Rational Homotopy Theory II [Reprint ed.] 9814651427, 9789814651424

Table of contents :
Content: Sullivan Model and Rationalization of a Non-Simply Connected Space
Homotopy Lie Algebra of a Space and Fundamental Group of the Rationalization, Model of a Fibration
Holonomy Operation in a Fibration
Malcev Completion of a Group and Examples
Lusternik-Schnirelmann Category
Depth of a Sullivan Lie Algebra
Growth of Rational Homotopy Groups
Structure of Rational Homotopy Lie Algebras
Weighted Lie Algebras

Citation preview

RATIONAL HOMOTOPY THEORY II

9473_9789814651424_tp.indd 1

27/1/15 4:35 pm

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

RATIONAL HOMOTOPY THEORY II Yves Félix Catholic University of Louvain, Belgium

Steve Halperin University of Maryland, USA

Jean-Claude Thomas Université d’Angers, France

World Scientific NEW JERSEY

•

LONDON

9473_9789814651424_tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

27/1/15 4:35 pm

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Félix, Y. (Yves) Rational homotopy theory II / by Yves Félix (Catholic University of Louvain, Belgium), Steve Halperin (University of Maryland, USA), Jean-Claude Thomas (Universite d'Angers, France). pages cm Sequel to Rational homotopy theory (2001), but self contained--Introduction, following table of contents. Includes bibliographical references and index. ISBN 978-9814651424 (hardcover : alk. paper) 1. Homotopy theory. I. Halperin, Stephen. II. Thomas, J.-C. (Jean-Claude) III. Title. IV. Title: Rational homotopy theory two. QA612.7.F47 2015 514'.24--dc23 2014046652

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore

EH - Rational homotopy theory II.indd 1

28/1/2015 2:29:30 PM

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Preface

Sullivan’s seminal paper, Infinitesimal Computations in Topology, includes the application of his techniques to non-simply connected spaces, and these ideas have been used frequently by other authors. Our objective in this sequel to [18] is to provide a complete description with detailed proofs of this material. This then provides the basis for new results, also included, and which we complement with recent advances for simply-connected spaces. There do remain many interesting unanswered questions in the field, which hopefully this text will make it easier for others to resolve. This monograph may also be seen as an exploration of the minimal models introduced by Sullivan, to whom this work is gratefully dedicated.

v

page v

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Contents

Preface

v

Introduction

xi

1.

Basic definitions and constructions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

2.

Graded algebra . . . . . . . . . . . . . . . . . . . . . Differential graded algebra . . . . . . . . . . . . . . . Simplicial sets . . . . . . . . . . . . . . . . . . . . . . Polynomial differential forms . . . . . . . . . . . . . Sullivan algebras . . . . . . . . . . . . . . . . . . . . The simplicial and spatial realizations of a Λ-algebra Homotopy and based homotopy . . . . . . . . . . . . The homotopy groups of a minimal Sullivan algebra

1 . . . . . . . .

. . . . . . . .

. . . . . . . .

1 6 8 14 18 24 30 35

Homotopy Lie algebras and Sullivan Lie algebras

45

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

45 48 54 56 63 71 75

The homotopy Lie algebra of a minimal Sullivan algebra . The fundamental Lie algebra of a Sullivan 1-algebra . . . Sullivan Lie algebras . . . . . . . . . . . . . . . . . . . . . Primitive Lie algebras and exponential groups . . . . . . . The lower central series of a group . . . . . . . . . . . . . d L . . . . . . . . . . . The linear isomorphism (∧sV )# ∼ =U The fundamental group of a 1-finite minimal Sullivan algebra The homology Hopf algebra of a 1-finite minimal Sullivan algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The action of GL on πn (| ∧ V, d|, ∗) . . . . . . . . . . . . . 2.10 Formal Sullivan 1-algebras . . . . . . . . . . . . . . . . . . vii

81 84 87

page vii

January 8, 2015

14:59

9473-Rational Homotopy II

viii

3.

Rational Homotopy Theory II

Fibrations and Λ-extensions

91

3.1 3.2

91

3.3 3.4 3.5 3.6 3.7 4.

. . . . . .

main theorem . . . . . . . . . . . . . holonomy action of π1 (Y, ∗) on π∗ (F ) Sullivan model of a universal covering Sullivan model of a spatial realization

. . . . . . . . space . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

The loop cohomology coalgebra of (∧V, d) . . . . . . The transformation map ηL . . . . . . . . . . . . . . The graded Hopf algebra, H∗ (| ∧ U |; Q) . . . . . . . Connecting Sullivan algebras with topological spaces

Sullivan spaces . . . . . . . . . . . . . . . . . . . The classifying space BG . . . . . . . . . . . . . The Sullivan 1-model of BG . . . . . . . . . . . . Malcev completions . . . . . . . . . . . . . . . . . The morphism m|∧V,d| : (∧V, d) → AP L (| ∧ V, d|) When BG is a Sullivan space . . . . . . . . . . .

145 160 164 166 169

. . . .

. . . .

. . . .

Sullivan spaces 7.1 7.2 7.3 7.4 7.5 7.6

117 125 130 133 137 142 145

Loop spaces and loop space actions 6.1 6.2 6.3 6.4

7.

Holonomy of a fibration . . . . . . . . . . . . . . . . . . Holonomy of a Λ-extension . . . . . . . . . . . . . . . . Holonomy representations for a Λ-extension . . . . . . . Nilpotent and locally nilpotent representations . . . . . Connecting topological and Sullivan holonomy . . . . . . The holonomy action on the homotopy groups of a fibre

The The The The

93 95 101 107 111 114 117

The model of the fibre is the fibre of the model 5.1 5.2 5.3 5.4

6.

Fibrations, Serre fibrations and homotopy fibrations . . . The classifying space fibration and Postnikov decompositions of a connected CW complex . . . . . . . . . . . . . . Λ-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of minimal Sullivan models . . . . . . . . . . . . Uniqueness of minimal Sullivan models . . . . . . . . . . . The acyclic closure of a minimal Sullivan algebra . . . . . Sullivan extensions and fibrations . . . . . . . . . . . . . .

Holonomy 4.1 4.2 4.3 4.4 4.5 4.6

5.

RHT3

169 176 183 187 195

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

195 199 205 213 218 222

page viii

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Contents

8.

ix

Examples

227

8.1 8.2 8.3 8.4 8.5 8.6

Nilpotent and rationally nilpotent groups . . . . . Nilpotent and rationally nilpotent spaces . . . . . . The groups Z# · · · #Z . . . . . . . . . . . . . . . . Semidirect products . . . . . . . . . . . . . . . . . Orientable Riemann surfaces . . . . . . . . . . . . . The classifying space of the pure braid group Pn is livan space . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Heisenberg group . . . . . . . . . . . . . . . . 8.8 Seifert manifolds . . . . . . . . . . . . . . . . . . . 8.9 Arrangement of hyperplanes . . . . . . . . . . . . . 8.10 Connected sum of real projective spaces . . . . . . 8.11 A final example . . . . . . . . . . . . . . . . . . . .

9.

. . . . . a . . . . . .

. . . . . . . . . . . . . . . Sul. . . . . . . . . . . . . . . . . .

Lusternik-Schnirelmann category 9.1 9.2 9.3 9.4 9.5 9.6 9.7

The LS category of topological spaces and cochain algebras . . . . . . . . . . . . . . . The mapping theorem . . . . . . . . . . . . Module category and the Toomer invariant cat = mcat . . . . . . . . . . . . . . . . . . cat = e(−)# . . . . . . . . . . . . . . . . . . Jessup’s Theorem . . . . . . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . .

237 238 239 240 241 242 245

commutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. Depth of a Sullivan algebra and of a Sullivan Lie algebra 10.1 10.2 10.3 10.4 10.5 10.6 10.7

227 227 229 231 232

Ext, Tor and the Hochschild-Serre spectral sequence The depth of a minimal Sullivan algebra . . . . . . . The depth of a Sullivan Lie algebra . . . . . . . . . . Sub Lie algebras and ideals of a Sullivan Lie algebra Depth and relative depth . . . . . . . . . . . . . . . . The radical of a Sullivan Lie algebra . . . . . . . . . Sullivan Lie algebras of finite type . . . . . . . . . .

11. Depth of a connected graded Lie algebra of finite type

245 248 249 250 260 261 265 267

. . . . . . .

. . . . . . .

. . . . . . .

267 272 276 279 287 295 298 301

11.1 Summary of previous results . . . . . . . . . . . . . . . . . 301 11.2 Modules over an abelian Lie algebra . . . . . . . . . . . . 304 11.3 Weak depth . . . . . . . . . . . . . . . . . . . . . . . . . . 307

page ix

January 8, 2015

14:59

9473-Rational Homotopy II

x

RHT3

Rational Homotopy Theory II

12. Trichotomy 12.1 12.2 12.3 12.4 12.5 12.6

313

Overview of results . . . . . . . . . . . . . . . . . . . The rationally elliptic case . . . . . . . . . . . . . . . The rationally hyperbolic case . . . . . . . . . . . . . The gap theorem . . . . . . . . . . . . . . . . . . . . Rationally infinite spaces of finite category . . . . . . Rationally infinite CW complexes of finite dimension

. . . . . .

. . . . . .

. . . . . .

13. Exponential growth 13.1 13.2 13.3 13.4 13.5 13.6

The invariant log index . . . . . . . . Growth of graded Lie algebras . . . . Weak exponential growth and critical Approximation of log index L . . . . Moderate exponential growth . . . . Exponential growth . . . . . . . . . .

329 . . . . . . . . degree . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

14. Structure of a graded Lie algebra of finite depth 14.1 14.2 14.3 14.4 14.5 14.6

Introduction . . . . . . . . . . . . . . The spectrum . . . . . . . . . . . . . Minimal sub Lie algebras . . . . . . The weak complements of an ideal . L-equivalence . . . . . . . . . . . . . The odd part of a graded Lie algebra

313 317 317 318 319 325

. . . . . .

. . . . . .

. . . . . .

