320.2) (7.2.320.3)

oK  ,β (H) ∈ Hom(BK  C [1], K). δ1 (oK  ,β (H)) = 0.



Proof. We first prove (7.2.320.1). Using (7.2.315) and the fact that H0 and h are filtered An ,K  homomorphisms, we have

(7.2.321)

  ,β  ◦ (hk1 ,β1 ⊗ · · · ⊗ hk ,β ) ◦ Δ−1 H ,β  ◦ m

= m,β  ◦ (Hk1 ,β1 ⊗ · · · ⊗ Hk ,β ) ◦

 -



(hmj,1 ,βj,1 ⊗ · · · ⊗ hmkj ,βj,mk ) ◦ Δ

j

kj −1

j

j=1

.

Here the sum in the left hand side is taken over all ,  ,  , β  , β  , ki , βi such that        +  =  + 1, ki = K , β + β + βi = right hand β. The sum in the  side is taken over all , β  , ki , βi , mj,i , βj,i such that i,j mj,i = K  , β  + βj + βj,i = β. By the same way as in the proof of Lemma 7.2.193, the term involving at least one ha,β(a) with (a, β(a)) = (1, β0 ) cancels in (7.2.321) by induction hypothesis, that is H is filtered An,K homomorphism for (n, K) < (n , K  ). It implies (7.2.320.1). (7.2.320.2) follow from (7.2.317.2) and (7.2.317.3) and induction hypothesis. (7.2.317.3) follows from definition and induction hypothesis.  

We derive from Lemma 7.2.319 that there exists Cor ∈ Hom(BK  C [1], K) such that (7.2.322)

oK  ,β (H) = δ1 (Cor),



Cor ◦ h⊗K 1,β0 = 0.

We put HK  ,β = HIK  ,β + Cor. It is straightforward to check that it has the required properties. The proof of Proposition 7.2.312 is complete. 

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Proof of Lemma 7.2.310. We put C0 = C([0, 1]s × L0 , λ0 ), =C a , C

 s=s = Ca,r , C 0

C  = C ([0, 1]s × L0 , (λ0 , λ0 )),  r=r = Ca ([0, 1]s × L , (λ , λ )). C 1 0 1 0

We define 



Hr=1 = j[0,1]s ◦ Ea ◦ fid×φ ext ,

Hr=0 = j[0,1]s ◦ fid×φ ◦ ea , Hs=s0 = Fas=s0 ◦ Evals=s0

which are the right hand sides of (7.2.311.1)-(7.2.311.3) respectively. We also define H0 = Inclr ◦ j[0,1]s ◦ fid×φ



as in (7.2.311.4). Finally we define h : C([0, 1]s × L0 , λ0 ) → C([0, 1]s × L0 , (λ0 , λ0 )) to be the inclusion given in (7.2.301.5) Lemma 7.2.300. We will check (7.2.313.1), (7.2.313.2), (7.2.313.3). Combining (7.2.305.4), (7.2.305.2), (7.2.301.1), we obtain 



Evals=s0 ◦ j[0,1]s ◦ fid×φ ◦ ea = j ◦ f φ ◦ Evals=s0 ◦ ea 

= j ◦ f φ ◦ js=s0 ◦ Evals=s0 . On the other hand, by composing Evals=s0 with (7.2.306.1), we obtain 

Evalr=0 ◦ Fas=s0 ◦ Evals=s0 = j ◦ f φ ◦ js=s0 ◦ Evals=s0 and hence (7.2.323)

Evals=s0 ◦ Hr=0 = Evalr=0 ◦ Hs=s0 .

Applying (7.2.305.4), (7.2.305.3), (7.2.301.2) and (7.2.306.2) this time for r = 1, we similarly prove (7.2.324.)

Evals=s0 ◦ Hr=1 = Evalr=1 ◦ Hs=s0 .

Combining (7.2.323) and (7.2.324), we have proved (7.2.313.1). To check (7.2.313.2) we calculate (7.2.325)





Evalr=r0 ◦Inclr ◦ j[0,1]s ◦ fid×φ = j[0,1]s ◦ fid×φ .

We also use (7.2.305.1) to calculate 

j[0,1]s ◦ fid×φ ◦ ea = j[0,1]s ◦ fid×φ



on Ca ([0, 1]s × L0 , λ0 ). This coincides with (7.2.325). Next, using (7.2.301.6) and (7.2.305.1), we find that the following identity holds on C([0, 1]s × L0 , λ0 ): 



= j[0,1]s ◦ fid×φ . j[0,1]s ◦ Ea ◦ fid×φ ext Again this coincides with (7.2.325). Thus we checked (7.2.313.2). We finally check (7.2.313.3). Using (7.2.308), (7.2.305.4) and (7.2.301.1), we calculate (7.2.326)





Evals=s0 ◦Inclr ◦ j[0,1]s ◦ fid×φ = Inclr ◦ j ◦ fφ ◦ Evals=s0 .

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

557

On the other hand, since Evals=s0 (C([0, 1]s × L0 , λ0 )) ⊂ C(L0 , λ0 ), (7.2.306.3) implies that the following identity holds on C([0, 1]s × L0 , λ0 ): 

Fas=s0 ◦ Evals=s0 = Inclr ◦ j ◦ fφ ◦ Evals=s0 , which coincides with (7.2.326). (7.2.313.3) is proved. Now we can apply Proposition 7.2.312. Lemma 7.2.310 follows.



We have finished the discussion on the homotopy (7.2.298.1) and its s-parameterized version. We next proceed to the discussion on the homotopy (7.2.298.2) and its s-parameterized version. We recall the proof of Theorem 4.6.44. Lemma 4.6.45 says that if we choose an appropriate family of almost complex structures, the moduli spaces we use to define fψ◦φ1,s0 is identical to the moduli space we use to define fψ ◦ fφ1,s0 . In other words, the homotopy (7.2.298.2) is obtained by interpolating two different choices of (families of) almost complex structures, perturbations, etc., in the way we discussed in Subsection 7.2.10. Therefore we have the following: Lemma 7.2.327. There exist a filtered An ,K  algebra Cbs=s0 ([0, 1]r × L1 , (λ0 , λ1 )) and filtered An ,K  homomorphisms Inclr : C(L1 , λ1 ) → Cbs=s0 ([0, 1]r × L1 , (λ0 , λ1 )), and Evalr=r0 satisfying (7.2.175) and a filtered An,K homomorphism Fbs=s0 : C(L0 , λ0 ) → Cbs=s0 ([0, 1]r × L1 , (λ0 , λ1 )) such that Diagram 7.2.20 below commutes: φ ◦ψ0,s

f

C(L0 , λ0 )

−→

0

C(L1 , (λ0 , λ1 )) , ⏐ Evalr=1 ⏐

 s=s0

Fb

Cbs=s0 ([0, 1]r × L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 (

−→

C(L0 , λ0 ) 



fφ ext,s=s

ψ0,s 0

0 C(L0 , λ0 ) −−−−→ C s=s0 (L0 , (λ0 , λ0 )) −−−−−→ Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ))

f

Diagram 7.2.20 In the same way we can prove the following s-parameterized version. Lemma 7.2.328. There exist a filtered An ,K  algebra Cb ([0, 1]2 × L1 , (λ0 , λ1 )), filtered An ,K  homomorphisms Incl : C(L1 , λ1 ) → Cb ([0, 1]2 × L1 , (λ0 , λ1 )) and Evals=s0 , Evalr=r0 satisfying (7.2.189) and a filtered An,K homomorphism Hb : C(L0 , λ0 ) → Cb ([0, 1]2 × L1 , (λ0 , λ1 ))

558

7.

TRANSVERSALITY

such that we have Evals=s0 ◦Hb = Fbs=s0 . and the following diagram commutes: f{Φ0,s }s

C r=0 ([0, 1]s × L1 , (λ0 , λ1 )) , ⏐ Evalr=1 ⏐

−→

C(L0 , λ0 ) 

Cb ([0, 1]2 ×L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 (

H

b −→

C(L0 , λ0 )  f{ψ0,s }s

fid×φ



ext C(L0 , λ0 ) −−−−−→ C([0, 1]s × L0 , (λ0 , λ0 )) −− −−→Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 ))

Diagram 7.2.21 We remark that Φ0,s = φ ◦ ψ0,s . Hence Lemma 7.2.328 is a parameterized version of Lemma 7.2.327. It can be proved in the same way as the proof of Lemma 7.2.285. Our complex Cb ([0, 1]2 × L1 , (λ0 , λ1 )) is a subcomplex of S([0, 1]2 × L1 ) which was constructed during the proof of Lemma 7.2.285. We have thus completed the discussion on (7.2.280.1). 

Remark 7.2.329. We remark here that we use the fact that fid×φ is defined ext via the correspondence defined by the virtual fundamental chain of the moduli space (7.2.302). See Remark 7.2.303 (2). Discussion on (7.2.280.3): We remark that the homotopy (7.2.280.3) is a composition of the following two homotopies. ψ

(7.2.330.1)

1,s0 gψ1,s0 ◦φ = j ◦ fψ1,s0 ◦φ ∼ j ◦ fext,s=s ◦ fφ . 0

(7.2.330.2)

1,s0 ◦ fφ ∼ j ◦ fψ1,s0 ◦ j ◦ fφ = gψ1,s0 ◦ gφ . j ◦ fext,s=s 0

ψ

Let us first study (7.2.330.2). This part is parallel to (7.2.298.1). We first obtain ψ1,s0 and fext,s=s , such that Diagram 7.2.22 below commutes. 0

Cds=s0 (L1 , ((λ0 , λ1 ), λ1 ) C(L1 , λ1 ) ⏐ ⏐ (

ψ1,s 0

f

−−−−→

C s=s0 (L1 , (λ1 , λ1 )) ⏐ ⏐ (

←−−−− C(L1 , λ1 )

ψ1,s

0 fext,s=s

0 C(L1 , (λ0 , λ1 )) −−−−−→ Cds=s0 (L1 , ((λ0 , λ1 ), λ1 )) , ⏐φ ⏐f

C(L0 , λ0 )

Diagram 7.2.22

We next take its s-parameterized version and obtain the following: • Filtered An ,K  algebras C([0, 1]s × L1 , ((λ0 , λ1 ), λ1 )), C([0, 1]s × L1 , λ1 ). • Filtered An ,K  homomorphisms Incl : C(L1 , λ1 ) → C([0, 1]s × L1 , λ1 ) and Incl : C(L1 , λ1 ) → Cd ([0, 1]s × L1 , ((λ0 , λ1 ), λ1 )). {ψ } • Eval’s and fext1,s s .

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

559

They satisfy the following properties: (7.2.331.1) (7.2.331.2) (7.2.331.3)

Eval, Incl satisfy (7.2.175). ψ1,s0 {ψ } Evals=s0 ◦fext1,s s = fext,s=s . 0 Diagram 7.2.23 below commutes:

C(L1 , λ1 ) ⏐ ⏐ (

f{ψ1,s }s

−−−−→ C([0, 1]s ×L1 , (λ1 , λ1 )) ←−−−− C([0, 1]s × L1 , λ1 ) ⏐ ⏐ ( {ψ1,s }s

f

ext C(L1 , (λ0 , λ1 )) −− −−→ Cd ([0, 1]s ×L1 , ((λ0 , λ1 ), λ1 )) , ⏐φ ⏐f

C(L0 , λ0 )

Diagram 7.2.23 L1 , λ1 )

We remark that C([0, 1]s × is the same filtered An ,K  algebra as one we already introduced in Lemma 7.2.300. Lemma 7.2.332. There exist filtered An ,K  homomorphisms ed and Ed with the following properties: (7.2.333.1)

Diagram 7.2.24 commutes up to filtered An,K homotopy:

C(L1 , λ1 ) , ⏐ ⏐j

f{ψ1,s }s

−−−−→ C([0, 1]s × L1 , (λ1 , λ1 )) −−−d−→ C([0, 1]s × L1 , λ1 ) , ⏐E ⏐ d e

{ψ1,s }s

f

ext C(L1 , (λ0 , λ1 )) −− −−→ Cd ([0, 1]s × L1 , ((λ0 , λ1 ), λ1 )) , ⏐φ ⏐f

C(L0 , λ0 ) (7.2.333.2) (7.2.333.3) (7.2.333.4) (7.2.333.5) where

Diagram 7.2.24

ed = id on C([0, 1]s × L1 , λ1 ). Ed = id on C([0, 1]s × L1 , (λ1 , λ1 )). Evals=s0 ◦Ed = js=s0 ◦ Evals=s0 . Evals=s0 ◦ed = js=s0 ◦ Evals=s0 , js=s0 : Cds=s0 (L1 , ((λ0 , λ1 ), λ1 )) → C(L1 , λ1 )

js=s0 : C s=s0 (L1 , (λ1 , λ1 )) → C(L1 , λ1 ) are the homotopy inverses of the inclusions.

We can prove Lemma 7.2.332 by an obstruction theory. d (L , λ ) be a model of We next specify the homotopy in (7.2.333.1). Let C 1 1   [0, 1]r × C([0, 1]s × L1 , λ1 ). More precisely, we put  d (L , λ ) = C([0, 1]s × L , λ )[0,1]r . C 1 1 1 1 d (L , λ ) is a model of [0, 1]2 × C(L , λ ) in the sense of Definition 7.2.188. Then C 1 1 1 1 We have d (L , λ ))s=s = Cd,r (L , λ ), (C 1 1 1 1 0

d (L , λ ))r=r = C([0, 1]s × L , λ ), (C 1 1 1 1 0

560

7.

TRANSVERSALITY

where

Cd,r (L1 , λ1 ) = C(L1 , λ1 )[0,1]r is a model of [0, 1]r × C(L1 .λ1 ).  d (L , λ ) such that There exists Inclr : C([0, 1]s × L1 , λ1 ) → C 1 1 Evals=s0 ◦ Inclr = Inclr ◦ Evals=s0

(7.2.334)

as a map C([0, 1]s × L1 , λ1 ) → Cd,r (L1 , λ1 ). We remark that we have: Fds=s0 : C(L1 , (λ0 , λ1 )) → Cd,r (L1 , λ1 ) such that ψ

(7.2.335.1)

1,s0 , Evalr=0 ◦Fds=s0 = js=s0 ◦ js=s0 ◦ fext,s=s 0

(7.2.335.2)

Evalr=1 ◦Fds=s0 = js=s0 ◦ fψ1,s0 ◦ j,

(7.2.335.3)

Fds=s0 = Inclr ◦js=s0 ◦ fψ1,s0 Fds=s0

on C(L1 , λ1 ).

◦ f gives the homotopy (7.2.330.1). Existence of Fds=s0 In other words, satisfying (7.2.335.1) and (7.2.335.2) follows from homotopy commutativity of the diagram obtained by inverting the directions of 2 arrows in Diagram 7.2.22. And (7.2.335.3) follows from its exact commutativity on C(L1 , λ1 ). φ

Lemma 7.2.336. For any Fds=s0 satisfying the above equalities (7.2.335.1), (7.2.335.2) and (7.2.335.3), there exists a filtered An ,K  homomorphism d (L , λ ) Hd : C(L1 , (λ0 , λ1 )) → C 1 1 with the following properties: (7.2.337.1) (7.2.337.2) (7.2.337.3) (7.2.337.4)

{ψ

}

Evalr=0 ◦Hd = ed ◦ Ed ◦ fext1,s s , Evalr=1 ◦Hd = ed ◦ f{ψ1,s }s ◦ j, Evals=s0 ◦Hd = Fds=s0 , Hd = Inclr ◦ed ◦ f{ψ1,s }s , on C(L1 , λ1 ).

Proof. We will apply Proposition 7.2.312. We put C0 = C(L1 , λ1 ),

C  = C(L1 , (λ0 , λ1 )),

=C d (L , λ ). C 1 1

Let Hr=0 , Hr=1 , Hs=s0 be the right hand side of (7.2.337.1), (7.2.337.2), (7.2.337.3), respectively. We put H0 = Inclr ◦ ed ◦ f{ψ1,s }s . We define h : C(L1 , λ1 ) → C(L1 , (λ0 , λ1 )) to be the inclusion. We will check (7.2.313.1), (7.2.313.2), (7.2.313.3). By (7.2.333.4), (7.2.333.5), (7.2.331.2) and (7.2.335.1), we obtain {ψ

Evals=s0 ◦ ed ◦ Ed ◦ fext1,s

}s

ψ

1,s0 = js=s0 ◦ js=s0 ◦ fext = Evalr=0 ◦ Fds=s0 .

We also calculate, by using (7.2.333.5), (7.2.331.2) and (7.2.335.2), that Evals=s0 ◦ ed ◦ f{ψ1,s }s ◦ j = js=s0 ◦ fψ1,s0 ◦ j = Evalr=1 ◦ Fds=s0 . We thus proved (7.2.313.1).

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

561

By (7.2.333.1) and (7.2.333.3) we have the following equality on C(L1 , λ1 ): {ψ

ed ◦ Ed ◦ fext1,s

}s

= ed ◦ f{ψ1,s }s = Evalr=0 ◦ Inclr ◦ ed ◦ f{ψ1,s }s .

By the definition of j, we also have the following equality on C(L1 , λ1 ): ed ◦ f{ψ1,s }s ◦ j = ed ◦ f {ψ1,s }s = Evalr=1 ◦ Inclr ◦ ed ◦ f{ψ1,s }s . We thus proved (7.2.313.2). By (7.2.335.3) we have the following equality on C(L1 , λ1 ): Fds=s0 = Inclr ◦js=s0 ◦ fψ1,s0 . By (7.2.334), (7.2.333.5), (7.2.331.2) and (7.2.331.2), we have the following equality on C(L1 , λ1 ): Evals=s0 ◦ Inclr ◦ed ◦ f{ψ1,s }s = Inclr ◦Evals=s0 ◦ ed ◦ f{ψ1,s }s = Inclr ◦js=s0 ◦ fψ1,s0 . Thus (7.2.313.3) is proved. Lemma 7.2.336 now follows from Proposition 7.2.312.



