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Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part II [2]
 0821848372, 9780821848371

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AMS/IP

Studies in Advanced Mathematics S.-T. Yau, Series Editor

Lagrangian Intersection Floer Theory Anomaly and Obstruction, Part II

Kenji Fukaya Yong-Geun Oh Hiroshi Ohta Kaoru Ono

American Mathematical Society • International Press

Lagrangian Intersection Floer Theory

AMS/IP

Studies in Advanced Mathematics Volume 46.2

Lagrangian Intersection Floer Theory Anomaly and Obstruction, Part II

Kenji Fukaya Yong-Geun Oh Hiroshi Ohta Kaoru Ono

American Mathematical Society”International Press

Shing-tung Yau, General Editor 2010 Mathematics Subject Classification. Primary 53D12, 53D40; Secondary 14J32, 81T30, 37J10, 18D50, 55P62.

For additional information and updates on this book, visit www.ams.org/bookpages/amsip-46

Library of Congress Cataloging-in-Publication Data Lagrangian intersection floer theory : anomaly and obstruction / Kenji Fukaya . . . [et al.]. p. cm. — (AMS/IP studies in advanced mathematics ; v. 46) Includes bibliographical references and index. ISBN 978-0-8218-4831-9 (set : alk. paper) – ISBN 978-0-8218-4836-4 (pt. 1 : alk. paper) – ISBN 978-0-8218-4837-1 (pt. 2 : alk. paper) 1. Floer homology. 2. Lagrangian points. 3. Symplectic geometry. I. Fukaya, Kenji, 1959– QA665.L34 2009 516.36—dc22 2009025925

$06VRIWFRYHU,6%1 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society and International Press. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the International Press home page at URL: http://www.intlpress.com/ 10 9 8 7 6 5 4 3 2

20 19 18 17 16

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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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

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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

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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/046.2/01

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

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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

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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].

7.1. CONSTRUCTION OF THE KURANISHI STRUCTURE

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.

TRANSVERSALITY

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.

TRANSVERSALITY

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 )



; (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



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)



(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)



}

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



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



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



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

7.

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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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−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.

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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