15. Weight decompositions of a Sullivan Lie algebra

331 333 337 343 350 358 367

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

367 368 372 377 380 387 389

15.1 Weight decompositions . . . . . . . . . . . . . . . . . . . . 389 15.2 Exponential growth of L . . . . . . . . . . . . . . . . . . . 393 15.3 The fundamental Lie algebra of 1-formal Sullivan algebra 395 16. Problems

401

Bibliography

405

Index

409

page x

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

Rational homotopy theory assigns to topological spaces invariants which are preserved by continuous maps f for which H∗ (f ; Q) is an isomorphism. The two standard approaches of the theory are due respectively to Quillen [58] and Sullivan [61], and [62]. Each constructs from a class of CW complexes X an algebraic model MX , and then constructs from MX a CW complex XQ , together with a map ϕX : X → XQ . Both H∗ (XQ ; Z) and πn (XQ ) are rational vector spaces, and with appropriate hypotheses H∗ (ϕX ) : H∗ (X) ⊗ Q → H∗ (XQ ; Z) , πn (ϕX ) : πn (X) ⊗ Q → πn (XQ ),

and

n ≥ 2,

are isomorphisms. In each case the model MX belongs to an algebraic homotopy category, and a homotopy class of maps f : X → Y induces a homotopy class of morphisms Mf : MX → MY (in Quillen approach) and a homotopy class of morphisms Mf : MY → MX (in Sullivan’s approach). These are referred to as representatives of f . In Quillen’s approach, X is required to be simply connected and MX is a rational differential graded Lie algebra which is free as a graded Lie algebra. In this case H∗ (ϕX ) and π≥2 (ϕX ) are always isomorphisms. Here, as in [18], we adopt Sullivan’s approach, and in this Introduction provide an overview of the material in the monograph, together with brief summaries of the individual Chapters. Sullivan’s approach associates to each path connected space X a cochain algebra MX of the form (∧V, d) in which the free commutative graded algebra ∧V is generated by V = V ≥1 , and V = ⊕m ∧m V with ∧m V = xi

page xi

January 8, 2015

xii

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

V ∧ · · · ∧V (m factors). Additionally, each ∧V ≤k is preserved by d, and d also satisfies a “nilpotence” condition: (∧V, d) is called a minimal Sullivan algebra. A minimal Sullivan algebra determines a simplicial set h∧V, di with spatial realization | ∧ V, d|, and when (∧V, d) is the model of the CW complex X, this determines (up to homotopy) the map ϕX : X → XQ = | ∧ V, d| . This approach makes non-simply connected spaces accessible to rational homotopy theory. For example, if H1 (X; Q) is finite dimensional then π1 (XQ ) is the Malcev completion of π1 (X): π1 (ϕX ) induces an isomorphism limn π1 (X)/π1n (X) ⊗ Q ←−

∼ =

/ π1 (XQ ) ,

where (π1n (X)) denotes the lower central series of π1 (X). On the other hand, this approach also comes at the cost of a finiteness condition: If X is simply connected then H∗ (ϕX ) and π≥2 (ϕX ) are isomorphisms if and only if H∗ (X; Q) is a graded vector space of finite type. In the case of non-simply connected CW complexes X, two additional ingredients are required for rational homotopy theory: first, the action e Q) of the by covering transformations of π1 (X) on the cohomology H ∗ (X; universal covering space of X; second, a Sullivan representative ψ for a classifying map mapping X to the classifying space for π1 (X) and inducing e Q) has finite type, then an isomorphism of fundamental groups. If H ∗ (X; the groups π≥2 (X) ⊗ Q can be computed from a minimal Sullivan model of e and if the action of π1 (X) on H ∗ (X; e Q) is nilpotent, then this Sullivan X, model can be computed from ψ. Minimal Sullivan algebras (∧V, d) are equipped with a homotopy theory and a range of invariants analogous to those which arise in topology. Key among these are the graded homotopy Lie algebra L = {Lp }p≥0 and, when dim H 1 (∧V, d) < ∞, the group GL . Here Lp = Hom(V p+1 , Q), the Lie bracket is dual to the component d1 : V → ∧2 V of d, and an exponential map converts L0 to GL . The group GL acts by conjugation in L and also in H(∧V ≥2 , d) where (∧V ≥2 , d) is obtained by dividing by V 1 ∧ ∧ V . A third key invariant is the Lusternik-Schnirelmann category, cat (∧V, d), defined as the least m for which (∧V, d) is a homotopy retract of (∧V / ∧>m V, d). When (∧V, d) is the minimal Sullivan model of a CW complex X such that dim H 1 (X; Q) < ∞, then GL = π1 (XQ ), L≥1 = π≥1 (ΩXQ ), and there e Q) equivariant via π1 (ϕX ) with is a natural map H(∧V ≥2 , d) → H(X; respect to the actions of GL and the action by covering transformations of π1 (X). Finally, as in the simply connected case ( [18]) cat (∧V, d) ≤ cat X.

page xii

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xiii

Sullivan models (∧V, d) are constructed via a functor, AP L (inspired by the differential forms on a manifold) from spaces to rational cochain algebras, for which H(AP L (X)) and H ∗ (X; Q) are naturally isomorphic algebras. Then (∧V, d) is the unique (up to isomorphism) minimal Sullivan algebra admitting a morphism m : (∧V, d) → AP L (X) for which H(m) is an isomorphism. Moreover, any morphism ϕ : (∧W, d) → AP L (X) from an arbitrary minimal Sullivan algebra determines by adjunction a homotopy class of maps |ϕ| : X → | ∧ V, d|; in particular, |m| is homotopic to the map ϕX above. This second step can be applied to construct a minimal Sullivan model (∧V, d) → (A, d) for any commutative cochain algebra satisfying H 0 (A, d) = lk. While these may not be Sullivan models of a topological space, the homotopy machinery of minimal Sullivan algebras is established independently of topology, and so can be applied in this more general context. In particular, minimal Sullivan algebras become a valuable tool in the study of graded Lie algebras E = E≥0 with lower central series denoted by (E n ), provided that E0 acts nilpotently by the adjoint representation in each Ei , i ≥ 1, and that dim E0 /[E0 , E0 ] < ∞ , dim Ei < ∞ , i ≥ 1 , and ∩n E0n = 0 . Such Lie algebras are called Sullivan Lie algebras. Note that we may have dim E0 = ∞. For a Sullivan Lie algebra, E, limn C ∗ (E/E n ) is a minimal Sullivan −→ algebra, called the associated Sullivan algebra for E, and limn E/E n is ←− its homotopy Lie algebra. (Here C ∗ (−) is the classical Cartan-ChevalleyEilenberg cochain algebra construction.) In the reverse direction, if (∧V, d) is any minimal Sullivan algebra for which dim H 1 (∧V, d) < ∞ then its homotopy Lie algebra is a Sullivan Lie algebra whose associated Sullivan algebra is (∧V, d1 ) with d1 the component of d mapping V to ∧2 V . In summary, the interplay between spaces, minimal Sullivan algebras and graded Lie algebras is illustrated by the diagram Graded Lie algebras O L

Spaces

AP L

lim C ∗ (L/Ln )

− →n

/ Minimal Sullivan algebras

|.|

/ Spaces

page xiii

January 8, 2015

14:59

9473-Rational Homotopy II

xiv

RHT3

Rational Homotopy Theory II

A crucial technical tool in the theory of minimal Sullivan algebras is the conversion of cochain algebra morphisms to Λ-extensions (∧V, d) → (∧V ⊗ ∧Z, d) in which (∧V, d) is a minimal Sullivan algebra and d : Z p → ∧V ⊗ ∧Z ≤p satisfies a “nilpotence” condition; when V 1 6= 0 it may happen that Z 0 6= 0. Division by ∧+ V ⊗ ∧Z gives a quotient cochain algebra (∧Z, d) and the Λ-extension determines holonomy representations of L and, if dim H 1 (∧V, d) < ∞, of GL in H(∧Z, d). These Λ-extensions are the Sullivan analogues of fibrations, and (∧Z, d) is the Sullivan analogue of the fibre. The analogy is not merely abstract: suppose / (∧V ⊗ ∧Z, d) (∧V, d) 

α

AP L (Y )

AP L (p)



β

/ AP L (X)

is a commutative diagram in which Y is a CW complex and p is the projection of a fibration with fibre F . Then β factors to give a morphism γ : (∧Z, d) → AP L (F ), and, in this setting H(γ) : H(∧Z, d) → H ∗ (F ; Q) is equivariant via π1 (|ψ|) with respect to the holonomy representation of GL in H(∧Z, d) and π1 (Y ) in H(F ; Q). There are two important examples of Λ-extensions. First, if (∧V, d) is a minimal Sullivan algebra then (∧V 1 , d) → (∧V, d) is a Λ-extension, the Sullivan analogue of a classifying map for a CW complex X. If this morphism is a Sullivan representative for the classifying map, and if H ∗ (X; Q) has finite type and the covering space action of π1 (X) is nilpotent, then X is a Sullivan space and the quotient (∧V ≥2 , d) is a minimal Sullivan model e Many classical examples are Sullivan spaces, including all closed for X. orientable Riemann surfaces. Second, if (∧V, d) is any minimal Sullivan algebra, converting the augmentation (∧V, d) → lk yields a Λ-extension (∧V, d) → (∧V ⊗ ∧U, d) with H(∧V ⊗ ∧U, d) = lk; this is the acyclic closure of (∧V, d). Here the differential in the quotient ∧U is zero and so the holonomy representation is a representation of the homotopy Lie algebra L of ∧V in ∧U . The Λ-extension also determines a diagonal ∆ : ∧U → ∧U ⊗ ∧U which makes ∧U into a commutative graded Hopf algebra, and there is a natural homomorphism ηL : U L → Hom (∧U, lk) of graded algebras which converts right multiplication by L to the dual of the holonomy representation.

page xiv

January 8, 2015

14:59

9473-Rational Homotopy II

Introduction

RHT3

xv

In particular, in the case of the acyclic closure of the associated Sullivan algebra of a Sullivan Lie algebra, E, this yields a morphism U E → Hom(∧U, lk) which identifies ∧U as a sort of “predual” of U E. In this setting we define E depth E = least p (or ∞) such that TorU p (lk, ∧U ) 6= 0 .