We turn to the study of (7.2.330.1). The discussion of it is parallel to the discussion of (7.2.298.2). Namely we obtain the following two Diagrams 7.2.25 and 7.2.26 which strictly commute. ψ1,s

0 fext,s=s

fφ

0 C(L0 , λ0 ) −−−−→ C s=s0 (L1 , (λ0 , λ1 )) −−−−−→

 Fcs=s0

Ccs=s0 ([0, 1]r × L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 (

−→

C(L0 , λ0 ) 

ψ1,s ◦φ 0

f

C(L0 , λ0 )

Cds=s0 (L0 , ((λ0 , λ1 ), λ1 ) , ⏐ Evalr=1 ⏐

C(L1 , (λ0 , λ1 ))

−→

Diagram 7.2.25 {ψ1,s}s

fφ

f

ext −−→ Cd ([0, 1]s × L1 , ((λ0 , λ1 ), λ1 )) C(L0 , λ0 ) −−−−→ C([0, 1]s × L0 , (λ0 , λ1 )) −− , ⏐ Evalr=1 ⏐ 

C(L0 , λ0 )

H

c −→

 C(L0 , λ0 )

f{Φ1,s }s

−→

Cc ([0, 1]2 × L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 ( C r=1 ([0, 1]s × L1 , (λ0 , λ1 ))

Diagram 7.2.26 We have thus finished the discussion on each of the cases (7.2.280.1), (7.2.280.2), (7.2.280.3). We now glue them to obtain the required homotopy of homotopies H. We first recall the following construction which we already used in the proof of Proposition 4.2.37. Let Cj (j = 1, 2) be either models of [0, 1]r × C or of [0, 1]2 × C.

562

7.

TRANSVERSALITY

We assume that the target of Evalr=1 of C1 is equal to the target of Evalr=0 of C2 . Then we put C1 ∨r C2 = {(x, y) ∈ C1 ⊕ C2 | (Evalr=1 )1 (x) = (Evalr=0 )1 (y)}. C1 ∨r C2 has the structure of a filtered An,K algebra. (We can easily prove it by using the fact that (Evalr=r0 )k,β = 0 for (k, β) = (1, β0 ).) We define Evalr=r0 on C1 ∨r C2 by Evalr=0 (x, y) = Evalr=0 (x),

Evalr=1 (x, y) = Evalr=1 (y).

In the context of our application below, Cj is a model of [0, 1]2 ×(C0 , {Cj;r0 ,s0 }r0 ,s0 ) for each j = 1, 2 such that C1;r0 =1,s0 = C2;r0 =0,s0 . In our case, Incl : C0 → C1 ∨r C2 is defined by Incl(x) = (Incl(x), Incl(x)). If fi : C → Ci is a filtered An,K homomorphisms with Evalr=1 ◦f1 = Evalr=0 ◦f2 , then we define a filtered An,K homomorphism. f1 ∨r f2 : C → C1 ∨r C2 by (f1 ∨r f2 )k,β = (f1 )k,β ⊕ (f2 )k,β . We can glue more than two algebras and homomorphisms in the same way. Now we glue three of the five homotopies of homotopies we have constructed so far. Definition 7.2.338. We put (7.2.339.1)

 X (L ) =Cb ([0, 1]2 × L , (λ0 , λ )) C 1 1 1 ∨r CX ([0, 1]2 × L1 , (λ0 , λ1 )) ∨r Cc ([0, 1]2 × L1 , (λ0 , λ1 )).

(See Diagram 7.2.21, Lemma 7.2.285, Diagram 7.2.26.) We put C0 = C(L1 , λ1 ). We also define 0 Cs=s (L1 ) =Cbs=s0 ([0, 1]r × L1 , (λ0 , λ1 )) X

s=s0 ([0, 1]r × L1 , (λ0 , λ1 )) ∨r CX ∨r Ccs=s0 ([0, 1]r × L1 , (λ0 , λ1 )),

(7.2.339.2)

X (L ), F s=s0 : C(L0 , λ0 ) → Cs=s0 (L ) by and HX : C(L0 , λ0 ) → C 1 1 X X (7.2.340.1)

HX = Hb ∨r f{Φr,s }r,s ∨r Hc ,

(7.2.340.2)

s=s0 = Fbs=s0 ∨r f{Φr,s0 }r ∨r Fcs=s0 . FX

 X (L ) → Cs=s0 (L ) such that We can define Evals=s0 : C 1 1 X s=s0 . Evals=s0 ◦ HX = FX

(7.2.341)

We have the following commutative diagram {ψ1,s }s

fφ

f

ext −−−→ Cd ([0, 1]s ×L1 , ((λ0 , λ1 ), λ1 )) C(L0 , λ0 ) −−−−→ C([0, 1]s ×L0 , (λ0 , λ1 )) −− , ⏐ Evalr=1 ⏐ 

X (L ) C 1 ⏐ ⏐ Evalr=0 (

H

X −→

C(L0 , λ0 )  f{ψ0,s }s

fid×φ



ext C(L0 , λ0 ) −−−−−→ C([0, 1]s ×L0 , (λ0 , λ0 )) −− −−→ Ca ([0, 1]s ×L1 , ((λ0 , λ0 ), λ1 ))

Diagram 7.2.27

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

563

We next glue HX with Ha and Hd . For this purpose we compose HX with another homotopy equivalence J we will construct below. We first take a model of [0, 1]r × C(L1 , λ1 ). Then by Proposition 7.2.190 we  Y (L ) of [0, 1]2 × C(L , λ ) such that have model C 1

1

1

Y (L ))r=0 = C([0, 1]s × L , λ ) = (C Y (L ))r=1 , (C 1 1 1 1    [0,1]r  (CY (L1 ))s=s0 = C(L1 , λ1 ) . Lemma 7.2.342. There exist filtered An ,K  homotopy equivalences X (L ) → C Y (L ), J:C 1 1

s=s0 (L ) → (C Y (L ))s=s J(s=s0 ) : C 1 1 0 X

such that (7.2.343.1) (7.2.343.2)

Evals=s0 ◦J = J(s=s0 ) ◦ Evals=s0 Y (L ) J ◦ Incl = Incl : C(L , λ ) → C 1

1

1

and the following Diagram 7.2.28 commutes: d d Cd ([0, 1]s × L1 , ((λ0 , λ1 ), λ1 )) −→ C([0, 1]s × L1 , (λ1 , λ1 )) −→ C([0, 1]s × L1 , λ1 ) , , ⏐ ⏐ Evalr=1 ⏐ Evalr=1 ⏐

E

 X (L ) C 1 ⏐ ⏐ Evalr=0 (

e

Y (L ) C 1 ⏐ ⏐ Evalr=0 (

J

−→

a Ca ([0, 1]s × L1 , ((λ0 , λ0 ), λ1 )) −→

E

Cs (L1 , (λ0 , λ1 ))

j[0,1]

s −−−→ C([0, 1]s × L1 , λ1 )

Diagram 7.2.28

We remark that Diagrams 7.2.28 and (7.2.343) induce (7.2.344)

Y (L )s=s . Inclr = I(s=s0 ) ◦ Inclr : C(L1 , λ1 ) → C 1 0

Diagrams 7.2.28 and (7.2.343) also induce the following commutative diagram: js=s

js=s

Cds=s0 (L1 , ((λ0 , λ1 ), λ1 )) −−−−0→ C s=s0 (L1 , (λ1 , λ1 )) −−−−0→ C(L1 , λ1 ) , , ⏐ ⏐ Evalr=1 ⏐ Evalr=1 ⏐ J(s=s0 )

X (L )s=s C 1 0 ⏐ ⏐ Evalr=0 (

 Y (L )s=s C 1 0 ⏐ ⏐ Evalr=0 (

−→

js=s

Cas=s0 (L1 , ((λ0 , λ0 ), λ1 ) −−−−0→

C(L1 , (λ0 , λ1 ))

Diagram 7.2.29

−−−−→ C(L1 , λ1 ) j

564

7.

TRANSVERSALITY

Proof of Lemma 7.2.342. We first use Theorem 7.2.176 to construct J(s=s0 ) such that Diagrams 7.2.29 commutes and (7.2.344) holds. Next we will construct J by using Proposition 7.2.312. We put C0 = C(L1 , λ1 ),  X (L ), C =C Y (L ). C =C 1 1 We also define h = Incl, Hr=0 = j[0,1]s ◦ Ea ◦ Evalr=0 ,

Hr=1 = ed ◦ Ed ◦ Evalr=1 ,

Hs=s0 = J(s=s0 ) ◦ Evals=s0 , Y (L ). H0 = Incl : C(L , λ ) → C 1

1

1

Let us check (7.2.313). We have Evals=s0 ◦ Hr=0 = j ◦ js=s0 ◦ Evals=s0 ◦ Evalr=0 by (7.2.305.4), (7.2.305.3). On the other hand, we have Evalr=0 ◦ Hs=s0 = j ◦ js=s0 ◦ Evals=s0 ◦ Evalr=0 by the commutativity of Diagram 7.2.29. Thus we checked (7.2.313.1) for r = 0. We next calculate Evals=s0 ◦ Hr=1 = js=s0 ◦ js=s0 ◦ Evals=s0 ◦ Evalr=1 by (7.2.333.4), (7.2.333.5). On the other hand, Evalr=1 ◦ Hs=s0 = js=s0 ◦ js=s0 ◦ Evals=s0 ◦ Evalr=1 by the commutativity of the diagram 7.2.29. Thus we checked (7.2.313.1) for r = 1 also. Next using the fact Incls (C(L1 , λ1 )) ⊆ C([0, 1]s × L1 , λ1 ) and (7.2.305.1), we have Hr=0 ◦ h = j[0,1]s ◦ Ea ◦ Incls = Incls = Evalr=0 ◦ H0 . (7.2.313.2) follows. We next calculate Hs=s0 ◦ h = Is=s0 ◦ Inclr = Inclr = Evals=s0 ◦ H0 by (7.2.344). We thus proved (7.2.313.3). Now Lemma 7.2.342 follows from Proposition 7.2.312.  Completion of the proof of Proposition 7.2.275. We put  = C a (L , λ ) ∨r C Y (L ) ∨r C d (L , λ ), C 1 1 1 1 1 1 H = (Ha ◦ f{ψ0,s }s ) ∨r (J ◦ HX ) ∨r (Hd ◦ fφ ). Here

f{ψ0,s }s : C(L0 , λ0 ) → C([0, 1]s × L0 , (λ0 , λ0 )), fφ : C(L0 , λ0 ) → C(L1 , (λ0 , λ1 )).

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

565

Moreover we put (s=s0 )

C1

 =C 1,s=s0 ,

C0 = C([0, 1]s × L0 , λ0 ), I0 = j[0,1]s ◦ fid×φ

C1 = C([0, 1]s × L1 , λ1 ),

  C i,r=r0 = Ci ,



for r0 = 0, 1. We first check that H is well-defined. We calculate 

Evalr=1 ◦ Ha ◦ f{ψ0,s }s = j[0,1]s ◦ Ea ◦ fid×φ ◦ f{ψ0,s }s ext by (7.2.311.2). On the other hand, we have 

◦ f{ψ0,s }s Evalr=0 ◦ I ◦ HX = j[0,1]s ◦ Ea ◦ Evalr=0 ◦ HX = j[0,1]s ◦ Ea ◦ fid×φ ext by the commutativity of the diagrams 7.2.28 and 7.2.27. Moreover {ψ

Evalr=1 ◦ I ◦ HX = ed ◦ Ed ◦ Evalr=1 ◦ HX = ed ◦ Ed ◦ fext1,s

}s

◦ fφ

by the commutativity of the diagrams 7.2.28 and 7.2.27. On the other hand, {ψ



Evalr=0 ◦ Hd ◦ fφ = ed ◦ Ed ◦ fext1,s

}s

◦ fφ

by (7.2.337.1). Thus H is well defined. Next, let us check (7.2.276.1) - (7.2.276.6). (7.2.276.1)-(7.2.276.3) are easy to verify. We calculate Evalr=0 ◦ H = Evalr=0 ◦ Ha ◦ f{ψ0,s }s 

= j[0,1]s ◦ fid×φ ◦ ea ◦ f{ψ0,s }s = I0 ◦ g{ψ0,s }s , where the first and the third equalities follow from definition and the second equality follows from (7.2.311.1). (7.2.276.4) follows. We next calculate Evalr=1 ◦ H = Evalr=1 ◦ Hd ◦ fφ = ed ◦ f{ψ1,s }s ◦ j ◦ fφ = g{ψ1,s }s ◦ gφ , where the first and the third equalities follow from definition and the second equality follows from (7.2.337.2). (7.2.276.5) follows. We finally calculate Evals=s0 ◦ H = (Evals=s0 ◦ Ha ◦ f{ψ0,s }s ) ∨r (Evals=s0 ◦ J ◦ HX ) ∨r (Evals=s0 ◦ Hd ◦ fφ ) = (Fas=s0 ◦fψ0,s0 ) ∨r (J(s=s0 ) ◦(Fbs=s0 ∨r f{Φr,s0 }r ∨r Fcs=s0 )) ∨r (Fds=s0 ◦fφ ) {Φr,s0 }r

= g+

.

Here the first equality is the definition. The second equality follows from (7.2.311.3), (7.2.273), (7.2.343.1), (7.2.341), (7.2.340.2), (7.2.337.3), and the third equality is {Φ } the definition. In fact g+ r,s0 r is the composition of five homotopies, (7.2.298.1), (7.2.298.2), (7.2.280.3), (7.2.330.1), (7.2.330.2), and each of the five homotopies correspond each of the five factors glued in the third line of the above formula. (7.2.276.6) follows. The proof of Proposition 7.2.275 is complete. 

566

7.

TRANSVERSALITY

Proof of Proposition 7.2.277. The proof is similar to the proof of Proposition 7.2.275. In fact, we may regard Incls : C(Li , λi ) → C([0, 1]s × Li , λi ) as a homotopy between the identity to the identity. The construction of it (Proposition 7.2.162) was similar to the construction of f{ψr0 ,s }s in the proof of Proposition 7.2.160. In other words, to prove Proposition 7.2.277, we will apply the argument of the proof of Proposition 7.2.275 to the case when Ψr,s is independent of r, s and is φ . We however repeat the detail of the proof below for completeness. We consider the constant homotopy {φ }s given by φs = φ for s ∈ [0, 1]. It induces a filtered An,K homomorphism 

f{φ }s : C(L0 , λ0 ) → C([0, 1]s × L1 , (λ0 , λ1 ))

(7.2.345) such that





Evals=s0 ◦ f{φ }s = fφ .

(7.2.346)

We remark that we may need to add more simplices to C([0, 1]s × L1 , (λ0 , λ1 )) which was defined in Lemma 7.2.300, to define homomorphism (7.2.345). However we can actually perform the proofs of Proposition 7.2.277 and Proposition 7.2.275 simultaneously. Then we can fix a choice of C([0, 1]s ×L1 , (λ0 , λ1 )) so that (7.2.345) is defined at the stage where we prove Lemma 7.2.300. So for simplicity we assume we use the same C([0, 1]s × L1 , (λ0 , λ1 )) for both. Lemma 7.2.347. There exist a filtered An ,K  algebra C1 ([0, 1]2 × L1 , (λ0 , λ1 )) which is a model of [0, 1]2 × C(L1 , (λ0 , λ1 )) of An ,K  class, and a filtered An ,K  homomorphism  1 : C(L , λ ) → C1 ([0, 1]2 × L , (λ , λ )) H 0 0 1 0 1 such that  1 = Inclr ◦ fφ Evals=s0 ◦ H

(7.2.348)

and that Diagram 7.2.30 below commutes: f{φ

C(L0 , λ0 )

} s

C  ([0, 1]s × L1 , (λ0 , λ1 )) , ⏐ Evalr=1 ⏐

−−−−→

  H

C(L0 , λ0 )

C1 ([0, 1]2 × L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 (

1 −→

 fid×φ

Incl



C(L0 , λ0 ) −−−−s→ C([0, 1]s × L0 , λ0 ) −−−−→ C([0, 1]s × L1 , (λ0 , λ1 )) Diagram 7.2.30 is constructed in Lemma 7.2.300. C  ([0, 1]s × Here C([0, 1]s × is a model of [0, 1] × C(L1 , (λ0 , λ1 )) constructed geometrically on an appropriate countable subcomplex consisting of adapted singular simplices.

L1 , (λ0 , λ1 ))

L1 , (λ0 , λ1 ))

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

567

Proof. The proof is similar to that of Lemma 7.2.328. The maps in the bottom line of Diagram 7.2.30 is constructed in Lemma 7.2.300. The maps in the 1 top line of Diagram 7.2.30 is constructed in Subsection 7.2.10. To construct H satisfying (7.2.348) we consider the following commutative diagrams: Incl

C(L0 , λ0 ) −−−−s→

C(L0 , λ0 )

 fφ

C(L0 , λ0 )

fid×φ



−−−−→ C([0, 1]s × L1 , (λ0 , λ1 )) ⏐ ⏐ Evals=s0 (



C(L1 , (λ0 , λ1 )) , ⏐ Evalr=0 ⏐

−→

 fφ



Incl

C(L0 , λ0 ) −−−−→ C(L1 , (λ0 , λ1 )) −−−−r→ C([0, 1]r × L1 , (λ0 , λ1 )) Diagram 7.2.31 f{φ

C(L0 , λ0 )

} s

C  ([0, 1]s × L1 , (λ0 , λ1 )) ⏐ ⏐ Evals=s0 (

−−−−→

 fφ

C(L0 , λ0 )



C(L1 , (λ0 , λ1 )) , ⏐ Evalr=1 ⏐

−→

 fφ



Incl

C(L0 , λ0 ) −−−−→ C(L1 , λ1 ) −−−−r→ C([0, 1]r × L1 , (λ0 , λ1 )) Diagram 7.2.32 The commutativity of diagrams 7.2.31 and 7.2.32 show that we can construct  1 ‘on the boundary ∂[0, 1]2 × L ’. (Namely we can C1 ([0, 1]2 × L1 , (λ0 , λ1 )) and H 1 construct appropriate multisection of the moduli spaces there.) We can then extend it to the interior in the same way as the proof of Lemma 7.2.285. Namely we can extend those multisections. We remark that our proof here is geometric. In other words we do not use Proposition 7.2.312, which is purely algebraic. This is because it is hard to find a filtered An,K algebra that plays the role of C0 which appears in the assumption of Proposition 7.2.312. Instead we have used the fact that the space of parameters to perturb the moduli space is contractible.  Lemma 7.2.349. There exist a filtered An ,K  algebra C2 ([0, 1]2 × L1 , (λ0 , λ1 )) which is a model of [0, 1]2 × C(L1 , (λ0 , λ1 )) of An ,K  class, and a filtered An ,K  homomorphism  2 : C(L , λ ) → C2 ([0, 1]2 × L , (λ , λ )) H 0

0

1

such that (7.2.350)

 2 = Inclr ◦ fφ Evals=s0 ◦ H

0

1

568

7.