This generalizes the definition in [18] for Lie algebras E = E≥1 of finite type, because in this case U E = (∧U )# and Ext∗U E (lk, U E) is the dual of E TorU ∗ (lk, ∧U ). The invariant depth E plays an important role in the growth and structure theorems for the homotopy Lie algebra of a simply connected space of finite LS category. These were established after [18] appeared, and so are included here. The extent to which they may be generalized to non-simply connected spaces remains an open question. Although the present volume is a sequel to [18] it can be read independently, since all the definitions, conventions and results are stated here, whether or not they appear in [18], although we do quote proofs from [18] whenever this is possible. As in [18], we work where possible over an arbitrary field lk of characteristic zero, and with rare exceptions, definitions and notation are unchanged from [18]; in particular, V # denotes the dual of a graded vector space V . Also, for simplicity, the cohomology algebra H ∗ (X; lk) of a space X is denoted by H(X). That said, by and large the material in this monograph either is a nontrivial extension of, or is in addition to, the content of [18]. In particular, it includes: • the extension of Sullivan models from simply connected spaces to path connected spaces with general (not necessarily nilpotent) fundamental group G. • an analysis of L0 , the fundamental Lie algebra of (∧V, d). • a description of the holonomy action of π1 (B) on H ∗ (F ) in terms of Sullivan models. • a complete proof that under the most general possible hypotheses the Sullivan fibre associated with a fibration B ←− E is the Sullivan model of the fibre F of p, even when B is not simply connected. • an analysis of the minimal Sullivan model of a classifying space and the introduction of Sullivan spaces. • the definition of the depth of L for any Sullivan algebra and a homological analysis of its properties extending those provided in [18] when L0 = 0.

page xv

January 8, 2015

14:59

xvi

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

• complete proofs of the growth and structure theorems for the higher rational homotopy groups of a connected CW complex. We now proceed to a more detailed Chapter by Chapter description. Chapter 1: Basic definitions and constructions This Chapter provides the definitions and constructions which serve as the base for the theory. The focus is on three categories : topological spaces, simplicial sets, and commutative cochain algebras, and specifically in the last category on the cochain algebras (∧V, d) described above. Consistent with [18] these are called Sullivan algebras if V = V ≥1 . In general they are called Λ-algebras, and Λ-algebras with V 0 6= 0 appear naturally. A Λ-algebra is minimal if d(V n ) ⊂ ∧V ≤n . Homotopy and based homotopy are defined for continuous maps, morphisms of simplicial sets, and for cochain algebra morphisms from a Λalgebra; the various constructions are shown to preserve homotopy and based homotopy (Theorem 1.1, p. 33). In particular we show that a morphism from a Λ-algebra lifts up to (based) homotopy through a quasiisomorphism. The principal constructions in Chapter 1 then include the cochain algebra AP L (X) of polynomial differential forms on the singular simplices of a topological space X, and the minimal Sullivan model of a commutative cochain algebra, which is shown to be unique up to isomorphism. On the other hand, the homotopy groups πn (∧V, d) of a minimal Sullivan algebra are defined as follows:  n # (V ) , n≥2 πn (∧V, d) = GL , n = 1, where GL is the exponential group of the fundamental Lie algebra L0 = (V 1 )# . It is defined in Chapter 2, but only when dim H 1 (∧V, d) < ∞. A morphism ϕ : (∧V, d) → AP L (Y ) induces homomorphisms πn (ϕ) : πn (Y ) → πn (∧V, d), n ≥ 1 (Theorem 1.4, p. 37 for n ≥ 2 and Theorem 2.4, p. 75, for n = 1) with π1 (ϕ) defined only if dim H 1 (∧V, d) < ∞. In particular, associated with a Λ-algebra (∧V, d) is the spatial realization | ∧ V, d| of a simplicial set h∧V, di, and the adjoint to the identity of h∧V, di determines a morphism m|∧V,d| : (∧V, d) → AP L (| ∧ V, d)|). When

page xvi

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xvii

(∧V, d) is a minimal Sullivan algebra, Theorems 1.3, 1.4 (p. 37) and 2.4 (p. 75) show that ∼ =

πn (m|∧V,d| ) : πn (| ∧ V, d|) −→ πn (∧V, d), n ≥ 1 are isomorphisms of groups. Thus the minimal Sullivan model (∧V, d) of a space X computes the cohomology of X and the homotopy groups of | ∧ V, d|. Finally, the spatial realization of the adjoint of mX is a continuous map |mX | : |Sing X| → | ∧ V, d|, which, if X is a CW complex, is homotopy equivalent to the map ϕX : X → | ∧ V, d| described earlier. If X is simply connected then H ∗ (ϕX ; Q) is an isomorphism if and only if H ∗ (X; Q) is a graded vector space of finite type (Theorem 1.5, p. 42). Chapter 2: Homotopy Lie algebras and Sullivan Lie algebras This Chapter introduces the other major category central to the monograph: that of graded Lie algebras L = L≥0 . Associated with a graded Lie algebra are the following constructions: • The lower central series of the ideals Ln+1 = [L, Ln ], and the comple∼ = b b = lim L/Ln . We say L is pronilpotent if L → tion L L. ←−n • The universal enveloping algebra U L, the ideal IL ⊂ U L generated by d L, and the completion U L = limn U L/ILn , where ILn is the nth power of ←− d IL . The completion IbL = limn IL /ILn , is the augmentation ideal of U L. ←− • The Cartan-Chevalley-Eilenberg construction C∗ (L) = (∧sL, ∂), where ∂(sx∧sy) = (−1)deg x s[x, y], and the dual commutative cochain algebra C ∗ (L) = (⊗q (∧q sL)# , ∂ # ). With a minimal Sullivan algebra (∧V, d) is associated the quadratic Sullivan algebra (∧V, d1 ) defined by d1 : V → ∧2 V and d − d1 : V → ∧≥3 V , and its homotopy Lie algebra L = L≥0 , defined by Lk = (V k+1 )#

and hv, [x, y]i = (−1)deg y+1 hd1 v, x, yi.

The Sullivan condition implies that ∩n Ln = 0. Here the finiteness conditions dim H 1 (ΛV, d) < ∞

and

dim V i < ∞,

i≥1

are respectively equivalent to the conditions dim L0 /[L0 , L0 ] < ∞ and

dim Li < ∞,

i ≥ 1,

page xvii

January 8, 2015

14:59

xviii

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

and when they hold L is a Sullivan Lie algebra. Sullivan Lie algebras are important for much of the theory and for the applications: in particular, the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) satisfying the finiteness conditions above is a pronilpotent Sullivan Lie algebra (Theorem 2.1, p. 50). In this case the surjections L → L/Ln+1 are the duals of the inclusions Vn → V defined by V0 = V ∩ ker d1

2 and Vn+1 = V ∩ d−1 1 (Λ Vn ).

If L is a Sullivan Lie algebra then limn C ∗ (L/Ln ) is a quadratic Sulli←− b is the van algebra, the associated quadratic Sullivan algebra for L, and L homotopy Lie algebra for this Sullivan algebra. Now consider an arbitrary graded Lie algebra L = L≥0 . The diagonal b =U d ∆ : L → U L ⊗ U L, x 7→ x ⊗ 1 + 1 ⊗ x extends to a morphism ∆ L→ \ UL ⊗ U L. A sub Lie algebra PL ⊂ IbL and a subgroup GL ⊂ Ud L0 are then defined by b = x⊗1 ˆ + 1⊗x} ˆ , PL = {x ∈ IbL | ∆x

and

b + y) = (1 + y)⊗(1 ˆ + y)}. GL = {1 + y ∈ 1 + IbL | ∆(1 The exponential IbL → 1 + IbL and the logarithm 1 + IbL → IbL restrict to inverse bijections ∼ =

PL0 −→ GL

∼ =

and GL −→ PL0 .

When L is a pronilpotent Sullivan Lie algebra, then the inclusion L0 → U L extends to an isomorphism ∼ =

L 0 → PL 0 . In particular, if L is the homotopy Lie algebra of a minimal Sullivan algebra for which dim H 1 (∧V, d) < ∞, then L0 = (V 1 )# and Theorem 2.4 (p. 75) establishes that ∼ =

π1 (|ΛV, d|) −→ GL is an isomorphism of groups. On the other hand, by definition, ∼ Ln , πn+1 (| ∧ V, d|) = n ≥ 1. Moreover, the Whitehead product defines an action of π1 on πn for any topological space, reducing to conjugation when n = 1. On the other hand, the adjoint representation of L0 in L gives, via the exponential map, an adjoint representation of GL in L and the isomorphisms above are equivariant with respect to the adjoint representations of GL and the Whitehead product in π∗ (| ∧ V, d|).

page xviii

January 8, 2015

14:59

9473-Rational Homotopy II

Introduction

RHT3

xix

For any group G and subsets S, T ⊂ G, [S, T ] is the subgroup generated by the commutators [a, b] = aba−1 b−1 , a ∈ S, b ∈ T . The lower central series for G is the sequence of normal subgroups defined by Gn+1 = [G, Gn ]. In the case G = GL where L = L0 is a pronilpotent Sullivan Lie algebra, ∼ = the exponential map restricts to bijections Ln −→ GnL , n ≥ 1 and thus it follows that GL = limn GL /GnL (Theorem 2.2, p. 64). ←− Chapter 3: Fibrations and Λ-extensions The relative analogue of a Λ-algebra is an inclusion of commutative cochain algebras of the form (B, d) → (B ⊗ ∧Z, d) in which the restriction of the differential to 1⊗Z satisfies a relative version of the Sullivan condition defined in Chapter 1, with the additional condition that d(1 ⊗ Z) ⊂ (lk ⊗ ∧Z) ⊕ (B + ⊗ ∧Z). These are called Λ-extensions and are minimal if d(Z n ) ⊂ B ⊗ ∧Z ≤n . The quotient cochain algebra (∧Z, d) is a Λ-algebra, the Sullivan fibre of the Λ-extension. The Λ-extensions are the Sullivan analogues of fibrations, and satisfy analogous properties. In particular homotopy rel B of morphisms from (B ⊗ ∧Z, d) is defined, and given a commutative diagram of commutative cochain algebras (B, d)  (B ⊗ ∧Z, d)

α

/ (A, d)

ψ

 / (C, d) ,

there is a morphism ϕ : (B ⊗ ∧Z, d) → (A, d) extending α and such that η ◦ ϕ ∼ ψ rel B; moreover, ϕ is unique up to homotopy rel B. Every continuous map is homotopy equivalent to a fibration, which in turn is unique up to fibre homotopy equivalence. The analogue for commutative cochain algebras asserts that any morphism ϕ : (B, d) → (C, d) with H ◦ (B) = H ◦ (C) = lk factors through a minimal Λ-extension as '

(B, d) → (B ⊗ ∧Z, d) −→ (C, d) (Theorem 3.1, p. 102) and that the Λ-extension is unique up to isomorphism (Theorem 3.2, p. 110). This result is significantly more difficult than in the case considered in [18] when H 1 (ϕ) is injective. With that assumption one has Z = Z ≥1 whereas in the general case it can happen that Z 0 6= 0.

page xix

January 8, 2015

14:59

9473-Rational Homotopy II

xx

RHT3

Rational Homotopy Theory II

Thus ∧Z will not have finite type as a vector space and the usual degree arguments do not apply. Two applications of this result are relied on throughout the rest of this monograph. On the one hand, suppose B : (B, d) → lk is an augmented commutative cochain algebra. If H 0 (B) = lk then there is a Λ-extension ∼ =

(B ⊗ ∧U, d) −→ lk, the acyclic closure of (B, d). When (B, d) is a minimal Sullivan algebra ∼ = then d determines the isomorphism of degree 1, α : U −→ V satisfying d(1 ⊗ u) − α(u) ⊗ 1 ∈ V ∧ ∧+ (V ⊗ U ). In particular, the differential in the fibre is 0. The second application converts the analogy with fibrations into a genj p uine comparison. Suppose F −→ X −→ Y is a fibration with X and Y ∼ = path connected, and let mY : (∧V, d) −→ AP L (Y ) be a minimal Sullivan model. Here we obtain a commutative diagram. / (∧V ⊗ ∧Z, d)

(∧V, d) mY



mX '

'

AP L (Y )

lk⊗∧V −

mF



AP L (p)

/ AP L (X)

/ (∧Z, d)

AP L (j)

 / AP L (F ) .

Chapter 5 will show that mF is a quasi-isomorphism under the weakest possible hypotheses; in general this is not the case. For example in the case of the covering projection R → S 1 , F is the discrete set of integers, and (∧v ⊗ ∧Z, d) will be the acyclic closure of (∧v, 0) with deg v = 1. Thus ∧Z = ∧u with deg u = 0, which is not isomorphic to H(F ).

Chapter 4: Holonomy j

p

Suppose F −→ X −→ Y is a fibration defined over a based space (Y, ∗). Then for any loop σ : (I, ∂I) → (Y, ∗) the inclusion of F extends to a map I ×F → X covering σ, which then restricts to a map λσ : {1}×F → F ⊂ X. The correspondence [σ] → [λσ ] is an antihomomorphism to the group of homotopy classes of homotopy equivalences of F , written α 7→ hol α. This is the classical holonomy action of π1 (Y, ∗) on F . Analogously, suppose (∧V, d) → (∧V ⊗∧Z, d) → (∧Z, d¯) is a minimal Λextension of a minimal Sullivan algebra (ΛV, d) with homotopy Lie algebra

page xx

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xxi

L, and for which dimH 1 (∧V, d) < ∞. If x ∈ L then a derivation θ(x) in ¯ is defined by (∧Z, d) θ(x)Φ = −Σhvi , xiθi Φ , where vi is a basis of V 1 and d(1 ⊗ Φ) − 1 ⊗ d¯ Φ − Σvi ⊗ θi Φ ∈ (∧V )≥2 ⊗ ∧Z ,

Φ ∈ ∧Z .

The Sullivan condition for the Λ-extension implies that if x ∈ L0 then θ(x) acts locally nilpotently. It follows that exp θ(x) is defined, and is an automorphism of (∧Z, d¯). Since L0 is a Sullivan Lie algebra, exp : L0 → GL is a bijection, and a homomorphism, hol, from GL to the group, G(∧Z,d) of homotopy classes of automorphisms of (∧Z, d¯) is defined by hol(expL0 x) = [exp θ(x)] (Theorem 4.1, p. 126). This is the holonomy action of GL in (∧Z, d¯). These two definitions of holonomy actions are related as follows. Suppose that (∧V, d)

/ (∧V ⊗ ∧Z, d)

ϕY

ϕX

 AP L (Y )

AP L (p)

/ (∧Z, d) ϕF



/ AP L (X)

AP L (j)

 / AP L (F ) ,

is a commutative diagram connecting the Λ-extension above to the polynomial forms on the fibration Y ←− X ←− F . Then (Theorem 4.2, p. 137) for α ∈ π1 (Y ), ϕF ◦ hol( π1 (ϕY )α) ∼ AP L (hol α) ◦ ϕF , where hol α means any continuous map F → F representing hol α. Finally, recall that the derivations θ(x) of (∧Z, d¯) were defined for all ¯ x ∈ L. In particular they induce derivations θ(x) = H(θ(x)) in H(∧Z, d¯) and it turns out that this is a representation of L in H(∧Z, d¯) : the holonomy representation. In particular, when d¯ = 0 the holonomy representation is just the correspondence x 7→ θ(x). In general, it follows from the Sullivan d condition for the Λ-extension that θ extends to a representation of U L in H(∧Z, d).

page xxi

January 8, 2015

14:59

9473-Rational Homotopy II

xxii

RHT3

Rational Homotopy Theory II

Chapter 5: The model of the fibre is the fibre of the model As in Chapter 3, we consider a commutative diagram (∧V, d) mY



/ (∧V ⊗ ∧Z, d) mX '

'



AP L (Y )

/ AP L (X)

/ (∧Z, d) mF

 / AP L (F )

connecting a minimal Λ-extension of a minimal Sullivan algebra (∧V, d) to the polynomial forms on a fibration. We suppose X and Y are path connected, Y is a CW complex, and mY and mX are quasi-isomorphisms. If the holonomy action of π1 (Y, ∗) is locally nilpotent, and if one of H(Y ) or H(F ) is a vector space of finite type, then mF is a quasi-isomorphism (Theorem 5.1, p. 145). It follows from the Theorems of Chapter 4 that if dim H 1 (∧V, d) < ∞ and if mF is a quasi-isomorphism, then the holonomy action of π1 (Y, ∗) is locally nilpotent. Thus this hypothesis is necessary. Moreover, the Theorem fails for the trivial fibration Y × F if for some p and q dim H p (Y ) and dim H q (F ) are infinite. Thus a finiteness condition is also necessary. A proof of this Theorem was first published in [27] in 1977. The current proof is simpler, and more intuitive.

Chapter 6: Loop spaces and loop space actions Suppose (∧V ⊗ ∧U, d) is the acyclic closure of a minimal Sullivan algebra satisfying our two standard conditions dim H 1 (∧V, d) < ∞

and dim V i < ∞ ,

i ≥ 2.

This then determines a co-associative morphism of graded algebras ∆ : ∧U → ∧U ⊗ ∧U , which satisfies ∆ ◦ θ(x) = (id ⊗ θ(x)) ◦ ∆ where θ : L → Der ∧ U denotes the holonomy representation for the acyclic closure. The morphism ∆ dualizes to ∆# : (∧U )# ⊗ (∧U )# → (∧U )# , making (∧U )# into a graded algebra. The augmentation ε∧U : ∧U → lk # # defined by ε∧U (U ) = 0 dualizes to ε# ∧U : lk → (∧U ) and ε∧U is the identity # for the graded algebra (∧U ) . Moreover, ∆ realizes to a continuous map |∆| : | ∧ U | × | ∧ U | → | ∧ U | ,

page xxii

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xxiii

which makes |∧U | into a topological group. The natural map m|∧U | : ∧U → AP L (| ∧ U |) then induces a homomorphism H(m|∧U | ) : ∧U → H(| ∧ U |) which dualizes to a morphism H∗ (m|∧U | ) : H∗ (| ∧ U |) → (∧U )# of graded algebras. d On the other hand, let a ⊗ Φ 7→ a · Φ, a ∈ U L, Φ ∈ ∧U , denote the c d L in ∧U . Then (Theorem 6.1, p. 180) an holonomy representation, U θ of U isomorphism ∼ =

d ηL : U L −→ (∧U )# of graded algebras is defined by hηL (a), Φi = ε∧U (a · Φ). As observed in d Chapter 2, lk[GL ] ⊗ U L≥1 is a subalgebra of U L. When lk = Q there is a commutative diagram (Theorem 6.2, p. 184) of graded Hopf algebras, Q[GL ] ⊗ U L≥1

ηL ∼ = ρU

/ (∧U )# O H∗ (m|∧U | )

) H∗ (| ∧ U |) .

Thus H∗ (m|∧U | ) maps H∗ (| ∧ U |) isomorphically onto ηL (Q[GL ] ⊗ U L≥1 ). Next suppose (∧V ⊗ ∧Z, d) is a Λ-extension of (∧V, d). As with the acyclic closure, this determines a homotopy class of morphisms of Λalgebras, ∆∧Z : (∧Z, d) → (∧Z, d) ⊗ ∧U and thereby a homomorphism H(∆∧Z ) : H(∧Z, d) → H(∧Z, d) ⊗ ∧U . Composition of H(∆∧Z ) with evaluation with (∧U )# then gives a map H(∧Z, d) ⊗ (∧U )# → H(∧Z, d) , and (proof of Theorem 6.3, p. 185) the isomorphism ηL identifies this with d the holonomy representation of U L in H(∧Z, d). When lk = Q the homomorphism H∗ (m|∧U | ) converts this map to a representation H(∧Z, d) ⊗ H∗ (| ∧ U |) → H(∧Z, d) of the algebra H∗ (| ∧ U |), which ηL identifies with the holonomy representation of Q[GL ] ⊗ U L≥1 . Finally, these constructions for Sullivan models are connected to topological fibrations. For this we suppose a continuous map h : (X, ∗) → (Y, ∗)

page xxiii

January 8, 2015

14:59

9473-Rational Homotopy II

xxiv

RHT3

Rational Homotopy Theory II

q

i

has been converted to a fibration Y ← E ← F as described in §4.2; here X and Y are supposed to be path connected and Y is a CW complex. We then suppose given a commutative diagram of cdga morphisms (∧V, d) ϕY

m

 AP L (Y )

/ (∧Z, d)

/ (∧V ⊗ ∧Z, d)

mF

 / AP L (F )



/ AP L (E)

in which the upper row is the Λ-extension above. Note, however that no assumption is made about the morphisms ϕY , m and mF . In the case of the path space fibration Y ← P Y ← ΩY , we may choose the Λ-extension to be the acyclic closure (∧V ⊗ ∧U, d), thereby defining a morphism ϕΩ : (∧U, 0) → AP L (ΩY ). Then, on the one hand, as described in §4.2, the loop space ΩY acts on F via a map µX : F × ΩY → F ; on the other hand, ∆∧Z : (∧Z, d) → (∧Z, d) ⊗ ∧U . Here the algebra translates the topology (Theorem 6.4, p. 188): the diagram (∧Z, d) 

∆∧Z

/ (∧Z, d) ⊗ (∧U, 0)

mF

AP L (F )

AP L (µX )



mF ⊗ϕΩ

/ AP L (F × ΩY )

is homotopy commutative. Evaluation with H∗ (ΩY ) determines the holonomy representation H(F ) ⊗ H∗ (ΩY ) → H(F ) ,

γ ⊗ a 7→ a · γ

given by a · γ = (−1)deg a·deg γ hH(µX )γ, ai . Moreover, the morphism ϕΩ induces by adjunction a map |ϕΩ | : | ∧ U | ← |Sing ΩY |, and a homomorphism H∗ (|ϕΩ |) : H∗ (ΩY ) → H∗ (| ∧ U |) of graded Hopf algebras. When lk = Q, we have (Theorem 6.5, p. 193) H(mF )(H∗ (|ϕΩ |)a · β) = a · H(mF )β ,

a ∈ H∗ (ΩY ), β ∈ H(∧Z, d) ,

thus relating the holonomy representation of H(ΩY ) to the holonomy representation of H∗ (| ∧ U |) and hence to that of Q[GL ] ⊗ U L≥1 .

page xxiv

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xxv

Chapter 7: Sullivan spaces Suppose (X, ∗) is a path connected, based topological space with universal e Then (§1.8) a morphism ϕ : (∧V, d) → AP L (X) from a covering space X. minimal Sullivan algebra determines natural linear maps πk (ϕ) : πk (X) ⊗Z lk → πk (| ∧ V, d|) ,

k ≥ 2.