TRANSVERSALITY

and Diagram 7.2.33 below commutes: fφ



Incl

C(L0 , λ0 ) −−−−→ C(L1 , (λ0 , λ1 )) −−−−s→ C([0, 1]s × L1 , (λ0 , λ1 )) , ⏐ Evalr=1 ⏐   H

C(L0 , λ0 )

C1 ([0, 1]2 × L1 , (λ0 , λ1 )) ⏐ ⏐ Evalr=0 (

2 −→

 f{φ

C(L0 , λ0 )

} s

C  ([0, 1]s × L1 , (λ0 , λ1 ))

−−−−→ Diagram 7.2.33



Here C ([0, 1]s ×

L1 , (λ0 , λ1 ))

is the same as one in Lemma 7.2.347.

The proof is similar to the proof of Lemma 7.2.247 and is omitted. We next consider Diagram 7.2.34 below. Diagram 7.2.34 is homotopy commutative since it commutes exactly on the subcomplex C(L1 , λ1 ) ⊂ C(L1 , (λ0 , λ1 )), by (7.2.301.4). The next lemma specifies the homotopy. We put 3 = C([0, 1]s × L , λ )[0,1]r . C 1 1 2      C3 is a model of [0, 1] × C(L , λ ) of An ,K class. 1

1

C(L1 , (λ0 , λ1 )) ⏐ ⏐ (Incls

j

−−−−→

C(L1 , λ1 ) ⏐ ⏐ (Incls

j[0,1]s

C([0, 1]s × L1 , (λ0 , λ1 )) −−−−→ C([0, 1]s × L1 , λ1 ) Diagram 7.2.34 Lemma 7.2.351. There exists a filtered An ,K  homomorphism 3  3 : C(L , (λ , λ )) → C H 1 0 1 with the following properties: (7.2.352.1) (7.2.352.2) (7.2.352.3)

 3 = j[0,1] ◦ Incls , Evalr=0 ◦ H s  3 = Incls ◦ j, Evalr=1 ◦ H  3 = Inclr ◦ j. Evals=s ◦ H 0

Proof. The proof is purely algebraic. We use Proposition 7.2.312. We put C0 = C(L1 , λ1 ) and h : C(L1 , λ1 ) → C(L1 , (λ0 , λ1 )) = C  is the inclusion. We can  3.  apply Proposition 7.2.312 in an obvious way to obtain H We define (7.2.353)

X = C1 ([0, 1]2 × L , (λ , λ )) ∨r C2 ([0, 1]2 × L , (λ , λ )). C 1 0 1 1 0 1

 4 be a model of [0, 1]2 × C(L , λ ) of An ,K  class which satisfies Let C 1 1 4,s=s = [0, 1]r × C(L , λ ), C 1 1 0

 4,r=r = C = C([0, 1]s × L , λ ), C 1 1 1 0

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

569

that is (7.2.278.1). Here by abuse of notations, we denote by [0, 1]r × C(L1 , λ1 ) any of its model that was constructed in Proposition 7.2.190. By Theorem 4.2.34 there exists X )s s → [0, 1]r × C(L , λ ) id × j : (C 1 1 = 0 for i = 1, 2, such that it commutes with Incl and (7.2.354)

Evalr=r0 ◦ (id × j) = j : C(L1 , (λ0 , λ1 )) → C(L1 , λ1 ).

Lemma 7.2.355. There exists a filtered An ,K  homomorphism  X → C 4 I:C with the following properties: (7.2.356.1) (7.2.356.2) (7.2.356.3)

 I ◦ Incl = Incl,  = j[0,1] ◦ Evalr=r , Evalr=r0 ◦ I 0 s  Evals=s0 ◦ I = (id × j) ◦ Evals=s0 .

The proof is purely algebraic and easy by using Proposition 7.2.312. We define (7.2.357) (7.2.358)

=C  4 ∨r C 3 C  = (  1 ∨r H  3 ◦ fφ ).  2 )) ∨r (H H I ◦ (H

It is easy to check that they have the required properties.



We remark that we have discussed constructions of filtered A∞ algebras, filtered A∞ homomorphisms, filtered A∞ homotopies between them, in detail in this section. Our proofs are written in a way so that their generalizations to the case of filtered A∞ bimodules, filtered A∞ bimodule homomorphisms, and filtered A∞ homotopies between them become straightforward. The generalization to the homotopy unital versions is discussed in the next section. In Section 7.4, we will further generalize to handle the operators p, q, r. 7.2.14. Bifurcation vs cobordism method: an alternative proof. As we mentioned in Section 5.3, Floer’s original argument in [Flo88IV] to prove invariance of Floer homology is by a bifurcation argument. It was replaced in [Flo89I] and [Oh93] by a cobordism argument. The method we used so far to define filtered A∞ homomorphism etc in this book is to use the ‘top’ (time ordered product) moduli space and is closer to the cobordism argument. In this subsection we explain an alternative method which uses ‘twp’ (time-wise-product) moduli space and is based on the bifurcation argument. Let us consider the situation of Section 4.6 or of Subsection 7.2.9, where we constructed a filtered A∞ or An,K homomorphism associated to a symplectic diffeomorphism ψ : (M, L) → (M  , L ). The main idea was to use the ‘top’ moduli space Mmain k+1 ({Jρ,s0 }ρ : β; top(ρ)) and the correspondence via this moduli space to construct a filtered An,K homomorphism.

570

7.

TRANSVERSALITY

An alternative method we will explain below is to use ‘twp’ (time-wise-product) moduli space Mmain k+1 ({Jρ }ρ : β; twp(ρ)), which we define by the following formula.    (7.2.359) Mmain {ρ} × Mmain k+1 ({Jρ }ρ : β; twp(ρ)) = k+1 (M , L ; Jρ , β). ρ∈[0,1]

Here {Jρ }ρ is a family of almost complex structures on M  such J0 = ψ∗ J, J1 = J  . We define an evaluation map (7.2.360)

 k+1 ev = (ev1 , · · · , evk , ev0 ) : Mmain k+1 ({Jρ }ρ : β; twp(ρ)) → ([0, 1] × L )

in an obvious way. We consider a countable set Xλ,λ ([0, 1] × L ) of smooth singular simplices on [0, 1] × L as in Subsection 7.2.10 such that, every element P ∈ Xλ,λ ([0, 1] × L ), have the properties: (7.2.361.1) (7.2.361.2)

ψ∗−1 P|ρ=0 ∈ Xg(λ) ([0, 1] × L), P|ρ=1 ∈ Xg(λ ) ([0, 1] × L ).

We then put  Mmain k+1 ({Jρ }ρ : β; twp(ρ); P)

(7.2.362)

= Mmain k+1 ({Jρ }ρ : β; twp(ρ)) 

(ev1 ,··· ,evk )

× (P1 × · · · × Pk ).



Let C([0, 1] × L ; λ, λ ) be the free Λ0,nov module generated by Xλ,λ ([0, 1] × L ). We then use an appropriate perturbation s of (7.2.362) to obtain a structure of filtered An (λ,λ ),K  (λ,λ ) algebra on C([0, 1] × L ; λ, λ ). Namely we define    s . mk,β (P1 , · · · , Pk ) = (ev0 )∗ Mmain k+1 ({Jρ }ρ : β; twp(ρ); P) (As in other cases we already discussed we take simplicial decomposition of the right hand side to regard it as a singular chain.) Moreover we use (7.2.361.1), (7.2.361.2) to define Evalρ=ρ0 (ρ0 = 0, 1) and the diagram Evalρ=0

Evalρ=1

C(L, λ) ←−−−−− C([0, 1] × L ; λ, λ ) −−−−−→ C(L , λ ). −1

Diagram 7.2.35 to define Evalρ=0 . etc. is similar to the argument used in Subsection

We compose ev0 with ψ The construction of mk,β 7.2.10. So we omit the details. Since Evalρ=0 is a homotopy equivalence (Theorem 4.2.45), we obtain a filtered An (λ,λ ),K  (λ,λ ) homomorphism: (7.2.363)

C(L, λ) → C(L , λ ).

This is an alternative proof of Theorem 4.6.1. Actually to construct a filtered A∞ homomorphism from the system of filtered An (λ,λ ),K  (λ,λ ) homomorphisms, we need to show compatibility of them for different λ, λ . Since we use a family of almost complex structures and others to define An (λ,λ ),K  (λ,λ ) homomorphism (7.2.363), we need a homotopy of homotopies to construct a homotopy between filtered An,K homomorphisms (7.2.363). This can be carried out similarly as in Subsections 7.2.12, 7.2.13. To go one step further, that is to prove Theorem 4.6.25, we need to study a homotopy of homotopies of homotopies of filtered An,K structures. This can be carried out in a similar way as in Subsections 7.2.12, 7.2.13.

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

571

We have thus derived an An,K homomorphism (7.2.363), its A∞ version and homotopies between them by a bifurcation argument. There are following three disadvantages in the bifurcation argument compared to the cobordism argument: (7.2.364.1) As we mentioned above we need one more iteration of homotopies. (Namely in cobordism argument we only need up to homotopy of homotopies. In bifurcation argument we need homotopy of homotopies of homotopies.) So it is a bit more cumbersome to write down the details of the proof. (Although we already have enough machinery at hand to do so.) (7.2.364.2) There is some difficulty of constructing a filtered A∞ bimodule homomorphism in the situation of Subsection 5.3.2, using the bifurcation argument alone. We will elaborate this point later in this subsection. (7.2.364.3) The authors are unable to use the bifurcation argument in the study in the proof of Theorem X given in Section 7.5 at the time of writing of this book. On the other hand, the cobordism argument has following disadvantages compared to the bifurcation argument: (7.2.365.1) To study operators q and r (which are introduced in Section 3.8), we can obtain a better result by the bifurcation argument. We will explain this point in Section 7.4. (7.2.365.2) For the bifurcation argument, we only need the moduli spaces of the ‘twp’ type. So we never need to use the ‘top’ moduli space. By this reason some parts of the argument are simpler if we use the bifurcation argument. (7.2.365.3) We do not need to work out the geometric construction of Incl which we gave in Subsection 7.3.10. Since the geometric construction of Incl is rather delicate it would be desirable to avoid it. All the differences mentioned above are rather technical and it is very likely that anything we can prove by one of the two methods can also be proved by the other method, if we work a bit harder. ((7.2.364.2) seems to be the most serious point among them.) However at the time of writing this book the authors need to use both methods to get as best results as possible, because of the points (7.2.364.2), (7.2.364.3) and (7.2.365.1). Also since the whole construction is rather delicate it is preferable that we have several different proofs. We next prove that the A∞ homomorphisms obtained by the two methods are homotopic to each other. Let us consider the situation of Theorem 4.6.1 and use the same notations there. Proposition 7.2.366. The filtered A∞ homomorphism ψ∗ : C(L; Λ0,nov ) → C(L ; Λ0,nov ) obtained in Subsections 7.2.9 and 7.2.11 is homotopic to the one obtained in this subsection. Proof. We prove the An,K -version of the proposition only. We first take a trivial family ≡ J to obtain the following diagram. Evalρ=0

Evalρ=1

C(L, λ) ←−−−−− C([0, 1] × L; λ) −−−−−→ C(L, λ). Diagram 7.2.36 We next consider the two parameter family of almost complex structures {Jσ,ρ }0≤σ≤1,0≤ρ≤1 = {Jσ,ρ }σ,ρ

572

7.

TRANSVERSALITY

such that J0,ρ = ψ∗ J, J1,ρ = Jρ , Jσ,0 = ψ∗ J. We then use the moduli space Mmain k+1 ({Jσ,ρ }σ,ρ : β; top(σ), twp(ρ)) defined in Subsection 7.2.10 to obtain the vertical arrows of the following diagram, which commutes up to homotopy: Evalρ=0

C(L, λ) ←−−−−− ⏐ ⏐ (

Evalρ=1

C([0, 1] × L; λ) ⏐ ⏐ (

−−−−−→ C(L, λ) ⏐ ⏐ (

Evalρ=0

Evalρ=1

C(L, λ) ←−−−−− C([0, 1] × L ; λ, λ ) −−−−−→ C(L , λ ) Diagram 7.2.37 By construction, the third column is the homomorphism ψ∗ obtained in Subsections 7.2.9 and 7.2.11. The second line induces the homomorphism ψ∗ obtained in this subsection. The first column and the first line induce identity. Hence the proposition.  We next elaborate (7.2.364.2). Consider the situation of Subsection 5.3.2 and use the notation introduced at the beginning of Subsection 5.3.2. We first explain a trouble to realize a naive idea to use the bifurcation argument to prove Theorem 5.3.14. We put    Rpara = {ρ} × ψρ(1) (L(1) ) ∩ ψρ(0) (L(0) ) . ρ∈[0,1]

Note ∂Rpara ∼ = (L(1) ∩ L(0) ) − (L(1) ∩ L(0) ). We have a diagram  Evalρ=0 Evalρ=1   h C(Rh ; Q) ←−−−−− C(Rpara ; Q) −−−−−→ h C(Rh ; Q). Diagram 7.2.38 We might try to add ‘quantum effect’ to Diagram 7.2.38, to obtain Evalρ=0

Evalρ=1

C(L(1) ∩L(0) ;Λ0,nov ) ←−−−− C(Rpara ;Λ0,nov ) −−−−→ C(L(1) ∩L(0) ;Λ0,nov ) Diagram 7.2.39 and use this to prove Theorem 5.3.14. However, there are two troubles to do so: (7.2.367.1) Two arrows in Diagram 7.2.38 do not induce isomorphisms on homology. Therefore, we are unable to use Theorem 5.2.35 directly, even if we obtain Diagram 7.2.39. (7.2.367.2) To define an effect of pseudo-holomorphic strip to the bimodule structure of C(Rpara ; Λ0,nov ), we need to consider the union (over ρ) of the moduli spaces (1) (0) of pseudo-holomorphic strips connecting p, q ∈ ψρ (L(1) ) ∩ ψρ (L(0) ). However (1) (0) (1) there can be some ρ for which ψρ (L ) is not transversal to ψρ (L(0) ). In other words, the intersection sets bifurcate at some ρ. In such a case, the moduli space of pseudo-holomorphic strips is harder to study. We remark that Floer [Flo88IV] already studied the problem (7.2.367.2) and a more detailed treatment of the necessary analytic details are given in [Lee01]. Nevertheless this point has not been understood up to the level needed for our purpose, since the case of a families of Hamiltonian diffeomorphisms parameterized by two or more variables has not been not worked out yet. Such a family naturally arises in the study of homotopy of homotopies and etc.

7.2. MULTISECTIONS AND CHOICE OF COUNTABLE SET OF CHAINS

573

So far, the authors do not know how to handle (7.2.367.1). We believe that the study of the above mentioned problems is interesting in its own and will be useful for future applications to the Floer theory. However we do not discuss these points any further in this book. Instead, we will explain below a method that is a mixture of the bifurcation argument and the cobordism argument. The weakly filtered A∞ bimodule homomorphisms we obtain in this way is the one over a pair of filtered A∞ homomorphisms defined by the bifurcation method in this subsection. We use the notation of Subsection 5.3.2. Let {Jτ,t }τ,t be the two parameter family of almost complex structures introduced there. We put Jρ,t = Jχ−1 (ρ),t , where χ is as in (5.3.2). Using {Jρ,i }ρ i = 0, 1 we define Evalρ=0

Evalρ=1

C(L(i) ; Λ0,nov ) ←−−−−− C([0, 1]ρ × L(i) ; Λ0,nov ) −−−−−→ C(L(i) ; Λ0,nov ) Diagram 7.2.40 for i = 0, 1. (Actually we first need to prove an An,K -version and proceed same way as in the earlier subsections to go from the An,K to the A∞ -version. We omit this process since we did it already many times.) Using the homomorphisms Evalρ=i (i = 0, 1), we regard C(L(1) , L(0) ; Λ0,nov ) and C(L(1) , L(0) ; Λ0,nov ) as filtered C([0, 1]ρ ×L(0) ; Λ0,nov ) - C([0, 1]ρ ×L(1) ; Λ0,nov ) A∞ bimodules. We will construct a weakly filtered A∞ bimodule homomorphism C(L(1) ∩ L(0) ; Λ0,nov ) → C(L(1) ∩ L(0) ; Λ0,nov )

(7.2.368)

over C([0, 1]ρ ×L(0) ; Λ0,nov ) - C([0, 1]ρ × L(1) ; Λ0,nov ). To construct (7.2.368) we use a moduli space which is slightly different from the one used in Subsection 5.3.4. We describe it below. Let us first consider the moduli space (7.2.369)

◦

Mk1 ,k0 ((L(1) , ψ (1) ), (L(0) , ψ (0) ); {Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ))

which was introduced right after Remark 5.3.22. We compactify this to (7.2.370)

Mk1 ,k0 ({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ )) twp(τ )

as follows. (We remark that the moduli space (7.2.371)

Mk1 ,k0 ((L(1) , ψ (1) ), (L(0) , ψ (0) ); {Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ))

which we used in Subsection 5.3.4 is not a compactification of (7.2.369) since in general (7.2.369) is not dense therein.) Definition 7.2.372. The space (7.2.370) is the elements in (7.2.371) satisfying the condition χ−1 (ρa ) = τa for all a. Let us explain our compactification (7.2.370). An element of (7.2.371) is regarded as a pseudo-holomorphic strip contained in (7.2.370) (with different ki and w possibly) together with disc bubbles and the components which slide to τ → ±∞. We remark that each component Σa of a disc bubble is associated with a time allocation ρa . It is assumed that χ−1 (ρa ) is not greater than the R-component τa of the point (τa , i) in R × {0, 1}, where the bubbled disc Σa is rooted. The necessary condition for this element to be in (7.2.370) is that χ−1 (ρa ) = τa for all a.

574

7.

TRANSVERSALITY

Using the evaluation maps described in Subsection 5.3.2, we obtain an evaluation map ◦

ev : Mk1 ,k0 ({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ))

(7.2.373)

→ ([0, 1]k1 × (L(0) )k1 ) × ([0, 1]k0 × (L(0) )k0 ).

Note in Subsection 5.3.2, evtime,(i),j (i = 0, 1) is defined as a map to R. We compose it with χ−1 . They are the [0, 1] factors of the above evaluation map. We can extend ev to the moduli space (7.2.370). Here we take [0, 1] component of the ev to be 0 or 1 if the corresponding marked point is on the component which slides to τ → ±∞, respectively. Note the inverse image by ev of ∂(([0, 1]k1 × (L(0) )k1 ) × ([0, 1]k0 × (L(0) )k0 ))

(7.2.374)

lies in the codimension one boundary of our moduli space (7.2.370). So we add a collar type neighborhood to our moduli space and extend the map (7.2.173) to twp(τ )+

ev : Mk1 ,k0

(7.2.375)

({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ))

−→ ([0, 1]k1 × (L(0) )k1 ) × ([0, 1]k0 × (L(0) )k0 ).

Here + in (7.2.375) means that we extend our moduli space (7.2.370) by adding collars. We may assume that the map ev is of a product type on this collar. There also exists an evaluation map twp(τ )+

(7.2.376) (ev−∞ , ev+∞ ) : Mk1 ,k0

({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ )) → Rh × Rh .

(i)

Now let Pj (i = 0, 1) be chains in [0, 1]ρ × L(i) and S a chain in Rh . We then take a fiber product using the map (7.2.375) and the map ev−∞ in (7.2.376) to obtain (7.2.377)

twp(τ )+

Mk1 ,k0

 (1) , S, P  (0) ). ({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ); P (i)

We remark that in (5.3.24) we use the chains Pj on L(i) to take fiber product, where we are using chains on [0, 1]ρ × L(i) here. Now after taking an appropriate perturbation s, the map  (1) ⊗ S ⊗ P  (0) P   twp(τ )+  (1) , S, P  (0) )s −→ (ev+∞ )∗ Mk1 ,k0 ({Jτ,t }τ,t ; [h, w], [h , w ]; top(τ ); P defines an A∞ bimodule homomorphism (7.2.368). We can use the moduli space (7.2.377) to prove Theorem 5.3.14 instead of using the moduli space (5.3.24) used in Subsection 5.3.4. This is an alternative proof of Theorem 5.3.14. The proof of other theorems in Section 5.3 can be modified in an appropriate way using this alternative homomorphism. 7.3. Construction of homotopy unit 7.3.1. Statement of the result and idea of its proof. The purpose of this section is to show the following:

7.3. CONSTRUCTION OF HOMOTOPY UNIT

575

Theorem 7.3.1. Let m be the operators we defined using the fundamental chains constructed in Section 7.2 and X0 (L) the countable set of singular simplices constructed there. Then we find a countable set of singular simplices X (L) ⊃ X0 (L) and extend m there so that it defines a filtered A∞ algebra and e = P D([L]), the Poincar´e dual of the fundamental cycle of L, is a homotopy unit. (We remark that X0 (L) in Theorem 7.3.1 is different from X0 (L) in Subsection 7.2.4. In other words, we forget the generation in Subsection 7.2.4.) Before starting the proof, we explain the main idea of the proof and point out a difficulty to directly carry it out. For this purpose, we consider the following types of forgetful maps: Let Mk+1 (β) be the moduli space of pseudo-holomorphic discs with k + 1 marked points on the boundary and of homotopy class β. We regard it as a space with Kuranishi structure. For a = (a1 , · · · , a| a| ) ∈ {1, · · · , k}| a| with a1 < · · · < a| a| , let main forget a : Mmain k+1 (β) −→ Mk+1−| a| (β)

be the map forgetting the a1 · · · , a| a| -th marked points. By construction, the Kuranishi structure will be compatible with the forgetful maps. (See Lemma 7.3.8 for the precise meaning of the compatibility.) We say a smooth singular simplex P = (|P |, f ) of L is a degenerate simplex if there exists a smooth singular simplex P  = (|P  |, f  ) with dim |P  | = dim |P | − 1 and a simplicial map π : |P | → |P  | such that f = f  ◦ π. A linear combination of degenerate simplex is called a degenerate chain. Lemma 7.3.2. If we have a family of multisections sk+1,β of Mmain k+1 (β) which are compatible with the forgetful map forget a and if we have a countable set of singular simplices X (L) which has all the required transversality conditions, then we have  ≡0 (k, β) = (1, β0 ) (7.3.3) mk+1,β (P1 , · · · , Pi , L, Pi+1 , · · · , Pk ) ≡ ±P1 (k, β) = (1, β0 ) where ≡ means the equality modulo degenerate chains. Remark 7.3.4. (1) Lemma 7.3.2 means that P D([L]) is a unit modulo the subcomplex of degenerate chains of our filtered A∞ algebra. (2) As matter of fact, this lemma is a hypothetical lemma in that it is hard to construct the structure that satisfies the hypotheses of Lemma 7.3.2. Proof of Lemma 7.3.2. We put P + = (P1 , · · · , Pi , L, Pi+1 , · · · , Pk ),

P = (P1 , · · · , Pi , Pi+1 , · · · , Pk )

and use the moduli spaces main + Mmain k+2 (β; P ) = Mk+2 (β) ×Lk+1 (P1 × · · · × Pi × L × Pi+1 × · · · × Pk ) Mmain (β; P ) = Mmain (β) ×Lk (P1 × · · · × Pi × Pi+1 × · · · × Pk ). k+1

k+1

main +  As a set, we have ev0 (Mmain k+2 (β; P )) = ev0 (Mk+1 (β; P )). On the other hand, we have the following for (k, β) = (1, β0 ):

(7.3.5)

main +  dim Mmain k+2 (β; P ) = dim Mk+1 (β; P ) + 1.

576

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Here dim is the ‘virtual dimension’ i.e., the dimension in the sense of the Ku+ ranishi structure. The extra dimension of Mmain k+2 (β; P ) corresponds to the parameter parameterizing the location of the (i + 1)-th marked point in ∂Σ. Under the assumption of Lemma 7.3.2, we can find a perturbation (multisection) main +  sk+2,β,P + of Mmain k+2 (β; P ) which is compatible to one of Mk+1 (β; P ). (We omit the suffix parameterizing the decoration in sk+2,β,P + , since it is automatically determined by P + .) Then (7.3.5) implies that we may choose a triangulation of  + )sk+2,β,P + so that its virtual fundamental chain is a degenerate chain. Mk+2 (β; P Lemma 7.3.2 follows.  As we already mentioned, it seems difficult to find the perturbations of the moduli space Mmain k+1 (β) satisfying the assumption of Lemma 7.3.2 because of the transversality problem we have been studying in the last section. This is the reason why we construct a homotopy unit instead of the strict unit. We also remark that we do not use the notion of degenerate chain in the proof of Theorem 7.3.1. This is because it is not easy to construct the product structure mk such that products among degenerate chains again become degenerate. By a more careful argument at the end of this section, we will replace the ‘equality modulo degenerate chains’ in Condition 7.3.14 by the strict equality as a singular chain. To overcome this difficulty we proceed as follows. We consider two (different) multisections on the moduli space such that one is compatible with the forgetful map and the other is one we used in Section 7.2. We then take a homotopy between the two multisections. Using the homotopy we obtain the operators hk required in Definition 3.3.2. In fact, we need a [0, 1]k parameter family (instead of a homotopy) of multisections. 7.3.2. Proof of Theorem 7.3.1. To carry out the plan described at the end of the last subsection, we need to repeat some parts of argument developed in Section 7.2 again. Namely we need to again enlarge the countable set of singular chains and also need to overcome the trouble explained in Subsection 7.2.3. However they do not raise any new problem. To handle the first point, we are going to construct a sequence of Xg (L) inductively over g ∈ Z≥0 . (We remark that X0 (L) we start with is already constructed by a similar induction in Section 3.2. Namely X0 (L) = ∪g X0,g (L). Here X0,g (L) is the set of singular simplices of the g-th generation in the sense of Subsection 7.2.4. But in this section, we regard all the elements of X0 (L) being of the zero-th generation.) To handle the second point, we will prove that e is a homotopy unit with respect to the An,K structure. (Here (n, K) = (n(λ), K(λ)) → (∞, ∞) as λ → ∞.) Then the rest of the proof is similar to the argument of Subsection 7.2.8. (Actually we need to define the notion of homotopy unit of the An,K structure and to establish a generalization of Theorem 7.2.72 including homotopy unit. This work is similar to the discussion of Chapter 4 and so omitted.) Now we start the proof of Theorem 7.3.1 by defining an appropriate moduli space. Take a smooth simplicial decomposition of L and put

(7.3.6) [L] = ±Δnc where n is the dimension of L and right hand side is the sum of n dimensional simplicies of our simplicial decomposition. The sign ± in (7.3.6) is determined as

7.3. CONSTRUCTION OF HOMOTOPY UNIT

577

follows: We first fix an order of the set of vertices of our triangulation of L. We use it to identify each of the simplices Δnc to the standard n simplex. It induces an orientation to Δnc . The sign is + if and only if this orientation coincides with the orientation of Δnc induced by one on L. We may assume that all the simplices in our simplicial decomposition of L is contained in X0 (L). For a = (a1 , · · · , a| a| ) ∈ {1, · · · , k}| a| with a1 < · · · < a| a| , let main forget a : Mmain k+1 (β) −→ Mk+1−| a| (β)

(7.3.7.1)

be the map forgetting the a1 · · · , a| a| -th marked points. Let P = (P1 , · · · , Pk−| a| ) with Pi ∈ Xd(i) (L). Let P + be a k-tuple of chains obtained by inserting L into the ai -th spot of P for each of i = 1, · · · , |a|. (If a = (1, 3) and k = 4, for example, then P + = (L, P1 , L, P2 ), P = (P1 , P2 ).) Then (7.3.7.1) induces a map main +  forget a : Mmain k+1 (β; P ) −→ Mk+1−| a| (β; P ).

(7.3.7.2)

We have the following compatibility of Kuranishi structures. Let x be an  element of Mmain k+1−| a| (β; P ) and let (Vx , Ex , Γx , ψx , sx ) be a Kuranishi neighborhood of x in Mmain (β; P ). Namely Vx is a manifold on which a finite group Γx acts, k+1−| a|

Ex is a Γx equivariant bundle on Vx , sx is a Γx equivariant section of Ex , and ψx is main  a homeomorphism from s−1 x (0)/Γx to a neighborhood of x in Mk+1−| a| (β; P ). We may assume without loss of generality that 0 ∈ Vx , sx (0) = 0, and ψx ([0]) = x. Let + + + + + + + x+ ∈ Mmain a (x ) = x. Let (Vx+ , Ex+ , Γx+ , ψx+ , sx+ ) be a k+1 (β; P ) such that forget Kuranishi neighborhood of x+ . + + Lemma 7.3.8. We can take (Vx++ , Ex++ , Γ+ x+ , ψx+ , sx+ ) with the following properties:

(7.3.9.1) Vx++ ∼ = (Wx × Wx + ) × Vx where Wx ⊆ R| a| is an open set containing 0 + + + + and Wx+ is an open set of Rc(x ) containing 0. s+ x+ (0) = 0 and ψx+ ([0]) = x . + + ∼ Here c(x ) is a nonnegative integer depending on x and = denotes the piecewise smooth isomorphism.  ∼ (7.3.9.2) Γ+ x+ = Γx . It acts trivially on the first factor Wx × Wx+ . Moreover the + action of Γx+ to the second factor Vx coincides with the action of Γx . (7.3.9.3) The evaluation maps ev0+ : Vx++ → L, ev0 : Vx → L satisfy ev0+ = ev ◦ π where π : Vx++ ∼ = Wx × Vx → Vx is the projection to the second factor. + + (7.3.9.4) Ex+ |(Wx ×{0})×Vx ∼ = π ∗ Ex ⊕ Rc(x ) . (7.3.9.5) On the subspace (Wx × {0}) × Vx , the π ∗ Ex component of the Kuranishi map s+ x+ is the pull-back of sx . + (7.3.9.6) We consider the Rc(x ) -component of s+ x+ . Then, on the subspace (Wx × {0}) × Vx , the derivation of it with respect to Wx + direction coincides with the + + natural embedding T0 (Wx + ) → T0 (Rc(x ) ) = Rc(x ) . Proof. Represent x+ = (Σ+ , z+ , w+ ). Here Σ+ is a genus zero bordered semistable curve, z+ boundary marked points and w+ : (Σ+ , ∂Σ+ ) → (M, L) a pseudo-holomorphic map for a given almost complex structure on M . We denote by (Σ , z , w ) the triple obtained by forgetting |a| marked points. Here Σ may not be stable. Then x is represented by (Σ, z, w) where Σ is obtained by contracting the components of Σ corresponding to the unstable components of (Σ , z , w ).

578

7.

TRANSVERSALITY

When Σ = Σ, i.e., when (Σ , z , w ) is already stable, Lemma 7.3.8 is in fact obvious from the construction. And in this case, we have c(x+ ) = 0. (We remark that while we construct Kuranishi structures, we do not break various symmetries.) Next we consider the case Σ = Σ. Let Σ = ∪i∈I Σi be the decomposition of Σ into the irreducible components. Denote by I0 the subset of i ∈ I for which Σi is unstable. According to the general construction of Kuranishi structures in Section 7.1, Ex++ is a direct sum of some finite dimensional spaces Ei of smooth sections of (w+ )∗ T M ⊗ Λ(0,1) Σ+ such that all elements of Ei are supported on a compact subset of Σi respectively for i ∈ I. In the current case, we can choose our Kuranishi structures so that we have a natural identification  Ei ∼ = Ex i∈I\I0  via the contraction (Σ , z , w ) to (Σ, z, w). Now we put c(x+ ) = i∈I0 dim Ei and define Wx to be a small neighborhood of  0 in i∈I0 Ei . Recalling that the Kuranishi maps s+ x+ , sx are obtained linearizing ∂ followed by projecting to Ei ’s, (7.3.9) can be easily checked: For example, (7.3.9.6) follows by the fact that all the maps on the components Σi with i ∈ I0 are constant maps. 

Let sx be a multisection which is a perturbation of sx . We now explain how it + + + induces a multisection of (Vx++ , Ex++ , Γ+ x+ , ψx+ , sx+ ). We pull sx to Vx+ and obtain ∗ the π Ex -component of a multivalued perturbation of the Kuranishi map s+ x+ of + + + + + c(x+ ) (Vx+ , Ex+ , Γx+ , ψx+ , sx+ ) on (Wx × {0}) × Vx . (See (7.3.9.5).) We put R component to be zero on (Wx × {0}) × Vx . We then extend our multisection so that + + (7.3.9.6) is satisfied. This way we obtain a multisection of (Vx++ , Ex++ , Γ+ x+ , ψx+ , sx+ ), which we call the pull-back of sx . Now we are ready to state the main technical result to be used in the proof of Theorem 7.3.1. We use the same notation as in Subsection 7.2.4. For Pi ∈ Xgi (L) we put d(i) = gi . If P + = (P1+ , · · · , Pk+ ) with Pi+ ∈ Xg+ (L), we put d+ (i) = gi+ . # i (In case Pj+ = L we have d+ (j) = 0.) We put X(g) = g ≤g Xg (L) as usual. We note that L is not a singular simplex. So strictly speaking, we need to use simplicial decomposition of L: For example, we should write

P + = ±(Δnc , P1 , Δnc , P2 , Δnc , P3 ) c,c ,c

for (L, P1 , L, P2 , L, P3 ) for k = 6 and a1 = 1, a2 = 3, a3 = 5. Here Δc is as in (7.3.6). However by some abuse of notation, we will just write it simply as (L, P1 , L, P2 , L, P3 ) for the simplicity of notations. Proposition 7.3.10. For any δ > 0 and K > 0, there exist a countable set  of chains Xg (L) of L for g = 1, · · · , K and multisections sd,k,β,P of Mmain k+1 (β; P ) | a| with Pi ∈ X(g) (L) for (d, β) ≤ K, and multisections s + × + of [0, 1] a,d+ ,k,β,P main + +  M (β; P ) with Pi ∈ X(g) (L) for (d , β) ≤ K with the following properties: k+1

(7.3.11.1)

X(g) (L) has Properties 7.2.27.