Suppose X is simply connected, the graded vector space H(X) has finite type, and H(ϕ) is an isomorphism. Then, (Theorem 1.6, p. 43) the linear maps πk (ϕ) are isomorphisms. This may not be true if X is not simply connected, which motivates the definition: X is a Sullivan space if e has finite type, (i) dim H 1 (X) < ∞ and H(X) and (ii) the linear maps πk (mX ), k ≥ 2, induced by a minimal Sullivan model mX : (∧V, d) → AP L (X), are isomorphisms. Now for any path connected, based CW complex (X, ∗) there is (§4.2) a classifying space fibration (B, ∗) o

q

(X ×B M B, ∗) o

j

(F, ∗) , '

together with a based homotopy equivalence ` : (X, ∗) → (X ×B M B, ∗), in which (B, ∗) is a classifying space for π1 (X), and π1 (q) and πk (j), k ≥ 2, are isomorphisms. Suppose (∧V, d) is a minimal Sullivan model for X. Then the Example in §4.5 (p. 141) provides a commutative diagram / (∧V 1 ⊗ ∧V ≥2 , d)

(∧V 1 , d) m

mX '



AP L (B)

/ (∧V ≥2 , d)

AP L (q)

 / AP L (X ×B M B)

m

AP L (j)

 / AP L (F )

in which mX is the minimal Sullivan model for X ×B M B. When X is a Sullivan space then both m and m are quasi-isomorphisms. In fact (Theorem 7.2, p. 197) X is a Sullivan space if and only if the following conditions are satisfied: e i ≥ 2 are finite. (i) dim H 1 (X) and dim H i (X), e (ii) m is the minimal Sullivan model for X.

page xxv

January 8, 2015

14:59

9473-Rational Homotopy II

xxvi

RHT3

Rational Homotopy Theory II

The morphism m : (∧V ≥2 , d) → AP L (F ) belongs to the rational homotopy theory of simply connected spaces, which is laid out in detail in [18]. By contrast, m : (∧V 1 , d) → AP L (B) belongs to the rational homotopy theory of classifying spaces (whose universal covering spaces are contractible!), and this is the principal subject of the rest of Chapter 7. We denote by BG the classical functorial construction of a classifying space for a group G, and recall that Gn = [G, Gn−1 ] denotes the lower cen∼ = tral series for G. The natural isomorphism G/[G, G] → H1 (BG; Z) tensors to yield an isomorphism G/[G, G] ⊗Z lk ∼ = H1 (BG; lk), and so H 1 (BG; lk) is finite dimensional if and only if G/[G, G] ⊗Z lk is finite dimensional. As noted as the start of this Introduction, Sullivan’s theory requires finiteness hypotheses; thus throughout this Chapter attention is restricted to groups G satisfying dim G/[G, G] ⊗Z lk < ∞ . In this case (Theorem 7.3, p. 208) there is a commutative diagram ···

/ (∧Wn−1 , d)

···

 / AP L (B G/Gn+1 )

λn−1

'

/ (∧Wn , d)

/ ···

'

 / AP L (B G/Gn+2 )

/ ···

in which the vertical arrows are minimal Sullivan models and each Wn is a finite dimensional vector space concentrated in degree 1. Moreover each λn is an inclusion and d : Wn → ∧2 Wn−1 . Thus if (∧W, d) = limn (∧Wn , d) −→ then the diagram above induces a quasi-isomorphism '

(∧W, d) → lim AP L (B G/Gn ) . −→ n

Composed with the natural morphism limn AP L (B G/Gn ) → AP L (BG) −→ this provides (Theorem 7.4) a Sullivan 1-model m : (∧W, d) → AP L (BG) . When dim H 1 (X) < ∞ and B = BG the diagram at the start of this Introduction to Chapter 7 exhibits the minimal Sullivan model of X in the form (∧V, d) = (∧W ⊗ ∧V ≥2 , d) . Moreover if X is a Sullivan space then m ∼ = is a quasi-isomorphism and limn AP L (B G/Gn ) → AP L (BG) . −→ Moreover, if G is a group for which dim G/[G, G] ⊗Z lk < ∞, then the construction of (∧W, d) exhibits W as a filtered vector space W0 ⊂ W1 ⊂

page xxvi

January 8, 2015

14:59

9473-Rational Homotopy II

Introduction

RHT3

xxvii

· · · , and (Proposition 7.5, p. 211) this filtration coincides with the filtration of W defined via the differential in §2.1. In this case, (∧W, d) = lim C ∗ (L0 /Ln0 ) , −→ n where L0 = W # is the homotopy Lie algebra of (∧W, d). The Lie algebra L0 is also related to the Lie algebra ⊕k≥1 Gk /Gk+1 ([45]), with Lie bracket induced by the commutator map [ , ] : Gk × G` → Gk+` . Applying − ⊗Z lk constructs a Lie algebra LG and (Theorem 7.7, p. 217), LG ∼ = ⊕k≥1 Lk0 /Lk+1 0 with Lie bracket on the right induced by the Lie bracket [ , ] : Lk ⊗ L` → Lk+` . On the other hand, the Sullivan 1-model, m : (∧W, d) → AP L (BG), determines a homomorphism, π1 (m) : G → GL = π1 (∧W, d) , ∼ =

which (Theorem 7.5, p. 214) induces isomorphisms Gn /Gn+1 ⊗Z lk → GnL /Gn+1 L . Thus this exhibits GL as a Malcev-lk-completion of G. In particular (Theorem 7.6, p. 216) when lk = Q the correspondence (∧W, d) → π1 (∧W, d) is a bijection from the isomorphism classes of Sullivan 1-algebras satisfying dim H 1 < ∞ to the isomorphism classes of Malcev-complete groups G satisfying dim G/[G, G] ⊗ Q < ∞. In Chapter 1 we showed that for any minimal Sullivan algebra (∧V, d) with V 1 = 0 that the graded vector space V has finite type if and only if m|∧V,d| : (∧V, d) → AP L (| ∧ V, d|) was a quasi-isomorphism. This Theorem generalizes to the case V 1 6= 0 (Theorem 7.8, p. 218) : | ∧ V, d| is a Sullivan space if and only if H 1 (∧V, d) and each V i are finite dimensional, and if m|∧V,d| is a quasi-isomorphism. Finally, if K → G → G/K is a short exact sequence of groups in which BK and B(G/K) are Sullivan spaces, and if G/K acts nilpotently in each H k (BK), then (Theorem 7.9) BG is a Sullivan space; moreover, if BG1 and BG2 are Sullivan spaces, then (Theorem 7.10, p. 223) so is the classifying space of the coproduct G1 #G2 . Chapter 8: Examples This Chapter uses Theorems 7.9 and 7.10 to identify a number of spaces as Sullivan spaces. In particular, these arise classically as follows:

page xxvii

January 8, 2015

14:59

xxviii

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

(i) Rationally nilpotent groups G, defined by the condition H∗ (BGn+2 , Q) = Q, for some n. Then BG is a Sullivan space if dim G/[G, G] ⊗Z Q < ∞. (ii) Rationally nilpotent spaces (Y, ∗) defined by the conditions that π1 (Y, ∗) is rationally nilpotent and acts nilpotently in each πn (Ye ) ⊗ Q. These are Sullivan spaces if, in addition, H(Y ) has finite type. The remaining examples go beyond rational nilpotence to cases where the Lie algebra, L0 , may be infinite dimensional. Since by Theorem 7.9, B(Z# · · · #Z) is a Sullivan space, a comparison with a wedge of 3-spheres computes the minimal Sullivan model of S 1 ∨ · · · ∨ S 1 = B(Z# · · · #Z). In particular, if L is the free Lie algebra on r generators then the homotopy Lie algebra of BG is the inverse limit E = limn L/Ln . ←− Next, (Theorem 7.9) the classifying space of the semi-direct product K n G is a Sullivan space if BK and BG are Sullivan spaces and G acts nilpotently in K/[K, K] ⊗Z Q. The original example of a Sullivan model of non-nilpotent space is that of an orientable Riemann surface Sg , described by Sullivan ([62]). The proof relies on [13], in which the authors use Hodge theory and Galois descent to prove that Sg is formal. This result is improved here, where it is shown directly from a cochain model that Sg is intrinsically formal. This then gives the minimal Sullivan model (Theorem 8.1, p. 235) from which it follows that Sg is a Sullivan space. The remaining three examples in this Chapter with Sullivan classifying spaces are: the pure braid groups, the Heisenberg group, and the fundamental groups of those compact Seifert manifolds whose quotient by the action of S 1 is an orientable closed Riemann surface. Finally although Bπ1 (RP n+1 #RP n+1 ) = B(Z2 #Z2 ) is a Sullivan space, a simple model calculation shows that the connected sum RP n+1 #RP n+1 is not a Sullivan space. Chapter 9: Lusternik-Schnirelmann category A topological space X has LS category ≤ m (cat X ≤ m) if it is covered by m + 1 open sets each contractible in X. If X is a normal space, then (Theorem 9.1, p. 245) cat X ≤ m if and only if X is the retract of an mcone. Analogously, if (∧V, d) is a minimal Sullivan algebra, then the LS category of (∧V, d) is the least integer m (or ∞) (cat (∧V, d) = m) such that the inclusion (∧V, d) → (∧V ⊗∧Z, d) admits a retraction, where (∧V ⊗

page xxviii

January 8, 2015

14:59

9473-Rational Homotopy II

Introduction

RHT3

xxix

∧Z, d) is the Sullivan model of the surjection (∧V, d) → (∧V / ∧>m V, d). In particular, if (∧V, d) is the minimal Sullivan model of a path connected normal space X, then (Theorem 9.2, p. 247) cat (∧V, d) ≤ cat X . The hypothesis “finite category” has important implications for the growth and the structure of the homotopy Lie algebra of a minimal Sullivan algebra, and these are the subject of Chapters 10–15. This Chapter 9 extends the results of [18] to the non-simply connected case, beginning with the key mapping Theorem (Theorem 9.3, p. 248). It asserts that if ϕ : (∧V, d) → (∧W, d) is a surjection of minimal Sullivan algebras then cat(∧V, d) ≥ cat (∧W, d). Thus since dim H(∧V, d) < ∞ implies that cat (∧V, d) < ∞, the quotients (∧W, d) will also have finite LS category, although they will often have infinite dimensional cohomology. While finite LS category inherits to quotient Sullivan algebras it does not, usually, inherit to sub Sullivan algebras, as seen immediately from the example ∧(a, x), dx = a2 . There is, however, one important example where this does hold: if (∧V, d) is a minimal Sullivan algebra and dim V i < ∞, i ≥ 2, then (Proposition 9.6, p. 259) cat(∧V, d) < ∞ implies that for some N , H N (∧V 1 , d) = 0, and so cat (∧V 1 , d) < ∞. There are analogous invariants for (Z-graded) modules (M, d) over a minimal Sullivan algebra (∧V, d). Each module admits a (∧V, d)-semifree ' resolution, namely a quasi-isomorphism (P, d) → (M, d) from a semifree (∧V, d)-module (§1.2). Moreover, the quotient ρ : (P, d) → (P/∧k+1 V ·P, d) ζ