7.3. CONSTRUCTION OF HOMOTOPY UNIT

(7.3.11.2)

579

If g = (d; β), then the Q chain    sd,k,β,P ev0∗ Mmain,d k+1 (β; P )

is decomposed into elements of X(g) (L).  sd,k,β,P is in a δ-neighborhood of the original moduli space (7.3.11.3) Mmain,d k+1 (β; P )  Mmain,d k+1 (β; P ). (7.3.11.4) The multisections sd,k,β,P are compatible to one another in the sense of Compatibility Conditions 7.2.38 and 7.2.44 below. (7.3.11.5) If g = (d+ ; β), then the Q chain   + +  + ))sa,d+ ,k,β,P + ([0, 1]| a| × Mmain,d (β; P ev0∗ k+1 is decomposed into elements of X(g) (L). (7.3.11.6) For each fixed (t1 , · · · , t| a| ) ∈ [0, 1]| a| , the restrictions of the multisecmain,d+  + ) are compatible in the sense (β; P tions s + a| )} × Mk+1 + to {(t1 , · · · , t| a,d+ ,k,β,P of Compatibility Conditions 7.2.38 and 7.2.44. (7.3.11.7) Compatibility Conditions 7.3.13 and 7.3.14 below hold. (7.3.11.8) Compatibility Condition 7.3.17 below holds. + s+ (7.3.11.9) For each t ∈ [0, 1]| a| , the set ({t} × Mmain,d (β; P + )) a,d+ ,k,β,P + is in a +

k+1

(β; P + ) in a Kuranishi neighborhood. δ-close to the original moduli space Mmain,d k+1 The general idea to formulate Compatibility Conditions 7.3.13, 7.3.14, 7.3.17 is as follows: If ti goes to 1, then the multisection becomes a pull back with respect to the forgetful map. If ti goes to 0, then the multisection turns out to coincide with the one obtained by just inserting L into the i-th place. To describe Compatibility Condition 7.3.13, we prepare some notations. We split a = (a(1), · · · , a(|a|)) ∈ {1, · · · , k}| a| into a disjoint union a = a1 ! a2 and we remove L’s at the a1 (1), · · · , a1 (|a1 |)-th positions from P + and get P  . We define a 2  by a 2  (j) = a2 (j) − #{i | a1 (i) < a2 (j)}. We note that a 2  (j)-th chain of P  are L. We also define 2 I a11 , a2 : [0, 1]| a | → [0, 1]| a| by I a11 , a2 (t1 , · · · , t| a2 | ) = (s1 , · · · , s| a| ) where  1 if a(i) ∈ a1 , si = if a(i) ∈ a2 (j). tj Example 7.3.12. If a = {1, 3, 5}, a1 = {1, 5}, a2 = {3}, k = 6, then P = (P1 , P2 , P3 ), a 2  = {2}, P  = (P1 , L, P2 , P3 ), P + = (L, P1 , L, P2 , L, P3 ), I a11 , a2 (t) = (1, t, 1). We define a forgetful map main +  forget a1 , a 2 : Mmain k+1 (β; P ) −→ Mk+1−| a1 | (β; P )

580

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TRANSVERSALITY

in the same way as (7.3.7). It is compatible with Kuranishi structures in the same way as Lemma 7.3.8. Let d be the decoration corresponding to P  . We remark | a 2 |  × Mmain that s +  is a multisection of [0, 1] k+1−| a1 | (β; P ). a 2  ,d ,k,β,P  Compatibility Condition 7.3.13. If Mmain k+1−| a1 | (β; P ) is nonempty, then 1 the restriction of s + + to the image of I a1 , a2 × id is equal to the pullback of a,d+ ,k,β,P s + a1 , a2 .  by the map forget a 2  ,d ,k,β,P

 When the moduli space Mmain k+1−| a1 | (β; P ) is empty, we modify Compatibility Condition 7.3.13 as follows. Compatibility Condition 7.3.14. Let k = 2 and β = β0 = 0, and a = a1 be either {1} or {2}. Namely P = {P }, P  = ∅ and P + = (L, P ) or (P, L). In this case,   s + =0  + + main a,d ,2,β ,P + 0

{1}×M3

(β0 ;P )

and restriction of the evaluation map ev0 : [0, 1] × Mmain (β0 ; P + ) → L gives 3   + −1 main  + ) = P. ev0∗ (s+ + ) (0) ∩ ({1} × M (β ; P 0 3 + a,d ,2,β0 ,P

 For all the other cases where Mmain k+1−| a1 | (β; P ) is empty, we require that   + −1 main +  ev0∗ (s + ) (0) ∩ ({1} × M (β ; P )) 0 k+1 + a,d+ ,k,β ,P 0

is a degenerate chain. Remark 7.3.15. Let us describe + ev0∗ ((s a,d+ ,2,β0 ,P + )−1 (0) ∩ ({0} × Mmain (β0 ; P + )) 3

in the situation of Compatibility Condition 7.3.14. If m2,β0 is defined by using two diffeomorphisms ϕ0 and ϕ1 , then m2,β0 (P, L) = ϕ0 (P ) and m2,β0 (L, P ) = ϕ1 (P ). Hence Compatibility Condition 7.3.17 which we will discuss later implies   + −1 main  + )) = ϕi (P ). ev0∗ (s + ) (0) ∩ ({i} × M (β ; P 0 3 + + a,d ,2,β ,P 0

Thus Compatibility Condition 7.3.14 means that +

s + (Mmain (β0 ; P + ) a,d+ ,2,β0 ,P + ) P → ev0∗ 3

is a “chain homotopy” from id to (ϕi )∗ . We next describe Compatibility Condition 7.3.17. We split a into a disjoint union a1 ! a2 as before, and define 2

I a01 , a2 : [0, 1]| a | −→ [0, 1]| a| by I a01 , a2 (t1 , · · · , t| a2 | ) = (s1 , · · · , s| a| ) where  0 if a(i) ∈ a1 , si = if a(i) ∈ a2 (j). tj

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581

Example 7.3.16. If a = {1, 3, 5}, a1 = {1, 5}, a2 = {3}, k = 6, then P = (P1 , P2 , P3 ), P + = (L, P1 , L, P2 , L, P3 ), I a01 , a2 (t) = (0, t, 0). Compatibility Condition 7.3.17. The restriction of s + + to the ima,d+ ,k,β,P

age of I a01 , a2 × id is equal to s + +. a2 ,d+ ,k,β,P

In case a2 = ∅ (a = a1 ), the restriction of the multisection s + + to a,d+ ,k,β,P +

 + ) is s + (β, P {0}| a| × Mmain,d +. k+1 d ,k,β,P Proof of Proposition 7.3.10. To prove Proposition 7.3.10 we construct + sd,k,β,P and s + = (d+ , β). (We + by induction on g = (d, β) and g a,d+ ,k,β,P remark that we can use Lemma 7.2.47 to show that this induction is possible.) The construction of sd,k,β,P is similar to the argument in Subsection 7.2.5. Let us describe the construction of s + + . Compatibility Conditions 7.3.13, 7.3.14 a,d+ ,k,β,P and 7.3.17 determine the restriction of s+ on ∂([0, 1]| a| ) × Mmain (β; P + ) + a,d+ ,k,β,P

k+1

inductively. On the other hand, (7.3.11.6) determines the restrictions of s + + a,d+ ,k,β,P | a| main +  to [0, 1] × ∂(Mk+1 (β; P )), inductively. Therefore, we can extend it by Theorem A1.23.  Now, we are ready to define hk+1 . Let P i ∈ Bji C[1], P i = Pi,1 ⊗ · · · ⊗ Pi,ji (i = 1, · · · ,  + 1) and put j1 + · · · + j+1 = m with  ≥ 0. We put a = (j1 + 1, j1 + j2 + 2, · · · , j1 + · · · + j + ),  P = (P1,1 , · · · , P1,j1 , P2,1 , · · · , P,j , P+1,1 , · · · , P+1,j+1 ).  Here |a| =  and k =  + +1 i=1 ji . Definition 7.3.18. For  ≥ 1 we define (7.3.19)

h+1,β (P 1 ⊗ · · · ⊗P +1 )  sa,d+ ,k,β,P +  main,d+ | a| +  . = ev0∗ [0, 1] × Mk+1 (β; P )

See Section 8.10 for the orientation of the moduli space in the right hand side. We also define h1,β = mβ . We actually need to modify h+1,β slightly. We will discuss it later in this section. Before checking the required property (3.3.10), let us exhibit the definition by describing it in a special case. Let us consider the case h2,β0 . The moduli space we (β0 ). We recall that Mmain (β0 ) = L as a set. However use to define h2,β0 is Mmain 3 3 the evaluation map is not the diagonal embedding but is −1 (ev1 , ev2 , ev0 )(x) = (ϕ−1 0 (x), ϕ1 (x), x),

where ϕ0 , ϕ1 : L → L are generic diffeomorphisms close to the identity map. Let ϕti : L → L be a small isotopy such that ϕ1i (x) = x and ϕ0i (x) = ϕi (x) for each i = 0, 1. We define Mmain (β0 )+ = [0, 1] × L together with the evaluation map 3 (ev1 , ev2 , ev0 )(t, x) = ((ϕt0 )−1 (x), (ϕt1 )−1 (x), x).

582

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Then if P + = (P, L), k = 2 and a = {2}, we have  sa,d+ ,k,β,P + + [0, 1]| a| × Mmain,d (β; P + ) = Mmain (β0 )+ ×L2 (P × L). 3 3 (See Remark 7.3.15.) Therefore    (β0 )+ ×L2 (P × L) = ϕt0 (P ). h2,β0 (P ⊗ 1) = ev0∗ [0, 1] × Mmain 3 t∈[0,1]

We have (7.3.20)

δ(h2,β0 (P ⊗ 1)) = P − ϕ0 (P ) = P − m2,β0 (P ⊗ [L])

which is a part of the required formula (3.3.10) for h. We now check Formulas (3.3.10). The formula h1 = m is immediate from Definition 7.3.18. Let (3.3.10.1), We consider the boundary of the  us verify properties sa,d+(3.3.10.2). + ,k,β,P main,d+ | a| +  chain [0, 1] × Mk+1 (β; P ) . Let (ti , xi ) be a divergent sequence   s + + +  + ) a,d ,k,β,P . Then one of the following of elements of [0, 1]| a| × Mmain,d (β; P k+1 3 cases occurs: Let ti = (ti1 , · · · , ti ). Case 1 Case 2

tij goes to 1. tij goes to 0.

Case 3

 + ). xi goes to ∂Mmain,d (β; P k+1

+

Case 2 gives −hk ◦ bk+1,i by Compatibility Condition 7.3.17. (See the definition of bk+1,i in Subsection 3.3.1.) In a way similar to the argument of the proof of A∞ formula (the proof of Theorem 3.5.11 in Chapter 3), we can show that Case 3 together with the term (−1)n ∂(h+1,β (P 1 ⊗ · · · ⊗P +1 )) k+1  after summed over β ∈ Π(M ; L), gives (Recall that ∂ = =1 h ◦ hk−+2 . n (−1) m1,β0 .) Finally we examine Case 1 which requires the most careful study among the three. For the case P = 0, it follows from Compatibility Conditions 7.3.13, 7.3.14 and the argument used in the proof of Lemma 7.3.2 that the boundary arising from Case 1 is a degenerate chain. However we need to modify the definition of h given in (7.3.19) appropriately so that the boundary term arising from Case 1 becomes exactly 0 as a singular chain. We will come back to this point shortly. For the case P = 0, we have P 1 ⊗ · · · ⊗P +1 = P ⊗1 or 1⊗P . Hence by Condition 7.3.14, the boundary corresponding to Case 1 gives −P = P2 (P ⊗1) or P = P2 (1⊗P ). We have thus finished the proof the identities (3.3.10). Then using the An,K analog of Proposition 3.3.8, we find that P D([L]) is a homotopy unit of the An,K algebra. The rest of the proof of Theorem 7.3.1 going from An,K to A∞ is parallel to the argument used in Subsection 7.2.8 and so omitted. We finally explain the modification of h needed to wrap up the study of Case 1. For this purpose, we need to modify the moduli space  s + + +  + ) a,d ,k,β,P [0, 1]| a| × Mmain,d (β; P k+1

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583

by collapsing the fiber of the map (7.3.7) in a way similar to the proof of Proposition 7.2.170. To ensure that the moduli space after this collapsing still carries a triangulation which are compatible with various forgetful maps etc., we need to study the natural stratification of Kuranishi neighborhoods and the way how it behaves under the forgetful maps (7.3.7.1), (7.3.7.2). Let us first describe this stratification. We use the notation of Lemma 7.3.8. Definition 7.3.21. Let xi ∈ Vx , (i = 1, 2) be an element of Kuranishi neighborhood of x ∈ Mmain zi , wi ). (We k+1 (β). Assume that xi is represented by (Σi ,  remark that wi is not necessarily pseudo-holomorphic.) We write x1 ∼ x2 if there exists a homeomorphism u : (Σ1 , z1 ) → (Σ2 , z2 ) such that the restriction of w2 ◦ u to each of the irreducible components is homologous to the restriction of w1 to the corresponding irreducible component. We consider the equivalence classes with respect to ∼. We further identify two equivalence classes if they become the same class after resolving all the interior singular points. We call the decomposition to equivalence classes which we obtain above the disc stratification. Roughly speaking, two elements in Vx are contained in the same stratum if they have the same combinatorial type after resolving all the sphere bubbles. Disc stratification is preserved under the coordinate change of Kuranishi structure. Namely, two elements belonging to the same stratum will be sent to the same stratum by the coordinate change. In fact, this stratification coincides with that of the Kuranishi neighborhood as an orbifold with corners. We remark that strata themselves of the disc stratification are not closed sets but their closure define a stratification in the standard sense. To clarify this point we call the original strata as open strata (of the disc stratification) and their closures as closed strata (of the disc stratification). Each closed stratum is a smooth manifold with corners. Let us consider the forgetful map (7.3.7.2). We remark that it induces maps between Kuranishi neighborhoods in the same way as Lemma 7.3.8. Let x = forget a (x+ ) and forget a : Vx++ → Vx be the map induced to the Kuranishi neighborhoods. Lemma 7.3.22. (1) Each open (resp. closed) stratum of Vx++ is sent to an open (resp. closed) stratum of Vx . (2) The restriction of forget a to each open stratum is a C ∞ map. (3) Let Xc+ be an open stratum of Vx++ and Xc be the open stratum of Vx that contains forget a (Xc+ ). Then one of the following holds: (7.3.23.1) (7.3.23.2)

dim Xc − rank Ex < dim Xc+ − rank Ex++ . dim Xc+ < dim Vx++ .

The same holds for forget a(1) , a(2) . As we explain at the end of the Section A1.4, our map forget a is not necessarily smooth on Vx++ . This is the reason why we need some caution in stating this lemma. Proof. Statement (1) and (2) are obvious from construction. It remains to prove (3). We represent y ∈ Xc+ by (Σ+ , z+ , w+ ). By forgetting the marked points we obtain (Σ , z , w ). As we mentioned, this may not be stable.

584

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If (Σ , z , w ) is stable, then the dimension of the fiber of the map forget a : → Xc is positive. Moreover rank Ex = rank Ex++ . (See the proof of Lemma 7.3.8.) Therefore (7.3.23.1) holds in this case. If (Σ , z , w ) is not stable, Σ+ must have a disc component where w+ is trivial. Therefore (7.3.23.2) holds in this case.  Xc+

Now we can find a perturbation (multisection) that is transversal to zero and compatible with forgetful maps. We remark that the disc stratification of a Kuranishi neighborhood of each point of Mmain k+1 (β) induces a stratification on [0, 1]| a| × Mmain (β). (We take the stratification of [0, 1]| a| as manifolds with cork+1 ners.) Lemma 7.3.24. We may choose the system of the multisections s a,d+ ,k,β,P + in Proposition 7.3.10 such that the following holds in addition: (7.3.25.1) s a,d+ ,k,β,P + is smooth on each closed stratum X c of the Kuranishi neighborhood. (7.3.25.2) The restriction of s a,d+ ,k,β,P + to each open stratum Xc is transversal to 0. Proof. As we explain at the end of Section A1.4 we can restrict the class of our multisections to those which have exponential decay along the boundaries or along the corners with respect to the coordinate T used there. Then the pull-back of such a multisection will still be in the same class (and in particular smooth up to the boundary or the corner). In this way we can construct a system of multisections satisfying the compatibility both with the forgetful map and with (7.3.25.1). The rest of the proof is given during the proof of Proposition 7.3.10.  Consider the zero set  s + + +  + ) a,d ,k,β,P [0, 1]| a| × Mmain,d (β; P k+1 of the (system of) multisection obtained in Lemma 7.3.24. We write it as  s +  +) [0, 1]| a| × Mmain,d (β; P k+1 for the simplicity of notations. Lemma 7.3.24 implies that it has a triangulation compatible with disc stratification.  s +  +) Our next task is to choose a triangulation of [0, 1]| a| × Mmain,d (β; P k+1 that is compatible with the forgetful map and collapse a simplicial subcomplex of s  +  + ) so that the contribution of Case 1 above becomes [0, 1]| a| × Mmain,d (β; P k+1

zero as a singular chain. We have the decomposition ∂Mmain k+1 (β) =





k2 

main Mmain k1 +1 (β1 )ev0 ×evi Mk2 +1 (β2 )

β1 +β2 =β k1 +k2 =k+1 i=1

as a space with Kuranishi structure. We need some more notations. We will decompose a = (a(1), · · · , a(|a|)) ∈ {1, · · · , k}| a|

7.3. CONSTRUCTION OF HOMOTOPY UNIT

585

into a disjoint union a = a1 ! a 2 . Here the data a, i, k1 , k2 determine a1 , a 2 , m according to the formula ⎧ 1 j ≤ m, ⎪ ⎨ a (j) (7.3.26) a(j) = a 2 (j − m) + i − 1 m + 1 ≤ j ≤ m + |a 2 |, ⎪ ⎩ 1 a (j − |a 2 | + m) + k1 − 1 m + 1 + |a 2 | + 1 ≤ j. main, a For each a we prepare a copy of Mmain k+1 (β) and write it as Mk+1 (β). We consider the following commutative diagram: 1

2

a main, a Mmain, (β2 ) k1 +1 (β1 )ev0 ×evi Mk2 +1 ⏐ ⏐forget×forget (

a −−−−→ Mmain, k+1 (β) ⏐ ⏐forget (

main main Mmain a1 | (β1 )ev0 ×evi Mk2 +1−| a 2 | (β2 ) −−−−→ Mk+1−| k1 +1−| a| (β)

Diagram 7.3.1 We put [0, 1]k− a = {(t1 , · · · , tk ) | t a(i) = 1, i = 1, · · · , |a|} ∼ = [0, 1]k−| a| . Let us decompose a = a1 ! a 2 . Then, the forgetful map and I a11 , a 2 (see the paragraph right before Example 7.3.12) induce a map a k−| a| +  × Mmain forget : [0, 1]k− a × Mmain, k+1−| a| (β; P ). k+1 (β; P ) → [0, 1]

This map is compatible with our system of multisections. (Compatibility Condition 7.3.13.) Then Diagram 7.3.1 and the compatibility of our system of multisections induce the following commutative diagram. (Here ×L in the diagram denotes ev0 ×evi ): 1

1

a + s ([0, 1]k1 − a × Mmain, k1 +1 (β1 ; P1 )) 2

a + s −−−−→ ([0, 1]k− a × Mmain, k+1 (β); P ))

2

a ×L ([0, 1]k2 − a × Mmain, (β2 ; P2+ ))s k +1 ⏐ 2 ⏐forget×forget (

1  s ([0, 1]k1 −| a | × Mmain a1 | (β1 ); P1 ) k1 +1−|

×L ([0, 1]k2 −| a

2

|

⏐ ⏐forget (

 s −−−−→ ([0, 1]k−| a| × Mmain k+1−| a| (β); P ))

+ s × Mmain a 2 | (β2 ; P2 )) k2 +1−| Diagram 7.3.2

Lemma 7.3.27. We can find a system of triangulations of our perturbed moduli + spaces ([0, 1]| a| × Mmain,d (β; P + ))s such that the arrows in Diagram 7.3.2 are k+1 piecewise linear. We may choose this triangulation so that it is also compatible with the identification by Compatibility Condition 7.3.17.