'

factors as (P, d) → (P (k), d) → (P/ ∧k+1 V · P, d) in which (P (k), d) is also (∧V, d)-semifree. With this notation mcat (M, d) is the least integer k (or ∞) such that ζ admits a retraction. The Toomer invariant e(M, d) is then the least integer k such that H(ζ) or, equivalently H(ρ), is injective. With these definitions, e(M, d) ≤ mcat(M, d) ≤ mcat(∧V, d) ≤ cat(∧V, d) . The last inequality is improved in the fundamental result of Hess (Theorem 9.4, p. 250): if (∧V, d) is a minimal Sullivan algebra then mcat(∧V, d) = cat(∧V, d) . If X is a topological space with minimal Sullivan model (∧V, d) then the invariant e(X) = e(∧V, d) can be defined directly in terms of X ([63]). Originally it was thought that it might be true when X is simply connected that e(X) = cat XQ , where XQ was the rationalization of X. This would

page xxix

January 8, 2015

14:59

xxx

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

imply e(∧V, d) = cat (∧V, d), but e(∧V, d) is in general strictly less than cat(∧V, d). However, if H(∧V, d) has finite type then (Theorem 9.5, p. 260) the dual module (∧V, d)# satisfies
e (∧V, d)# = cat(∧V, d) . In particular, if H(∧V, d) satisfies Poincar´e duality then (∧V, d)# and (∧V, d) have isomorphic semifree resolutions. Thus in this case e(∧V, d) = cat(∧V, d) , and so the original guess is correct in this case. Finally, suppose a minimal Sullivan algebra decomposes as a Sullivan extension (∧V ⊗ ∧Z, d) and that elements y1 , · · · , yr of even degree in the homotopy Lie algebra L of (∧V ⊗ ∧Z, d) map to linearly independent elements in the homotopy Lie algebra of (∧V, d). If adL (y1 ), · · · , adL (yr ) are nilpotent then a theorem of Jessup (Theorem 9.6, p. 263) asserts that cat(∧V ⊗ ∧Z, d) ≥ cat(∧Z, d) + r . A variant of this result (Theorem 9.4, p. 250) asserts that if L is the homotopy Lie algebra of a minimal Sullivan algebra with finite category, and if dim L0 = ∞, then for some x, y ∈ L0 , (adL x)k y 6= 0 ,

k ≥ 0.

Chapter 10: Depth of a minimal Sullivan algebra and of a Sullivan Lie algebra As defined in Chapter 2, a quadratic Sullivan algebra is a Sullivan algebra of the form (∧V, d1 ) in which d1 : V → ∧2 V . These arise in two ways: first if (∧V, d) is any Sullivan algebra then the associated quadratic Sullivan algebra (∧V, d1 ) is defined by d − d1 : V → ∧≥3 V ; second if L is a Sullivan Lie algebra then limn C ∗ (L/Ln ) is the quadratic Sullivan algebra associated −→ to L. Let (∧V ⊗ ∧U, d1 ) be the acyclic closure of a quadratic Sullivan algebra (∧V, d1 ). Then the depth of (∧V, d1 ) is the least p (or ∞) such that H p,∗ (Hom∧V (∧V ⊗ ∧U, ∧V )) 6= 0 where the grading p corresponds to Hom(∧U, ∧p V ) under the standard isomorphism. If (∧V, d1 ) is associated with a Sullivan algebra (∧V, d) then (Theorem 10.1, p. 273) depth(∧V, d1 ) ≤ cat(∧V, d) .

page xxx

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xxxi

On the other hand, if L = L≥0 is a graded Lie algebra then, as in Chapd ter 2, U L = limn U L/ILn , where IL is the ideal generated by L. Generalizing ←− the definition in [18] when L = L≥1 has finite type, we set p d L) 6= 0 . depth L = least p (or ∞) such that ExtU L (lk, U

Now suppose L = L≥0 is a Sullivan Lie algebra and that (∧V, d1 ) is the associated quadratic Sullivan algebra. Then (Proposition 10.3, p. 286) the L depth of L is the least p (or ∞) such that TorU p (lk, ∧U ) 6= 0, where L acts in ∧U via the holonomy representation. In this case the pairing V × L → lk induces a morphism µ : (∧V, d1 ) → C ∗ (L), and if µ is a quasi-isomorphism then (Theorem 10.2, p. 278) depth L = depth(∧V, d1 ) . The main objective in Chapter 10 is the study of Sullivan Lie algebras, L, of finite depth, with the aid of Hochschild-Serre spectral sequences recalled in §10.1. In particular, suppose (∧V ⊗ ∧U, d1 ) is the acyclic closure of the associated quadratic Sullivan algebra (∧V, d1 ) for L. Then for a sub Lie algebra E ⊂ L, UE

depthL E := least p (or ∞) such that Torp (lk, ∧U ) 6= 0 . On the other hand, E determines the subspace KE ⊂ V defined by KE = { v ∈ V | hv, Ei = 0 } , and the ideal KE ∧ ∧ V is preserved by d1 . Division by this ideal gives a quadratic Sullivan algebra (∧Z, d1 ) and, if E is itself a Sullivan Lie algebra then there is a natural morphism from (∧Z, d1 ) to the associated quadratic Sullivan algebra (∧W, d1 ) for E. Furthermore, if this morphism is a quasiisomorphism then (Theorem 10.3, p. 288) depthL E = depth E . If E is an ideal then depthL E ≤ depth E, and it follows that the condition of finite depth inherits to ideals satisfying the hypotheses above. Again, suppose E ⊂ L is a sub Lie algebra. The Sullivan closure of E is the Lie sub algebra E ⊂ L defined by E = {x ∈ L | hKE , xi = 0 } , and E is Sullivan closed if E = E. Sullivan Lie sub algebras that are Sullivan closed are the natural Lie sub algebras of a Sullivan Lie algebras. For example, if L = I ⊕ J is the sum of two ideals, then each is a Sullivan closed Sullivan Lie algebra, and depth L = depth I+ depth J. On the other

page xxxi

January 8, 2015

14:59

9473-Rational Homotopy II

xxxii

RHT3

Rational Homotopy Theory II

hand if an ideal I ⊂ L is a Sullivan closed Sullivan Lie algebra satisfying I ⊃ I Ln0 + L≥k for some n and k, and if L acts locally nilpotently in TorU ∗ (lk, lk) while L0 acts nilpotently in I0 /[I0 , I0 ], then (Theorem 10.5, p. 294) depth L = dim (L/I)even + depth I . Moreover, if depth L < ∞ then the subspace N ⊂ Leven spanned by the elements x for which ad x is locally nilpotent satisfies (Corollary 10.10 to Theorem 10.4, p. 293) dim N ≤ depth L . Finally, as in [18] the radical, rad L, of a Sullivan Lie algebra, L, is the sum of its solvable ideals, and if depth L < ∞ then (Theorem 10.6, p. 295) dim rad L < ∞ and dim(rad L)even = depth rad L ≤ depth L .

Chapter 11: Depth of a connected graded Lie algebra of finite type This Chapter contains the results for the depth of a connected graded Lie algebra, L, of finite type required for the growth and structure theorems in Chapters 12–14. For instance, (Proposition 11.6, p. 304) if L is abelian, and for some module M , Ext∗U L (M, U L) 6= 0 then M contains a submodule of the form U (L≥n ) for some n. Generalizing the concept of trivial L-modules, an L-module M is called weakly locally finite if it is the increasing union of finite dimensional subspaces M (1) ⊂ M (2) ⊂ · · · such that M (k) is preserved by L≤k . Then the weak depth of L is the least m (or ∞) such that Extm U L (M, U L) 6= 0 for some weakly locally finite module M . By definition, weak depth L ≤ depth L. Now suppose E ⊂ L is a sub Lie algebra with centralizer Z. If for some weakly locally finite L-module, M , the restriction map ExtU L (M, U L) → ExtU E (M, U L) is non-zero, then (Proposition 11.7, p. 306) Z is finite dimensional. It follows that if L has finite depth then there are elements x1 , · · · , xr of odd degrees such that for all sufficiently large odd integers n, the map r X [xi , −] : Ln → ⊕ri=1 Ln+deg xi i=1

is injective.

page xxxii

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Introduction

xxxiii

Chapter 12: Trichotomy For a path connected topological space (X, ∗) there are exactly three, mutually exclusive, possibilities (the trichotomy of the title) for the integers rk πk (X) = dim πk (X) ⊗ Q, k ≥ 2. X (i) rk πk (X) < ∞ . k≥2

(ii)

X

rk πk (X) = ∞ but rk πk (X) < ∞ for k ≥ 2 .

k≥2

(iii) For some k ≥ 2 , rk πk (X) = ∞ . When X is a CW complex with finite LS category then the following results hold for the sequence rk(πk (X)): Case (i). X

rk πk (X) < ∞ .

k≥2

e Q) = 0, In this case, X is rationally elliptic and for some N H >N (X; e denoting the universal cover. Moreover, (Theorem 12.1 quoting [18]), X P rk πk (X) = 0, k ≥ 2N and k rk πk (X) ≤ 2 cat X. Case (ii). X

rk πk (X) = ∞ , but rk πk (X) < ∞ ,

k ≥ 2.

k≥2

In this case X is rationally hyperbolic, the name chosen because the k (X)) rk πk (X) grow exponentially. More precisely, set αk (X) = log rk (π . k Then α(X) := lim supk αk (X) > 0, and, if α(X) < ∞, then (Theorem 12.2) for some R lim

n→∞

max

n≤k≤n+R

αk (X) = α(X) .

Moreover, in the special case that dim X = N then (Theorem 12.5) α(X) < ∞ and there are constants β, γ, K > 0 such that α(X) − γ

log n β ≤ max αk (X) ≤ α(X) + , n+2≤k≤N n n

n≥K.

These results follow from the corresponding results for minimal Sullivan algebras, which are established in Chapter 13. Case (iii). Some rk πk (X) = ∞.

page xxxiii

January 8, 2015

14:59

9473-Rational Homotopy II

xxxiv

RHT3

Rational Homotopy Theory II

In this case X is rationally infinite and (Theorem 12.7) for some K, R > 0, and all n ≥ K, rk πk (X) = ∞ ,

some k ∈ [n, n + R] .

The analogous results for the minimal Sullivan model of X also holds (Theorem 12.6) but does not imply or follow from the topological result. Finally, if dim X = N , then (Theorem 12.8) the result for rk πk (X) holds with R = N .

Chapter 13: Exponential growth Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) of finite type and satisfying V = V ≥2 and dim V = ∞. Set log dim V k log dim H k (∧V, d) and α = lim sup ; αH = lim sup k k k k since dim Lk = dim V k+1 it follows that α = lim supk notation, (Proposition 13.2, p. 332) αH ≤ α

log dim Lk . k

With this

and αH < ∞ ⇔ α < ∞ .

There are three sets of hypotheses, each stronger than the next, which then provide increasing strong assertions about exponential growth for the integers dim V k . These are H1 :

cat(∧V, d) < ∞ and αH < ∞ .

H2 :

cat(∧V, d) < ∞ and αH < α .

H3 :

H >N (∧V, d) = 0 .

The corresponding conclusions (Theorems 12.9, 12.10 and 12.11) are then C1 :

For some R ,  lim max n→∞

C2 :

C3 :

log dim Lk , n≤k≤n+R k

 = α.

For some R, K, β, γ > 0 ,   log n log dim Lk β α−γ ≤ max ≤α+ , n≤k≤n+R n k n

n≥K.

For some γ, β, K > 0   log n log dim Lk β α−γ ≤ max ≤α+ , n≥K. n+2≤k≤n+N n k n

page xxxiv

January 8, 2015

14:59

9473-Rational Homotopy II

Introduction

RHT3

xxxv

The proofs all have the same basic structure, and so are presented as a single proof, except that the technique for estimating the error varies depending on the hypotheses. A detailed outline of the strategy is provided in §13.1.

Chapter 14: Structure of a graded Lie algebra of finite depth Let L = L≥1 be a graded Lie algebra of finite type and finite depth m. Then the ideals I ⊂ L have the same properties, and the set of ideals has a strong structure, analogous to the structure of the ideals in a finite dimensional Lie algebra concentrated in degree zero. In particular, set log dim Ik αI = lim sup k k and let S = { αI | I an infinite dimensional ideal in L } . Then, (Theorem 14.2, p. 369) S is a finite set with cardinality ≤ m. The size of infinite dimensional graded vector spaces V = V≥1 and W = W≥1 of finite type can be compared by setting V 6 W if for some fixed positive numbers λ, q and N k+q X dim Vk ≤ λ dim Wi , k≥N. i=k

With this, an equivalence relation, ∼L , on the set of ideals I ⊂ L is defined by I ∼L J ⇔ I ∩ K 6 J ∩ K and J ∩ K 6 I ∩ K , for all ideals K ⊂ L. The set of equivalence classes is a distributive lattice under 6, and (Theorem 14.1, p. 368) there are at most 2m equivalence classes with equality holding if and only if L ∼L L(1) ⊕ · · · ⊕ L(m) where each I(j) is an ideal of depth 1. Finally, if I ⊂ L is an ideal, then (Theorem 14.4, p. 377) L contains a sub Lie algebra E, called a weak complement of I, and satisfying (i) E ∩ I = 0, (ii) L/E 6 I, and (iii) depth E ≤ depth F for all subalgebras F of E such that E/F 6 I. In this case depth E + depth I = depth (E ⊕ I) ≤ depth L .

page xxxv

January 8, 2015

14:59

9473-Rational Homotopy II

xxxvi

RHT3

Rational Homotopy Theory II

Chapter 15: Weight decompositions of a Sullivan Lie algebra As defined in §7.4, a weighted graded Lie algebra is a graded Lie algebra L = L≥0 satisfying L = ⊕p≥1 L(p), where the subspaces L(p) satisfy [L(p), L(q)] ⊂ L(p + q) ,

p, q ≥ 1 .

n

In particular, each L ⊂ ⊕p≥n L(p) and so if L0 /[L0 , L0 ] and each Lk , k ≥ 1 are finite dimensional then L is a Sullivan Lie algebra. If L is any graded Lie algebra and G is any group then, as defined in §7.4, the Lie algebras L(L) = ⊕n Ln /Ln+1 and LG = ⊕n Gn /Gn+1 ⊗ Q are weighted Lie algebras with brackets induced by respectively the commutators [x, y] and [a, b] = aba−1 b−1 . Now suppose a quadratic Sullivan algebra (∧V, d1 ) is both the associated quadratic Sullivan algebra for a minimal Sullivan algebra (∧V, d) and the associated quadratic Sullivan algebra for a Sullivan Lie algebra L. If H(∧V, d) has finite type and cat (∧V, d) < ∞, and if L0 has a weight decomposition, then (Theorem 15.1) depth L = depth (∧V, d) ≤ cat (∧V, d) . In the particular case of an infinite dimensional Sullivan Lie algebra, L, concentrated in even degrees, if the associated weighted Lie algebra L satisfies dim H(C∗ (L)) < ∞, then (Theorem 15.2): αL = lim sup n≥1

log dim Ln /Ln+1 n

satisfies 0 < αL < ∞, and there are constants b, c and K such that 1 nαL e ≤ dim Ln /Ln+1 ≤ eb enαL , n≥K. nc Finally, if (∧V, d) is a 1-formal Sullivan algebra as defined in §2.10, and if dim H 1 (∧V, d) < ∞, then V 1 = ⊕n≥1 V 1 (n) and L = ⊕n≥1 L(n) with L(n) = (V 1 (n))# is a weighted Lie algebra. Here L is a sub Lie algebra of the fundamental Lie algebra L0 of (∧V, d) and L(n) = Ln0 /Ln+1 . In this 0 1 1 case (Theorem 15.3) depth L = depth(∧V , d) ≤ cat(∧V , d) < ∞, and the has the same growth properties as in Theorem 15.2. sequence dim Ln0 /Ln+1 0 Moreover, if L is the free Lie algebra on L(1) then the inclusion L(1) → L extends to a surjection ρ : L → L of weighted Lie algebras. If K is the ideal generated by ker ρ(2) then (Theorem 15.4) ρ induces an isomorphism from the Chen Lie algebra L/K onto L.

page xxxvi

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Chapter 1

Basic definitions and constructions

In this Chapter we review and extend the standard material in [18] from graded algebra, simplicial sets, and topology required for rational homotopy theory. Throughout this entire text all vector spaces and algebras will be defined over a field lk of characteristic zero.

1.1

Graded algebra

A graded vector space is a family of vector spaces V = {V i }i∈Z or a family of vector spaces X = {Xi }i∈Z . There is a canonical isomorphism between these categories sending V → X where Xi = V −i ,

i ∈ Z.

In our review of definitions below we shall largely focus on the “superscript category” leaving it to the reader to translate to the “subscript category”. There are constructions which use graded vector spaces from both categories, and for these one must convert the objects in one of the two categories to the other; constructions are always “compatible with conversion”. One could, of course, simply convert the “subscript category” to the “superscript category” and work entirely in the latter, but there are historical reasons for not doing so; the singular chain complex of a space S is always written {Ci (S)}i≥0 while the cochain complex is always written {C i (S)}i≥0 and this tradition is so deeply embedded in the subject that we have preferred to continue to use both superscripts and subscripts. Suppose then that V = {V i }i∈Z is a graded vector space. We write V >n = {V i }i>n , V + = V >0 and define V ≥n , V ≤n , V j + 1. The simplicial objects in a category C form a category whose morphisms are the sequence of morphisms Cn → Cn0 in C commuting with the face and degeneracy morphisms. If C has products so does the corresponding simplicial category: K × L = {Kn × Ln }n≥0 ,

∂i = ∂i × ∂i ,

sj = sj × sj .

We will primarily be concerned with the category of simplical sets in which case morphisms are called simplicial maps. If K is a simplicial set then an element σ ∈ Kn is an n-simplex of K; if n = 0 then σ is a vertex of K. A pair (K, σ) with σ a vertex of K is called a pointed simplicial set. A simplex is degenerate if it is in the image of some sj , otherwise it is non-degenerate. If En ⊂ Kn is a sequence of inclusions in which each ∂i restricts to maps En → En−1 , then by adding simplices to En of the form sj1 ◦· · ·◦sjp (En−p ) we obtain a simplicial subset of K whose non-degenerate simplices are all in the sets En . This is the simplicial subset generated by E = {En }. We identify a vertex v ∈ K0 with the simplicial set it generates, and denote this by {v}. Now suppose L ⊂ K is an inclusion of simplicial sets. Then the simplicial set K/L together with a simplicial map ρ : K → K/L are defined as follows: let Kn − Ln be the complement of Ln in Kn ; then  σ if σ 6∈ Kn (K/L)n = (Kn − Ln ) ∪ ∗n , ρn σ = ∗n if σ ∈ Ln and the set of maps ∂i , sj in K/L are the unique maps such that ∂i ρ = ρ∂i and sj ρ = ρsj .

page 8

January 8, 2015

14:59

9473-Rational Homotopy II

Basic definitions and constructions

RHT3

9

Example: The simplicial set ∆[n]. Among simplicial sets an important role is played by the simplicial sets ∆[n] defined as follows. Let e0 , e1 , · · · be the standard basis of R∞ and let ∆n denote the Euclidean simplex with vertices e0 , · · · , en . In particular we identify (∆1 , e0 , e1 ) with (I, 0, 1) where I is the unit interval. Then ∆[n]k consists of the linear maps heσ(0) , · · · , eσ(k) i : ∆k → ∆n which map ei 7→ eσ(i) where σ is any order preserving map: 0 ≤ σ(0) ≤ · · · ≤ σ(k) ≤ n . The face and degeneracy maps in ∆[n] are given by ∂i heσ(0) , · · · , eσ(k) i = heσ(0) , · · · , ed σ(i) , · · · , eσ(k) i

(1.1)

and sj heσ(0) , · · · , eσ(k) i = heσ(0) , · · · , eσ(j) , eσ(j) , · · · , eσ(k) i . Here “ b ” means delete. Almost by definition, the non-degenerate simplices of ∆[n] are the simplices heσ(0) , · · · , eσ(k) i with σ(0) < · · · < σ(k); when k = n − 1 these have the form he0 , · · · , ebi , · · · , en i; here he0 , · · · , ebi , · · · , en i is the ith face of ∆[n]. Definition. The fundamental simplex of ∆[n] is the identity map cn = he0 , · · · , en i : ∆n → ∆n . In particular, when n = 0, for each k, ∆[0]k consists of a single simplex written as ∗k ; these are all degenerate except for ∗0 , which we denote by ∗. When n = 1, ∆[1] has three non-degenerate simplexes, c1 ∈ ∆[1]1 and ∂1 c1 and ∂0 c1 in ∆[1]0 . The boundary, ∂∆[n], is the simplicial set generated by the {∆[n]k }k i

 if k < j  tk and sj tk = tk + tk+1 if k = j .  tk+1 if k > j

Note that when lk ⊂ R then (AP L )n is the lk-subalgebra of the standard cochain algebra of differential forms in ∆n generated by the coordinate functions t0 , · · · , tn . Thus (AP L )n is called the cochain algebra of polynomial differential forms in ∆n . Definition. If K is a simplicial set then the cochain algebra of polynomial forms on K, AP L (K), is defined by AkP L (K) = Simpl (K, AkP L ) = {Φσ ∈ (AkP L )n | σ ∈ Kn , n ≥ 0 } in which the Φσ are required to satisfy ∂i Φσ = Φ∂i σ

and sj Φσ = Φsj σ .