586

7.

TRANSVERSALITY

Proof. We first consider the horizontal arrow of Diagram 7.3.2. By the definition of disc stratification, it is easy to see that the restriction of it to each stratum is a local diffeomorphism, strata-wise. We next use Lemma 7.3.8 to see that the restriction of the vertical arrow to each stratum is a submersion. Therefore we can construct triangulation inductively over the stratum.  a  s We next define an equivalence relation ∼ on ([0, 1]| a| × Mmain, k+1 (β; P )) . a Definition 7.3.28. Let (t, x), (t , x ) ∈ [0, 1]| a| × Mmain, k+1 (β). We say that they are equivalent and write (t, x) ∼ (t , x ) if there exist a1 , a 2 as in Compatibility 2 a 1  Condition 7.3.13 and (t− , x− ) ∈ [0, 1]| a | × Mmain, a1 , a 2 ( t− ) = k+1−| a 2 | (β) such that I  t = t and forget a1 , a 2 (x) = forget a1 , a 2 (x ) = x− . This equivalence relation induces an equivalence relation on the Kuranishi neigha  borhoods of [0, 1]| a| × Mmain, k+1 (β; P ). Then, by Compatibility Condition 7.3.13, an a  s equivalence relation ∼ on ([0, 1]| a| × Mmain, k+1 (β; P )) , is induced. We denote the  set of ∼ equivalence classes by:  s a  )) (7.3.29) [0, 1]| a| × Mmain, (β; P . k+1 Coll

It follows from Lemma 7.3.27 that (7.3.29) has a triangulation. We also have an evaluation map  s a  )) ev : [0, 1]| a| × Mmain, (β; P −→ Lk+1−| a| k+1 Coll

which is piecewise smooth. a  s We use (7.3.29) instead of ([0, 1]| a| × Mmain, k+1 (β; P )) and obtain the same conclusion as Proposition 7.3.10. (Compatibility Condition 7.3.13 disappears, since a  s it is replaced by the fact that the space ([0, 1]| a| ×Mmain, k+1 (β; P ))Coll is well-defined.) This is the moduli space we actually use to define h. Then by Lemma 7.3.22 (3) the boundary contribution corresponding to Case 1 becomes exactly zero. We finally discuss the case Mmain k+1−| a| (β) is empty, (that is the case of Compatibility Condition 7.3.14). In this case, we consider the evaluation map ev0 : Mmain k+1 (β0 ) −→ L

(7.3.30)

at the 0-th marked point and the following commutative Diagram 7.3.3 instead of Diagram 7.3.1. 1

2

a main, a a Mmain, (β0 ) −−−−→ Mmain, k1 +1 (β0 )ev0 ×evi Mk2 +1 k+1 (β0 ) ⏐ ⏐ ⏐ev ⏐ev ( 0 ( 0

L

L Diagram 7.3.3

where the map ev0 in the first vertical arrow is applied to the second factor a2 Mmain, (β2 ). k2 +1

7.3. CONSTRUCTION OF HOMOTOPY UNIT

587

We then use Diagram 7.3.3 instead of Diagram 7.3.1 to obtain a diagram similar to Diagram 7.3.2. In this way we find a perturbation s and its zero set a main, a + s + s Mmain, k+1 (β0 ; P ) . We then define an equivalence relation on Mk+1 (β0 ; P ) in a similar way as Definition 7.3.28. We remark that the fiber of (7.3.27) is of positive dimension unless k = 2. So we can discuss in the same way to complete the proof of Theorem 7.3.1. The proof of Theorem 7.3.1 is now complete.  In the same way we can prove that various filtered A∞ homomorphisms and homotopy between them (which we constructed in Chapter 4 and Section 7.2) are homotopy-unital. Also we can show that the filtered A∞ bimodules and homomorphisms between them we constructed in Chapter 5 are homotopy-unital. 7.3.3. Proof of (3.8.36). In this subsection, we prove (3.8.35), (3.8.36) in Theorem 3.8.32. Since the proof of (3.8.35) is a straightforward generalization of the proof of (3.8.33), (3.8.34) and Theorem 7.3.1, we focus on the proof of (3.8.36). Let Q1 , · · · , Q1 be smooth singular simplices on M . We consider the fundamental class of M and represent it

(7.3.31) [M ] = ±Δ2n a a∈A

by using simplicial decomposition of M in the same way as (7.3.6). We put (7.3.32.1) (7.3.32.2)

 = Q1 ⊗ · · · ⊗ Q ∈ B C(M ; Λ0,nov ), Q 1 1 + = Q  ⊗ M ⊗ · · · ⊗ M ∈ B C(M ; Λ0,nov ), Q 2 /0 1 . 2

where  = 1 + 2 . More precisely, we need to put

2n + =  ⊗ Δ2n Q ±Q a1 ⊗ · · · ⊗ Δa . a1 ,··· ,a2 ∈A

2

Let P ∈ Bk+1−|a| C(L; Λ0,nov )[1] and P + ∈ Bk+1 C(L; Λ0,nov )[1] be as in Subsection 7.3.2. We consider the fiber product (7.3.33)

 + × P + ) = Mk+1; (L; β; Q  + , P + ). Mk+1; (L; β) ×M  ×Lk (Q

We will define a family of multisections ss1 ,··· ,s2 ,t1 ,··· ,t|a| on it parameterized by [0, 1]2 ×[0, 1]| a| . We require that they satisfy the following compatibility conditions: (7.3.34.1) When ti → 1 or ti → 0, the compatibility condition is similar to Compatibility Conditions 7.3.13, 7.3.14 and 7.3.17. (7.3.34.2) When si → 1 our multisection converges to a pull-back of the multi +− , P + ) where Q  +− = Q  ⊗ M ⊗(2 −1) . section s∗ on Mk+1; (L; β; Q    (7.3.34.3) We put Q = Q ⊗ M . Then, when si → 0 the pull-back of the  + , P + ) converges to the multisecmultisection ss1 ,··· ,s ,t1 ,··· ,t|a| on Mk+1; (L; β; Q  + , P + ). (Here we define Q  + = tion ss1 ,··· ,si−1 ,si+1 ,··· ,s2 ;t1 ,··· ,t|a| on Mk+1; (L; β; Q  + .)   ⊗ M ⊗(2 −1) = Q Q  + , P + ) our multisections are com(7.3.34.4) At the boundary of Mk+1; (L; β; Q patible with the fiber product of the similar moduli spaces, in the same way as the other similar cases in Section 7.2.

588

7.

TRANSVERSALITY

We also assume that they are invariant under the permutation of the interior marked points. We require one additional compatibility condition, which describes the behavior of our multisection when an interior marked point becomes an exterior marked point in the limit. Let ((Σ, z, z+ ), w) be an element of Mk+1; (L; β). Here Σ is a bordered Riemann surface of genus zero, z+ = (z1+ , · · · , z+ ) are the interior marked points, and z = (z0 , z1 , · · · , zk ) are the boundary marked points. w : (Σ, ∂Σ) → (M, L) is a pseudo-holomorphic map of homology class β. We glue Σ with D2 by identifying zi ∈ ∂Σ with 1 ∈ ∂D2 . Let Σ+ be the bordered Riemann surface of genus zero which we obtain in this way. Σ+ has k − 1 exterior marked points z \ {zi }. We define  + 1 marked points of Σ+ as z+ ∪ {0}. Here 0 ∈ D2 . We extend w to w+ : Σ+ → M so that w+ = w(zi ) on Σ+ \ Σ = D2 . The map ((Σ; z, z+ ), w) → ((Σ+ , z \ {zi }, z+ ∪ {0}), w+ ) induces  + , P + ) −→ Mk;+1 (L; β; Q  ++ , P +− ), Ii : Mk+1; (L; β; Q where  ++ = Q  + ⊗ M, Q

P +− = P1 ⊗ · · · ⊗ Pi−1 ⊗ Pi+1 ⊗ · · · ⊗ Pk .

Let i = a(j). Now the additional compatibility condition we require is:  ++ , P +− ) pulls back (7.3.34.5) ss1 ,··· ,s2 +1 ;t1 ,··· ,tj−1 ,tj+1 ,··· ,t|a| on Mk;+1 (L; β; Q + +  to ss1 ,··· ,s2 ;t1 ,··· ,tj−1 ,s2 +1 ,tj+1 ,··· ,t|a| on Mk+1; (L; β; Q , P ) by Ii . M Pi−1

Pi−1

Pi+1

Q Q Q Q

Q

L

Pi+1

Q Q Q

Pk

Pk

ev0 P

P1

ev0 P

P1

Figure 7.3.1 In the same way as Proposition 7.3.10, we can construct a family of multisections ss1 ,··· ,s2 ;t1 ,··· ,t|a| which are transversal to zero and satisfies (7.3.34.1)(7.3.34.5). More precisely speaking we need to choose countable set of smooth singular simplices etc. in the same way as Proposition 7.3.10. We omit it. Now we define  ˜ ⊗2 , P1 ⊗ · · · ⊗ f ⊗ · · · ⊗ f ⊗ · · · ⊗ Pk ) q++ ,k+1 (Q ⊗ f 2 3

 + , P + )s . = T ω(β) eμL (β)/2 ev0∗ M(L; β; Q β

We require (3.8.36.1)-(3.8.36.3) as a part the definition of q++ . The fact that q++ satisfies (3.8.33) follows from (7.3.34.1)-(7.3.34.4). (3.8.36.5) follows from (7.3.34.5). The detail is similar to the proof of Proposition 7.3.10 and is omitted. This completes the proof of (3.8.36).

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

589

7.4. Details of the construction of the operators p, q and r This section is a continuation of Section 3.8. We write M(L; β) etc. in place ˜ in this section. of M(β), since we also use M(M ; β) 7.4.1. Details of the construction of p. We first prove Proposition 3.8.27. It is a combination of the argument of Section 3.8 and Subsection 7.1.4 (Theorem 7.1.43 for example), where we studied the moduli space Mmain k+1, (L; β) in the case k + 1 ≥ 1 (namely the case when there is at least one boundary marked point). In this subsection, we will discuss Mmain 0, (L; β). The main new point in this case is that, as we already pointed out in Subsection 3.8.3, the standard stable-map-compactification Mmain 0, (L; β) may not be compact for a class β ∈ H 2 (M, L; Z) whose image ∂(β) in the natural exact sequence i

∂

∗ H2 (M ) −→ H2 (M, L) −→ H1 (L)

is zero. A detailed study of this phenomenon is in order. First of all, we remark that the definition of moduli spaces of bordered Riemann surfaces involves the genus and the number of boundary components. In particular, when we define a compactification of the moduli space of smooth stable maps, the domain of added elements in the compactification is required to carry the same genus and the same number of boundary components as the smooth ones nearby. (In this book we consider the case when the genus is 0 and the number of boundary components is one.) Now consider a stable map (Σ, w) = ((|Σ|, z, z+ ), w) ∈ Mmain k, (L; β). (We assume k = 0 for simplicity.) We attempt to define its domain by forgetting all boundary marked points and some of the interior marked points by an order-preserving injection + : {1, · · · ,  } → {1, · · · , }, and obtain a nodal bordered Riemann surface of genus zero Σ = (|Σ|, ∅, z+ ). We then describe the types of unstable components D of (Σ , w). It turns out that the following case should be treated separately from the cases (7.1.41.1), (7.1.41.2), (7.1.41.3) which were handled in Section 7.1: (7.4.1) Disc with one interior singular point but no other singular or marked points. For this configuration, we leave this configuration as it is without shrinking D. Then, after shrinking all the other types of unstable components in the standard way, we will lead to either an element of Mmain 0, (L; β) or to a component of the type (7.4.1) with sphere bubbles attached. In this way, we obtain a forgetful map Forget∅, + :Mmain k, (L; β) (7.4.2)

→ Mmain 0, (L; β) ∪



˜ ×M L). (M +1 (M ; β)

˜ β˜ : i∗ (β)=β

As in Subsection 3.8.3, we simply write the target space in (7.4.2) as −1 Mmain 0, (L; β) ∪ (M +1 (M ; i∗ (β)) ×M L).

590

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TRANSVERSALITY

−1 We can then define a topology on Mmain 0, (L; β) ∪ (M +1 (M ; i∗ (β)) ×M L) in the same way as Definition 7.1.42. We can prove that −1 Mmain 0, (L; β) ∪ (M +1 (M ; i∗ (β)) ×M L)

is compact and Hausdorff in the same way as [FuOn99II] Sections 11,10. −1 Now we equip the space Mmain 0, (L; β) ∪ (M+1 (M ; i∗ (β)) ×M L) with a Ku ranishi structure with boundary. (Hereafter we put  =  to simplify the notation.) Construction of a Kuranishi neighborhood at each point of Mmain 0, (L; β) is already discussed in detail in Section 7.1. In this subsection we construct a Kuranishi ˜ ×M L with β˜ ∈ i−1 (β). neighborhood at a point on M+1 (M ; β) ∗ + + ˜ be a stable map of genus Let (Σ, w) = ((Σ, z0 , · · · , z ), w) ∈ M+1 (M ; β) zero (sphere) with  + 1 marked points. For simplicity, we discuss the case when Σ is smooth (that is diffeomorphic to S 2 ) and it has two marked points z0+ , z1+ . We assume w(z0+ ) ∈ L. For simplicity we also assume that there exists a family ˜ parameterized by u ∈ U and assume that this family is (Σu , wu ) ∈ M2 (M ; β) ˜ transversal. (In particular, we assume dim U = the virtual dimension of M2 (M ; β).) Moreover we assume that the map ev0 : U → M , ev0 (u) = wu (z0+ ) is transversal to L and that the automorphism group of (Σu , wu ) is trivial. (In general, we need an obstruction bundle and an action of a finite group. We omit the discussion about the obstruction bundle, since it can be handled in the same way as other well established ˜ cases.) We are going to find a family of elements of Mmain 0,1 (L; β)∪(M2 (M ; β)×M L) parameterized by (U ev0 ×M L) × [0, ). (Here (U ev0 ×M L) × {0} is identified with ˜ ×M L.) an open set in M2 (M ; β) 2 We consider (D , z0 , z + ) where z0 ∈ ∂D2 , z + = 0 ∈ D2 . For u ∈ U with 2 + + ev0 (u) ∈ L, we consider Σ+ u = Σu z0+ #z + D and let wu : Σu → M be the pseudoholomorphic map defined by

 wu+ (z)

=

wu (z)

z ∈ Σu

wu (z0+ )

z ∈ D2 .

We thus obtain a map I : U → Mmain 1,1 (L; β),

+ I(u) = (Σ+ u , wu ).

Since Σ+ u has an interior singular point, we can construct its neighborhood in Mmain (L; β) parameterized by D2 () (a small disc in C2 of diameter ) by smooth1,1 ing the singularity using a partition of unity (see [FuOn99II] Section 12). Hence we have the map (7.4.3)

I+,pre : (U ev0 ×M L) × D2 () → M ap1,1 (L; β).

Here M ap1,1 (L; β) is the set of “stable maps” which satisfy the same condition as Mmain 1,1 (L; β) except we do not require the map to be pseudo-holomorphic. We next use the implicit function theorem to obtain (7.4.4)

I+ : (U ev0 ×M L) × D2 () → Mmain 1,1 (L; β).

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

591

β z0 Figure 7.4.1 Lemma 7.4.5. I+ is an open embedding. Proof. We assumed that Mmain 1,1 (L;β) is transversal to L. Moreover there is main no symmetry of elements of Mmain 1,1 (L; β). Therefore the gluing of M1,1 (L;β) and ˜ results in a space diffeomorphic to a fiber product plus gluing parameter. M2 (M ; β) The lemma follows. (See [FuOn99II] Section 13 for more detail.)  We next consider the forgetful map main −1 Forget : Mmain 1,1 (L; β) → M0,1 (L; β) ∪ (M2 (M ; i∗ (β)) ×M L)

on the image of I+ . By the construction of the map I+,pre it is easy to see the following lemma. Let Forget : M ap1,1 (L; β) → M ap0,1 (L; β) ∪ (M ap2 (M ; i−1 ∗ (β)) ×M L) be the obvious forgetful map. We consider the S 1 action on D2 (). Lemma 7.4.6. For θ ∈ S 1 , we have Forget(I+,pre ((u, x), θz)) = Forget(I+,pre ((u, x), z)) where (u, x) ∈ U ev0 ×M L, z ∈ D2 (). Now we consider the process to go from I+,pre to I+ , which is performed by the implicit function theorem. Obviously the construction is invariant under the S 1 action. Then we have Forget(I+ ((u, x), θz)) = Forget(I+ ((u, x), z)). Thus the image of Forget on the image of I+ is diffeomorphic to U × D2 ()/S 1 ∼ = U × [0, ). We have thus constructed the Kuranishi neighborhood. The rest of the proof of Proposition 3.8.27 is the same as the other arguments which are now well-established.  We have thus constructed the Kuranishi structure we need. We will now construct a system of multisections (perturbations) of the moduli space Mmain k,1 (L; β) (k = 0, 1, 2, · · · ) to carry out the construction of the operator p. Namely we present details of the proofs of Theorem 3.8.9 and of Lemma 3.8.25. We remark that before we perturb Mmain k,1 (L; β), we have completed the construction of the perturbation of the moduli space Mmain k,0 (L; β; P1 , · · ·, Pk ) as well as the choices of the countable set of singular simplices X (L). (We also fix the ˜ in order to make our perturbation consistent with the perturbation of Mk (M ; β) sphere bubbles.) We put (7.4.7)

main Mmain k,1 (L; β; P1 , · · · , Pk ) = Mk,1 (L; β) ×Lk (P1 × · · · × Pk ).

592

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It has a Kuranishi structure. For the case k = 0 we consider Mmain 0,1 (L; β) and need to compactify it to  ˜ ×M L). (7.4.8) Mmain (M2 (M ; β) 0,1 (L; β) ∪ ˜ β˜ : i∗ (β)=β

From now on, we write Mmain 0,1 (L; β) for this compactification (7.4.8) to simplify the notation. We will construct a system of multisections on Mmain k,1 (L; β; P1 , · · · , Pk ) and on main M0,1 (L; β) by an induction over ω[β]. We remark that the multisections of Mmain k,1 (L; β; P1 , · · · , Pk ) may depend on main the choice of the multisections on Mk ,1 (L; β  ; P1 , · · · , Pk  ) for k > k (but ω[β  ] < ω[β]). Hence we have to construct them for various k’s simultaneously. Remark 7.4.9. We remark that while we construct a series of multisections on Mmain k,1 (L; β; P1 , · · · , Pk ), we do not need such an inductive argument as delicate as those given in Section 7.2. The main difference is the following: For the study of m, we need to apply m again to mk (P1 , · · · , Pk ). On the other hand pk (P1 , · · · , Pk ) is a singular simplex of the ambient symplectic manifold and hence we do not need to apply m or p to the singular chain pk (P1 , · · · , Pk ) again. We say (β, k) > (β  , k ) if one of the following holds: (7.4.10.1) (7.4.10.2)

ω[β] > ω[β  ]. ω[β] = ω[β  ] and k > k .

We construct a multisection of Mmain k,1 (L; β; P1 , · · · , Pk ) by the induction with respect to this order. We remark that Mmain 0,1 (L; β0 ) is empty, since there is no stable map in it and ˜ M2 (M ; β0 ) is empty. Therefore p0,β0 = 0. We do not need to perturb Mmain 1,1 (L; β0 ). Namely we put M1,1 (L; β0 ; P ) = P for any P . Now assume that we have constructed the perturbations (or multisections)    of Mmain any (β  , k ) such that either (β  , k ) < (β, k) or 1 , · · · , Pk ) for k ,1 (L; β ; P      dim Pi < dim Pi . Under these hypotheses, we will construct (β , k ) = (β, k), the perturbation of Mmain k,1 (L; β; P1 , · · · , Pk ). We consider the boundary ∂Mmain k,1 (L; β; P1 , · · · , Pk ). This is decomposed into various components. See Figures 7.4.2, 7.4.3. (7.4.11.1) Mmain k,1 (L; β; P1 , · · · , ∂Pi , · · · , Pk ).   (7.4.11.2) Mmain k−j+i,1 (L; β−β ; P1 , · · · , Pi−1 , Qi,j (β ), Pj+1 , · · · , Pk ), where 1 ≤ i ≤  j + 1 ≤ k + 1 and Qi,j (β  ) = ev0∗ (Mmain j−i+2,0 (L; β ; Pi , · · · , Pj )). main  (7.4.11.3) Mj−i+2,1 (L; β −β ; Pi ,· · ·, Pj , Rj+1,i−1 (β  )), where 2 ≤ i ≤ j + 1 ≤ k + 1  and Rj+1,i−1 (β  ) = ev0∗ (Mmain k−j+i,0 (L; β ; Pj+1 ,· · ·, Pk , P1 ,· · ·, Pi−1 )). ˜ ×M L in case k = 0. (7.4.11.4) M2 (M ; β) We remark that, before perturbation, there is a cyclic symmetry, that is the Zk action generated by: main cyc : Mmain k,1 (L; β; P1 , · · · , Pk ) → Mk,1 (L; β; P2 , · · · , Pk , P1 ).

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

593

We recall that we want to construct perturbations (multisections) preserving the cyclic symmetry. We note that for the moduli spaces appeared in (7.4.11) the perturbations have been already given by the induction hypothesis.

Pi

Pi−1

β

P1 β−β  z0 Pk

Pj Pj+1

Q i,j (β  )

Pk

Figure 7.4.2 The case (7.4.11.2). P P Pj+1 j β

P1

β

z0 Pi−1 Pi Rj+1,i−1 (β  )

Figure 7.4.3 The case (7.4.11.3).    Lemma 7.4.12. If the perturbations of Mmain k ,1 (L; β ; P1 , · · · , Pk ) constructed in the earlier stages preserve the cyclic symmetry, then the induced perturbation on ∂Mmain k,1 (L; β; P1 , · · · , Pk ) also preserves the cyclic symmetry.

Proof. Let us consider the components of ∂Mmain k,1 (L; β; P2 ,· · ·, Pk , P1 ). They main are obtained by replacing P1 , · · · , Pk in ∂Mk,1 (L; β; P1 , · · · , Pk ) by P2 , · · · , Pk , P1 , respectively. If we apply this replacement to   Mmain k−j+i,1 (L; β − β ; P1 , · · · , Pi−1 , Qi,j (β ), Pj+1 , · · · , Pk ),

then we obtain the component (7.4.13.1)

  Mmain k−j+i,1 (L; β − β ; P2 , · · · , Pi , Qi+1,j+1 (β ), Pj+2 , · · · , Pk , P1 ),

if j < k and (7.4.13.2) if j = k.

  Mmain i,1 (L; β − β ; P2 , · · · , Pi , Ri+1,1 (β )),

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If we apply the same replacement to   Mmain j−i+2,1 (L; β − β ; Pi , · · · , Pj , Rj+1,i−1 (β )),

then we obtain (7.4.13.3)

  Mmain j−i+2,1 (L; β − β ; Pi+1 , · · · , Pj+1 , Rj+2,i (β ))

if j < k − 1 and (7.4.13.4)

  Mmain k−i+1,1 (L; β − β ; Pi+1 , · · · , Pk , Q1,i (β ))

if j = k − 1. By the induction hypothesis, we find that the sum of (7.4.13.1), (7.4.13.2), (7.4.13.3), (7.4.13.4) is equal to the sum of components of the forms (7.4.11.2) and (7.4.11.3). Namely the sum of components of the forms (7.4.11.2) and (7.4.11.3) is cyclically symmetric. (We can check that the sign is as asserted. See Section 8.10.) The case of (7.4.11.1) is obvious. There is nothing to show in the case of (7.4.11.4).  Remark 7.4.14. We remark that we have used only cyclic symmetry of the    moduli space Mmain k ,1 (L; β ; P1 , · · · , Pk ) in the proof of Lemma 7.4.12 above. No main   extra symmetry of Mk ,0 (L; β ; P1 , · · · , Pk  ) is required. Now, by Lemma 7.4.12, we have a perturbation (multisection) on the boundary of Mmain k,1 (L; β; P1 , · · · , Pk ) preserving the cyclic symmetry. We can extend this symmetry to a multisection on Mmain k,1 (L; β; P1 , · · · , Pk ) in the following way: We first take the quotient of Mmain (L; β; P1 , · · · , Pk ) by the Zk action (see Definition k,1 A1.45 and Lemma A1.49) and obtain a space with Kuranishi structure. Then we can extend the multisection on the boundary by Theorem A1.23. We lift this to Mmain k,1 (L; β; P1 , · · · , Pk ) and obtain a required perturbation. Remark 7.4.15. This is the point where we need to work with the Q-coefficient even for the monotone case. Namely we need our multisection to be Zk invariant and the Zk action in general has a fixed point. We have thus completed construction of the perturbation that is cyclically symmetric and compatible at their boundaries. With this perturbation, we can then define the operator p as we discussed in Section 3.8. Then the description of the boundary (7.4.11) above implies (3.8.10.1), (3.8.10.2) and (3.8.10.3). To prove (3.8.10.4), (3.8.10.5) and (3.8.10.6), we also need to establish the relation of p with homotopy unit. The argument is a straightforward combination of the argument above and the one in Section 7.3. So we omit it. The proof of Theorem 3.8.9 is now complete.  Remark 7.4.16. The argument given above to achieve the cyclic symmetry of p is much simpler than what we might need for m. This is because the image of p lies in the set of chains on the ambient manifold M to which we do not need to apply the relevant operators p again. On the other hand for the case of m, we need  ◦m  = 0 on the chosen to apply m twice to make sure it satisfies the A∞ -relation m countable set of chains of L. (See Section 7.2.) In fact, the procedure of constructing m described in Section 7.2, which was designed to ensure the A∞ -relation, is bound to destroy the cyclic symmetry.

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

595

7.4.2. Construction of q I: An,K version. In this subsection, we discuss Proposition 3.8.66. In fact we first need to modify the construction to resolve a trouble described in Subsection 7.2.3. Namely we need to stop the construction at some finite stage and to use homological algebra. We use this modified construction to complete the proof of Theorem 3.8.32. The modified version is the following Lemma 7.4.17 whose proof is similar to the argument of Section 7.2. Let X1 (L) be a countable set of singular simplices on L we start with. We  assume that we have a system of multisections of Mmain k+1 (L, β; P ) for Pi ∈ X1 (L), (β, k)  (n, K). (See Definition 7.2.63 for .) We put X1,00 (L) = X1 (L). We assume that this system defines a filtered An,K structure and also assume that it satisfies Property 7.2.27. Let X (M ) be a countable set of singular simplices of M satisfying Assumption 3.8.28. To construct X1+ (L) in Proposition 3.8.66, we need to find a sequence of countable sets X1+,g (L) of chains where X1+,0 (L) ⊃ X1+,00 (L) = X1 (L). We put X1+,(g0 ) (L) =

g0 

X1+,g (L).

g=0

We say P is of the g-th generation if it is in X1+,g (L). We define the decorated   moduli space Mmain,d k+1, (L; β; Q, P ) in the same way as in Section 7.2. We remark that the decoration d is {1, · · · , k} → Z≥0 . (There is no decoration corresponding to Q’s since only P ’s have generations.) Lemma 7.4.17. For any δ > 0 and K > 0, there exist X1+,g (L) for g ≤ K and a system of multisections sd,k,β,Q, P for (β, d) ≤ K,  < K and Pi ∈ X1,(K) (L) such that sd,k,β,Q, P is transversal to 0. Moreover the following holds. (7.4.18.1) X1+,g (L) satisfies Properties 7.2.27.  P    sd,k,β,Q, (7.4.18.2) If Pi ∈ X1+,d(i) (L) (i = 1, · · · ), ev0∗ (Mmain,d ) is k+1, (L; β; Q, P ) decomposed into elements of X1+,(g) (L) where g = (d, β). (7.4.18.3) The multisections sd,k,β,Q, P are compatible in a sense similar to those in Section 7.2. (7.4.18.4) If  = 0 and Pi ∈ X1+,00 (L) = X1 (L), then sd,k,β,Q, P coincides with the system of multisections we start with.  P    sd,k,β,Q, ) lies in a δ neighborhood of the moduli (7.4.18.5) ev0∗ (Mmain,d k+1, (L; β; Q, P ) main,d   (L; β; Q, P ). space M k+1,

The proof of Lemma 7.4.17 is by the same induction process as we already discussed several times in Section 7.2. We omit the detail. Lemma 7.4.17 implies an “An,K version” of Theorem 3.8.62. (We state it precisely as Lemma 7.4.113 later.) To prove the A∞ -version using the “An,K version”, we need some homological algebra, which is considerably different from those used in Section 7.2. We discuss the homological algebra in Subsections 7.4.3 and 7.4.4 and complete the proof of Theorem 3.8.62 in Subsection 7.4.5. In the introduction, we stated Theorem Y which is related to but more involved than Theorem 3.8.62. We prove it in Subsection 7.4.6. We use the “cobordism argument” in Subsection 7.4.5 and the bifurcation argument in Subsection 7.4.6. (See Subsection 7.2.14 about those two methods.) The result of Subsection 7.4.6

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7.

TRANSVERSALITY

implies the one in Subsection 7.4.5. We describe them both to show that both methods work to prove Theorem 3.8.62. Remark 7.4.19. We thank C.-H. Cho who pointed us out that we need some homological algebra for the proof of Theorem 3.8.62 which is considerably different from the ones used in Section 7.2. 7.4.3. Construction of q II: q is an L∞ homomorphism. In this subsection we translate the conclusion of Theorem 3.8.32 into the language of filtered L∞ homomorphisms. The definitions of filtered L∞ algebra and L∞ homomorphism are in Section A3. Let (C, m) be a filtered A∞ algebra and (C amb , δ) a cochain complex over Λ0,nov . In our application to the proof of Theorem 3.8.32 we put C amb = C(M ; Λ0,nov ), C = C(L; Λ0,nov ). We define a trivial filtered L∞ structure on C amb [1] by  l1 (y) = (−1)deg y δ(y), (7.4.20) k = 1. lk = 0 We do not have a good explanation of the reason why the sign (−1)deg y is  taken instead of (−1)deg y . The only reason is then the proof of Proposition 7.4.34 works. We next consider the Hochschild complex: (7.4.21)

CH(C, C) =

∞ 

Homfilt (Bk (C[1]), C[1]).

k=0

Here Homfilt is the set of all Λ0,nov module homomorphisms ϕ such that ϕ(F λ (Bk (C[1]))) ⊂ F λ C[1]. (We call such a homomorphism filtered homomorphism from now on.) We will define a structure of differential graded Lie algebra (D.G.L.A.) on CH(C, C) as follows. The grading is defined by the degree of the homomorphism (and not by k in the right hand side of (7.4.21)). To define a differential and a bracket we need some lemmas and notations. (The discussion below is a filtered version of Subsection 4.4.5.) ˆ ˆ Definition 7.4.22. We define Der(B(C[1]), B(C[1])) to be the set of all filtered homomorphisms ˆ ˆ ϕˆ : B(C[1]) → B(C[1]) that satisfy  id) + (id ⊗  ϕ)) (7.4.22.1) ((ϕ ⊗  ◦Δ=Δ◦ϕ  and ˆ (7.4.22.2) the B0 (C[1])-component of ϕ(x) ˆ is zero for any x ∈ B(C[1]). (Note ˆ0 (C[1]).) ϕ(1) ˆ may be nonzero for 1 ∈ B  (n;1) (n;n) We recall Δn−1 x = a xa ⊗· · ·⊗xa . Here a runs over an index set which depends on n and x. ˆ ˆ For each ϕ = (ϕ0 , ϕ1 , · · · ), we define ϕ  : B(C[1]) → B(C[1]) by

 (3;1) (−1)deg ϕ deg xa x(3;1) ⊗ ϕ(x(3;2) ) ⊗ x(3;3) (7.4.23) ϕ(x)  = a a a a

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

597

ˆ ˆk (C[1]). where ϕ : B(C[1]) → C[1] is a homomorphism which restricts to ϕk on B In the same way as Lemma 4.4.43 we can prove the following: Lemma 7.4.24. The map ϕ → ϕˆ defines an isomorphism ˆ ˆ CH(C, C) ∼ B(C[1])). = Der(B(C[1]),

(7.4.25.1)

In case C is (strictly) unital we define (7.4.25.2)

ˆ ˆ CH unit (C, C) ∼ B(C[1])) = Derunit (B(C[1]),

to be the set of all ϕ such that ϕ(x1 ⊗ e ⊗ x2 ) = 0 ˆ for any x1 , x2 ∈ B(C[1]). ˆ ˆ From now on we identify CH(C, C) with Der(B(C[1]), B(C[1])) by the isomorphism (7.4.25). We also write ˆ ˆ DerB(C[1]) : = Der(B(C[1]), B(C[1])), unit unit ˆ ˆ DerB (C[1]) : = Der (B(C[1]), B(C[1])). Definition 7.4.26. For ϕ, ϕ ∈ DerB(C[1]), we put δ(ϕ) = dˆ ◦ ϕ − (−1)deg ϕ ϕ ◦ dˆ

(7.4.27)

where dˆ is as in Definition 3.2.16, and we put   1 ϕ ◦ ϕ − (−1)deg ϕ deg ϕ ϕ ◦ ϕ . (7.4.28) [ϕ, ϕ ] = − 2 The reason why there is minus sign in (7.4.28) which is rather unusual, will be justified by Proposition 7.4.34, Theorem 7.4.41 and Lemma 7.4.113. We also remark that 12 appears here since we defined Ek (C[1]) in Section 3.8 as the set of symmetric group invariant part of Bk (C[1]). In several other literature, Ek (C[1]) is defined as the quotient of Bk (C[1]) by the ideal generated by elements of the form · · · x ⊗ y · · · − (−1)deg



x deg y

(· · · y ⊗ x · · · ).

The difference between x ⊗ y ± y ⊗ x [x ⊗ y] (the equivalence class of x ⊗ y) is and⊗k  1 ⊗k 1 x x . We also recall that we put e = x and not e = x . k k k! 2 If we identify ∞ 

ϕi = (ϕik )k∈Z≥0 ∈

Homfilt (Bk (C[1]), C[1]) ∼ = DerB(C[1]),

k=0

then δ(ϕ) = ((δϕ)k ) is (δϕ)k (x) = (7.4.27’)

c

−

(−1)deg ϕ deg

c

and [ϕ1 , ϕ2 ] = (([ϕ1 , ϕ2 ])k ) is



x(3;1) c

(−1)deg ϕ+deg



m(x(3;1) ⊗ ϕ(x(3;2) ) ⊗ x(3;3) ) c c c

x(3;1) c

ϕ(x(3;1) ⊗ m(x(3;2) ) ⊗ x(3;3) ) c c c

598

7.

(7.4.28’)

TRANSVERSALITY

([ϕ1 , ϕ2 ])k (x) 2  (3;1) 1

(−1)deg ϕ deg xc +1 ϕ1 (x(3;1) ⊗ ϕ2 (x(3;2) ) ⊗ x(3;3) ) = c c c 2 c 1  (3;1) 2 1

+ (−1)deg ϕ (deg xc +deg ϕ ) ϕ2 (x(3;1) ⊗ ϕ1 (x(3;2) ) ⊗ x(3;3) ). c c c 2 c

We remark that for x = x1 ⊗ · · · ⊗ x we put deg x =



deg xi = deg x − .

Lemma 7.4.29. The right hand sides of the formulas (7.4.27), (7.4.28) are contained in DerB(C[1])). (DerB(C[1]), δ, [ , ]) is a differential graded Lie algebra. The proof is straightforward and standard which we omit. Then the filtered version of the prescription provided in Example A3.4 coming later in Section A3 makes DerB(C[1])) into a filtered L∞ algebra. We are going to study a strict and filtered L∞ homomorphism ˆˆqo : C amb [1] → DerB(C[1]).

(7.4.30)

Here the symbol o stands for “opposite”. See (7.4.39.1). From now on we assume all the filtered L∞ homomorphisms are strict, filtered and gapped. o By the definition of strict and filtered L∞ homomorphisms, ˆˆq is a sequence of filtered homomorphisms (7.4.31)

ˆqo : E (C amb [2]) → (DerB(C[1]))[1],

 = 1, 2, · · ·

of degree 0 satisfying an appropriate relation, which we will discuss later in Proposition 7.4.34. (We remark that we have operators qo for  ≥ 1 but not for  = 0, o ˆq is strict.) since ˆ Under the identification CH(C, C) ∼ = DerB(C[1])), having a sequence of filtered homomorphisms (7.4.31) is equivalent to having a family of filtered homomorphisms (7.4.32)

qo,k : E (C amb [2]) ⊗ Bk (C[1]) → C[1],

 = 1, 2, · · · , k = 0, 1, · · ·

of degree 1. We put qo0,k = mk : Bk (C[1]) → C[1].

(7.4.33)

To state the main result of this subsection, we need some more notations. (See also Subsection 3.8.4.) The coboundary operator δ on C amb induces a coderivation δo  amb [2]) by on E(C δo (y1 ⊗ · · · ⊗ yk ) =

k

(−1)deg y1 +···+deg yi y1 ⊗ · · · ⊗ δyi ⊗ · · · ⊗ yk . i=1

This is the coderivation induced by l1 . (See (7.4.20).) Let y ∈ E(C amb [2]) and x ∈ B(C[1]). We put



Δk−1 y = y(k;1) ⊗ · · · ⊗ y(k;k) , Δk−1 x = x(k;1) ⊗ · · · ⊗ x(k;k) . c1 c1 c2 c2 c1 ∈C1

c2 ∈C2

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

599

Here C1 depends on k and y and C2 depends on k and x. Let C1 ⊂ C1 be the set of c1 such that y(k;j) ∈ / E0 (C amb [2]), j = 1, · · · , k. c1 o ˆq is a strict and filtered L∞ homomorphism if and only Proposition 7.4.34. ˆ if qo satisfies the identity

(−1)deg y qo,k (δo (y) ⊗ x)

(2;2)  (3;1)  (3;1) (2;1) (2;1) (2;2) + (−1)deg yc1 deg xc2 +deg xc2 +deg yc1 +deg yc1 deg yc1

(7.4.35)

c1 ∈C1 ,c2 ∈C2 (3;3) qo (y(2;1) ⊗ (x(3;1) ⊗ qo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 )) = 0.

We remark that in (7.4.35) and from now on we omit the suffix (c1 ), k(c2 ) in (2;2) (3;2) qo(c1 ),k(c2 ) (yc1 ⊗ xc2 ) etc., where ∈ E(c1 ) (C amb [2]), y(2;2) c1

x(3;2) ∈ Ek(c2 ) (C amb [2]). c2

0

Proof. The condition that ˆˆq is a filtered L∞ homomorphism is written as

ˆqo (l1 (y)) = l2 (ˆqo (y(2;1) qo (y(2;2) qo (y)), (7.4.36) c1 ), ˆ c1 )) + l1 (ˆ c1 ∈C1

since lk (y) = 0 for y ∈ Ek (C amb [2]), k = 1. (We remark that the summation in (7.4.36) is taken for c1 ∈ C1 .) By (7.4.28’), Example A3.4 and the graded cocommutativity of Δ

(7.4.37)

y(2;1) ⊗ y(2;2) = c1 c1

c1



(2;1)

(−1)deg yc1

deg y(2;2) c 1

y(2;2) ⊗ y(2;1) c1 c1

c1

on E(C amb [2]), we obtain the following equality: (7.4.38.1)

l2 (ˆqo (y(2;1) qo (y(2;2) c1 ), ˆ c1 ))(x)

c1 ∈C1

=

1 2 +

=

1 2 +

=



(3;3) (−1)∗1 (i) ˆqo (y(2;1) ⊗ (x(3;1) ⊗ ˆqo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 ))

c1 ∈C1 ,c2 ∈C2

1 2



(3;3) (−1)∗2 (i) ˆqo (y(2;2) ⊗ (x(3;1) ⊗ ˆqo (y(2;1) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 ))

c1 ∈C1 ,c2 ∈C2



(3;3) (−1)∗1 (i) ˆqo (y(2;1) ⊗ (x(3;1) ⊗ ˆqo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 ))

c1 ∈C1 ,c2 ∈C2

1 2



(3;3) (−1)∗3 (i) ˆqo (y(2;1) ⊗ (x(3;1) ⊗ ˆqo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 ))

c1 ∈C1 ,c2 ∈C2



c1 ∈C1 ,c2 ∈C2

(3;3) (−1)∗1 (i) ˆqo (y(2;1) ⊗ (x(3;1) ⊗ ˆqo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 )).

600

7.

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Let us explain the above calculation. We remark deg ˆqo (y) ≡ deg(y) + 1 mod 2. The sign ∗1 (i) is ∗1 (i) = deg y(2;2) deg x(3;1) + deg x(3;1) + (deg y(2;1) + 1) deg y(2;2) + 1. c1 c2 c2 c1 c1 (Here we have +1 in the above formula because of the minus sign in (7.4.28).) Next the sign ∗2 (i) is ∗2 (i) = deg y(2;1) deg x(3;1) + deg x(3;1) c1 c2 c2 + (deg y(2;1) + 1) deg y(2;2) + (deg y(2;1) + 1)(deg y(2;2) + 1). c1 c1 c1 c1 (2;1)

(2;2)

Let ∗2 (i) be the formula obtained by exchanging the role of yc1 and yc1 in the above formula. We prove the second equality by using (7.4.37), where ∗3 (i) is ∗3 (i) = ∗2 (i) + deg y(2;1) deg y(2;2) = ∗1 (i). c1 c1 The third equality follows. By (7.4.27’) and Example A3.4 we have (−1)deg y+1 l1 (ˆqo (y))(x)

 (3;1)  (3;1) (3;3) (−1)deg y deg xc2 +deg xc2 m(x(3;1) ⊗ ˆqo (y ⊗ x(3;2) = c2 c2 ) ⊗ xc2 ) c2 ∈C2

+



(−1)deg y+deg



x(3;1) o c2

(3;3) ˆq (y ⊗ (x(3;1) ⊗ m(x(3;2) c2 c2 ) ⊗ xc2 )).

c2 ∈C2

(We use deg ˆqo (y) ≡ deg(y) + 1 mod 2 in the above calculation.) Therefore, using (7.4.33) and the identity

y(2;1) ⊗ y(2;2) = 1 ⊗ y + y ⊗ 1, c1 c1 c1 ∈C1 \C1

we obtain

(7.4.38.2)

(−1)deg y+1 l1 (ˆqo (y))(x)

(2;2)  (3;1)  (3;1) (2;1) (2;2) (−1)deg yc1 deg xc2 +deg xc2 +deg yc1 (deg yc1 +1) = c1 ∈C1 \C1 ,c2 ∈C2 (3;3) ˆqo (y(2;1) ⊗ (x(3;1) ⊗ ˆqo (y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 )).

(7.4.35) follows from (7.4.36), (7.4.38.1), (7.4.38.2) and + 1) deg y(2;2) + deg y ≡ deg y(2;1) + deg y(2;1) deg y(2;2) (deg y(2;1) c1 c1 c1 c1 c1

mod 2. 

We remark that (7.4.35) is almost the same as (3.8.33). The only difference (2;1) (2;2) is the sign (−1)deg y in the first term and (−1)deg yc1 deg yc1 in the second term. We can eliminate them by taking (7.4.39.1)

q,k (y1 ⊗ · · · ⊗ y )(x) = qo,k (y ⊗ · · · ⊗ y1 )(x)

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

601

in place of qo,k and (7.4.39.2)

 1 ⊗ · · · ⊗ yk ) = δ(y

k

(−1)deg y1 +···+deg yi−1 y1 ⊗ · · · ⊗ δyi ⊗ · · · ⊗ yk

i=1

in place of δo . In fact, if we put Op(y1 ⊗ · · · ⊗ y ) = y ⊗ · · · ⊗ y1 , then we have  Op(δo (y)) = (−1)deg y (δ(Op(y))).

(7.4.39.3)

Therefore if we put y = Op(y ) to the first term of (7.4.35), it becomes   ) ⊗ x). q,k (δ(y On the other hand, we have

(2;1) ΔOp(y) = Op(y(2;2) c1 ) ⊗ Op(yc1 ) c1

=



(2;1)

(−1)deg yc1

deg y(2;2) c1

(2;2) Op(y(2;1) c1 ) ⊗ Op(yc1 ).

c1

Here we use the graded cocommutativity to show the second equality. Therefore if we put y = Op(y ) to the second term of (7.4.35), it becomes

(2;2)  (3;1)  (3;1) (2;1) (−1)deg yc1 deg xc2 +deg xc2 +deg yc1 c1 ∈C1 ,c2 ∈C2 (3;3) q(y(2;1) ⊗ (x(3;1) ⊗ q(y(2;2) ⊗ x(3;2) c1 c2 c1 c2 ) ⊗ xc2 )).

Therefore (7.4.35) implies (3.8.33). Thus we have: Corollary 7.4.40. There exists a bijection between the set of the following objects. (1) A strict and gapped filtered L∞ homomorphism o ˆ ˆq : C(M ; Λ0,nov )[1] → DerB(C(L; Λ0,nov )[1]).

(2) A family of operators q,k,β as in Theorem 3.8.32 that satisfy (3.8.33) and (3.8.34.2). We remark that we equip C(M ; Λ0,nov )[1] with the trivial filtered L∞ structure. ˆ 7.4.4. Construction of q III: the homotopy invariance of Der(B(C[1]),   ˆ B(C[1])). Let (C, m), (C , m ) be gapped filtered A∞ algebras. We regard differential graded Lie algebras DerB(C[1]), DerB(C  [1]) as filtered L∞ algebras. Theorem 7.4.41. If (C, m) is gapped homotopy equivalent to (C  , m ) as filtered A∞ algebras, then DerB(C[1]) is strict and gapped homotopy equivalent to DerB(C  [1]) as a filtered L∞ algebra. The unital version also holds.

602

7.

TRANSVERSALITY

Remark 7.4.42. (1) This theorem seems to be known to experts, at least for the unfiltered version. In fact Theorem 7.4.41 may be regarded as a systematic way to say the homotopy equivalence of deformation functor, which was known before. (2) If C has an inner product and m satisfies the cyclic symmetry (see Subsection 3.6.4) in addition, then we can define the structure of differential graded Lie algebra on Homfilt (B cyc (C[1]), Λ0,nov ). Then we also have results similar to Theorem 7.4.41. In fact we can also prove its ‘higher genus analog’, at least in the case when mk = 0 for k = 1, 2. This will be related to the higher genus generalization of the story of this book, to string topology [ChSu99], and to perturbative ChernSimons gauge theory [AxSi91I,II]. We will explore these relationships elsewhere. (See also Remark 7.4.97.) (3) It seems that there is no natural differential graded Lie algebra homomorphisms between DerB(C[1]) and DerB(C  [1]) that induces a homotopy equivalence. So it seems inevitable to use the language of L∞ algebra to prove a result similar to Theorem 7.4.41. Proof. If C and C  are both canonical, then it follows from Proposition 5.4.5 ˆ ˆ  [1]) that is both a cochain that there exists an isomorphism ϕˆ : B(C[1]) → B(C map and a coalgebra homomorphism. It follows that DerB(C[1]) is isomorphic to DerB(C  [1]) as differential graded Lie algebras and hence the theorem holds in this case. Therefore it suffices to prove the theorem for the case C  = H(C, m1 ) ⊗Q Λ0,nov is a canonical model of C. (Here C ⊗Q Λ0,nov = C.) Let H = H(C, m1 ), H = H ⊗Q Λ0,nov and define a filtered A∞ structure on H as in the proof of Theorem 5.4.2. We will construct a filtered L∞ homomorphism (7.4.43)

ˆ go : DerB(C[1]) → DerB(H[1]).

By definition go can be identified with a family of filtered homomorphisms (7.4.44)

ˆk (H[1]) → H[1] go,k : E ((DerB(C[1]))[1]) ⊗ B

of degree 1. go,k is further decomposed into

go,k = T E(β) eμ(β)/2 go,k,β β∈G

where G = {(E(β), μ(β)) ∈ R≥0 × 2Z | β ∈ G} is a submonoid of R≥0 × 2Z satisfying Condition 3.1.6 and (7.4.45)

go,k,β : E (DerB(C[1])[1]) ⊗ Bk (H[1]) → H[1]

is a Q-linear map of degree 1. (We assume C and H are G-gapped as before.) The definition of (7.4.46) will be given by a summation over trees, which is similar to Subsection 5.4.4. Let Ya ∈ Hom(Bka (C[1]), C[1]) ⊂ Hom(B(C[1]), C[1]) ∼ = DerB(C[1]). We put (7.4.46)

Sym(Y1 , · · · , Y ) =

1

(−1)∗(σ) Yσ(1) ⊗ · · · ⊗ Yσ() ! σ∈S

7.4. DETAILS OF THE CONSTRUCTION OF THE OPERATORS P, Q AND R

where ∗(σ) =



603

deg Yi deg Yj .

iσ(j)

We will define a family g,k,β of maps by describing its image g,k,β (Sym(Y1 , · · · , Y ))(x) ∈ H[1] for x ∈ Bk (H[1])). Hereafter we omit Sym( ) and write go,k,β (Y1 , · · · , Y )(x) for simplicity. We now explain how we modify g to obtain go later in Proposition 7.4.85. Consider a connected planar tree Γ = (|Γ|, Y (·), β(·)) decorated with some extra data described below: (7.4.47.1)

The set of vertices C 0 (Γ) of Γ is divided into a disjoint union 0 0 (Γ) ∪ CM (Γ) ∪ CY0 (Γ). C 0 (Γ) = Cext

0 0 (Γ), CM (Γ), CY0 (Γ) is called an exterior vertex, a vertex of type M A vertex in Cext and a vertex of type Y , respectively. (7.4.47.2) Each exterior vertex has exactly one edge. They are called exterior edges. Other edges are called interior. (7.4.47.3) There are exactly k + 1 exterior vertices. They are enumerated by 0, 1, · · · , k. This enumeration respects the cyclic order induced by the counterclockwise orientation of the plane. (7.4.47.4) To each vertex v of type M , we assign an element β(v) ∈ G. If β(v) = β0 = (0, 0), then we require v to have at least 3 edges. (7.4.47.5) There exist exactly  vertices v of type Y . To each such v we assign Y (v), one among Y1 , · · · , Y . And this assignment v → Y (v) induces a bijection : CY0 (Γ) ∼ = {Y1 , · · · , Y }. (7.4.47.6) If v is of type Y and Y (v) = Ya , then we denote by ka + 1 the number of edges adjacent to v.

We put Y = (Y1 , · · · , Y ). Definition 7.4.48. We denote by G(k, Y) the set of all Γ satisfying (7.4.47). We remark that in case  = 0 (that is Y is empty) G(k, ∅) coincides with the set G+ k+1 of the trees we used in Subsection 5.4.4. Now for each Γ ∈ G(k, Y) we define two homomorphisms f(Γ) : Bk (H[1]) → C[1], g(Γ) : Bk (H[1]) → H[1]   with deg f(Γ) = (deg Ya − 1) and deg g(Γ) = 1 + (deg Ya − 1) by induction over the number of interior vertices of Γ. (Here an interior vertex is by definition a vertex which is one of the types, M or Y .) As before we fix an embedding H → C and denote by H its image. We then choose an idempotent ΠH : C → C, Π2H = ΠH with Image ΠH = H which induces a decomposition C = Image ΠH ⊕ Ker ΠH .

604

7.

TRANSVERSALITY

We identify H with the subspace H of C and consider a chain-homotopy G : C → C as in Lemma 5.4.28, i.e., the map satisfying m1 ◦ G + G ◦ m1 = −(1 − ΠH ). Using this identification, we will also regard f(Γ) as a map defined on C by setting its value to be zero on Ker ΠH . We first consider the case where there is only one interior vertex v. Let k + 1 be the number of edges of v. If v is of type Y , we put  f(Γ) = G ◦ Y (v) : Bk (H[1]) → C[1] (7.4.49.1) g(Γ) = ΠH ◦ Y (v) : Bk (H[1]) → H[1]. We next consider the case where v is of type M . We then put  f(Γ) = G ◦ mk,β(v) : Bk (H[1]) → C[1] (7.4.49.2) g(Γ) = ΠH ◦ mk,β(v) : Bk (H[1]) → H[1]. Here mk,β are defined by mk =



T E(β) eμ(β)/2 mk,β .

β∈G

Now we consider the case when Γ has at least two interior vertices. Let v0 be the 0-th exterior vertex. It has a unique edge e0 . We define v0 by ∂e0 = {v0 , v0 }. We remove e0 , v0 , v0 from Γ and obtain Γ1 , . . . , Γn as the closures of the connected component of the complement. We have Γi ∈ G(mi , Y(i) ). Here m1 + · · · + mn = k and ⎧ n  ⎪ ⎪ ⎪ Y(i) = Y if v0 is of type M, ⎪ ⎨ i=1 (7.4.50) n ⎪  ⎪ ⎪ ⎪ Y(i) ∪ {Y (v0 )} = Y if v0 is of type Y. ⎩ i=1

Here the left hand sides of (7.4.50) are disjoint union. We enumerate Γ1 , . . . , Γn such that it respects counter-clockwise orientation of the plane. (See Figure 5.4.2.) We now put  ˆ ···⊗ ˆ  f(Γn )) f(Γ) = G ◦ Y (v0 ) ◦ (f(Γ1 ) ⊗ (7.4.51) ˆ ···⊗ ˆ  f(Γn )) g(Γ) = ΠH ◦ Y (v0 ) ◦ (f(Γ1 ) ⊗ if v0 is of type Y, and  ˆ ···⊗ ˆ  f(Γn )) f(Γ) = G ◦ mn,β(v0 ) ◦ (f(Γ1 ) ⊗ (7.4.52) ˆ ···⊗ ˆ  f(Γn )) g(Γ) = ΠH ◦ mn,β(v0 ) ◦ (f(Γ1 ) ⊗ ˆ  is defined by if v0 is of type M. Here ⊗ ˆ  An )(u1 ⊗ · · · ⊗ un ) = (−1)∗ A1 (u1 ) ⊗ · · · ⊗ An (un ), ˆ ···⊗ (A1 ⊗  where ∗ = i