The differential is defined in the obvious way: if Φ = {Φσ } ∈ AkP L (K) then (dΦ)σ = d(Φσ ) . The assignment K AP L (K) is a contravariant functor from simplicial sets to commutative cochain algebras: if Φ = {Φσ } ∈ AP L (K) and if α : L → K is a map of simplicial sets then τ ∈ L.

(AP L (α)Φ)τ = Φα(τ ) ,

This functor, precomposed with the functor Sing yields the contravariant functor X

AP L (X) ,

ϕ

AP L (ϕ)

from topological spaces to commutative cochain algebras. Definition. If X is a topological space then AP L (X) is the algebra of polynomial differential forms on X.

page 14

January 8, 2015

14:59

9473-Rational Homotopy II

Basic definitions and constructions

RHT3

15

Now recall from §1.3 that cn denotes the fundamental simplex of ∆[n]. Proposition 1.6. (i) An isomorphism AP L (∆[n]) → (AP L )n is given by Φ 7→ Φcn . In particular, (AP L )(∆[0]) = (AP L )0 = lk. (ii) H((AP L )n , d) = lk. (iii) If α : L → K is an inclusion of simplicial sets then AP L (α) is surjective. proof: (i) is obvious; (ii) is just [18], Lemma 10.7(ii). (iii) follows from ([18], Lemma 10.7(iii)) applied to [18], Proposition 10.4(ii).  Given a zero simplex v0 ∈ K0 and a point x0 ∈ X, then x0 ∈ Sing 0 X and so by Proposition 1.6(i) the inclusions v0 ,→ K0 and x0 ,→ X define augmentations εK : AP L (K) → lk

and

εX : AP L (X) → lk .

Now suppose L ⊂ K is an inclusion of simplicial sets. We set AP L (K, L) = {Φ ∈ AP L (K) | Φ|L = 0 } . Thus we have the short exact sequence of differential graded objects 0 → AP L (K, L) → AP L (K) → AP L (L) → 0 , in which AP L (K, L) is an ideal in AP L (K). Thus a commutative graded algebra AP L (K, L)∗ is defined by AP L (K, L)∗ = AP L (K, L) ⊕ lk · 1 , and the inclusion AP L (K, L)∗ ,→ AP L (K) may be identified with the morphism AP L (ρ) : AP L (K/L) → AP L (K) . In particular, if Y ⊂ X is the inclusion of a subspace in a topological space X we write AP L (X, Y ) = AP L (Sing X, Sing Y ) and AP L (X, Y )∗ = AP L (Sing X, Sing Y )∗ . Finally, if K is a simplicial set we denote by Sn (K) the lk-vector space with basis Kn and by Dn (K) the subspace with basis the degenerate simPn plices in Kn . The operator ∂ = i=0 (−1)i ∂i maps Dn (K) to Dn−1 (K) and therefore defines a differential, ∂, in the graded vector space C∗ (K) with

page 15

January 8, 2015

14:59

9473-Rational Homotopy II

16

RHT3

Rational Homotopy Theory II

Cn (K) = Sn (K)/Dn (K). The dual, (C ∗ (K), d) = Hom((C∗ (K), ∂), lk) is the simplicial cochain complex for K. When K = Sing X then these are denoted by C∗ (X) and C ∗ (X); they are respectively the singular chain complex and the singular cochain algebra on X. (For the multiplication in C ∗ (K) see [18], §10). Thus H(C ∗ (X)) is the singular cohomology of X with coefficients in lk. We write H(C ∗ (X)) = H ∗ (X; lk). Now recall from Proposition 1.6 (p. 15) that when lk = Q, AP L (∆[n]) = (AP L )n is a sub cochain algebra of Hthe ordinary differential forms on ∆n . In particular, the classical integral n : AnP L (∆[n]) → R is defined, and as shown in §10(e) of [18], it takes values in Q. Thus for any simplicial set, K, and any topological space X, we can tensor with lk to obtain the linear maps, I I ∗ : AP L (K) → C (K) and : AP L (X) → C ∗ (X) , K

X

defined by I (

I Φ)(σ) =

K

I Φσ

and

n

I =

X

. Sing X

(Note that by definition C ∗ (X) = C ∗ (Sing X).) Proposition 1.7. ([18], Theorem H H10.9). If K is a simplicial set and X is a topological space, then K and X are natural quasi-isomorphisms of cochain complexes inducing natural isomorphisms of graded algebras ∼ =

H(AP L (K)) −→ H ∗ (C ∗ (K))

∼ =

and H ∗ (AP L (X)) −→ H ∗ (X : lk) .

Corollary 1.3. The morphisms C ∗ (λK ) : C ∗ (|K|) → C ∗ (K) and AP L (λK ) : AP L (|K|) → AP L (K) are quasi-isomorphisms. proof: The first assertion follows because, as observed in [18], p. 241, C∗ (λK ) is a quasi-isomorphism. The second assertion follows from this and the Proposition.  Note that for any simplicial set K we have a commutative square of natural isomorphisms H(|K|) O

∼ =

∼ =

H(AP L (|K|))

/ H(C ∗ (K)) O ∼ =

∼ =

/ H(AP L (K))

page 16

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Basic definitions and constructions

17

Convention: We shall always identify these four graded algebras, and denote all four simply by H|K|. In particular, for a topological space X and with K = Sing X we identify H ∗ (X; lk) = H(AP L (X)) and denote this by H(X). Example. ∆[n] and ∆[n]/∂∆[n]. It follows by an easy verification that evaluation at cn defines an isomorphism ∼ =

AP L (∆[n]) −→ (AP L )n . Note as well that the inclusions ti 7→ (AP L )n extend to an isomorphism ∼ =

∧(t1 , · · · , tn , dt1 , · · · , dtn ) −→ (AP L )n . On the other hand, for n ≥ 1, ∆[n]/∂∆[n] is the simplicial set with two non-degenerate vertices : the single 0-simplex, denoted c0 , and the image cn of the fundamental simplex cn of ∆[n]. In particular ( |∆[n]/∂∆[n]|, c0 ) = (S n , ∗) , and H∗ (S n ; Z) = H∗ (∆[n]/∂∆[n]; Z) = C∗ (∆[n]/∂∆[n]; Z) = Z cn ⊕ Z c0 . We call cn the fundamental cycle of ∆[n]/∂∆[n] and the fundamental homology class of S n . Next note that the isomorphism H ∗ (−, lk) = Hom(H∗ (−, ; Z), lk) determines a bilinear map H n (AP L (∆[n]/∂∆[n]); lk) × Hn (∆[n]/∂∆[n]; Z) → lk . Moreover, by definition, AP L (∆[n]/∂∆[n]) = lk ⊕ {Φ ∈ (AP L )n | ∂i Φ = 0 , 0 ≤ i ≤ n } . We show now that the class [dt1 ∧· · ·∧dtn ] ∈ H n (AP L (∆n /∂∆[n])) satisfies hcn , [dt1 ∧ · · · ∧ dtn ]i = 1 = h(−1)n [dt1 ∧ · · · ∧ dtn ], cn i

(1.4)

and therefore we call (−1)n [dt1 ∧ · · · ∧ dtn ] the fundamental cohomology class of ∆[n]/∂∆[n] and of S n . For the proof of the equality above, consider the short exact sequence 0 → C∗ (e1 ) → C∗ (∆[1], e0 ) → C∗ (∆[1], ∂∆[1]) → 0 . The connecting homomorphism satisfies δ[c1 ] = [e1 ]. Dually the exact sequence 0 ← AP L (e1 ) ← AP L (∆[1], e0 ) ← AP L (∆[1], ∂∆[1]) ← 0

page 17

January 8, 2015

14:59

9473-Rational Homotopy II

18

RHT3

Rational Homotopy Theory II

has a connecting homomorphism δ ∗ : [e1 ]∗ 7→ [dt1 ] and so h[dt1 ], c1 i = hδ ∗ [e1 ]∗ , c1 i = −h[e∗1 ], δ[c1 ]i = −1 . In particular, when n = 1 we denote t1 simply by t, thereby identifying AP L (∆[1]) = ∧(t, dt) and dt as a cocycle in AP L (∆[1], ∂∆[1]) satisfying h−dt, c1 i = 1 . The rest of the proof follows in the same way by induction from the short exact sequences C∗ (λ0 )

0 → C∗ (∆[n − 1], ∂∆[n − 1]) −→ C∗ (∆[n], J) → C∗ (∆[n], ∂∆[n]) → 0 in which J ⊂ ∆[n] is the subsimplicial set generated by all the faces containing e0 .  1.5

Sullivan algebras

Definition. A cochain algebra of the form (∧V, d) in which V = V ≥0 satisfies the Sullivan condition if V is the union of an increasing family of subspaces V (0) ⊂ V (1) ⊂ · · · ⊂ V (r) ⊂ · · · such that d = 0 in V (0) and for p ≥ 1, d : V p (r) → ∧(V

n. It follows from the long exact homotopy sequence that πk (F (n)) = 0 when k ≤ n and πk (j(n)) is an isomorphism for k > n. Clearly when n = 1 these are just classifying space fibrations. Postnikov decompositions are constructed in a similar way to classifying space fibrations: adjoin cells of dimension ≥ n + 2 to X to successively kill homotopy groups of degree k for k = n + 1, n + 2, · · · . This produces a

page 94

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Fibrations and Λ-extensions

95

map h(n) : X → Y for which πk (Y, ∗) = 0, k ≥ n + 1 and πk (h(m)) is an isomorphism for k ≤ n. Converting h(m) to a fibration as described in §3.1 then yields the desired Postnikov decomposition.

3.3

Λ-extensions

The analogue in Sullivan’s model theory of a (Serre) fibration X → Y is a Λ-extension of commutative cochain algebras, as described below. Definition. A commutative cochain algebra morphism (B, d) → (B ⊗ ∧Z, d), b 7→ b ⊗ 1, satisfies the Sullivan condition if (i) H 0 (B, d) = lk, (ii) d : 1 ⊗ Z → (lk ⊕ B ≥1 ) ⊗ ∧Z, and (iii) Z is the increasing union of subspaces 0 = Z(0) ⊂ · · · ⊂ Z(r) ⊂ · · · for which d : 1 ⊗ Z p (r) → B ⊗ ∧(Z

m+q(m−1) >i(m−1) d : 1 ⊗ Zq → ∧ V ⊕ ∧ V ⊗ Zq−i ⊕ (1 ⊗ (∧2 Z)q−1 ) . i=0

proof: We introduce a second gradation in ∧V (the left grading) by setting V p = V 1,p . Then a minimal Sullivan algebra (∧V, d1 ) is defined by d1 : V → ∧2 V and (d − d1 ) : V → ∧≥3 V . The differential d1 is called the quadratic part of d and it increases the left degree by 1. Exactly as in the construction of a Sullivan model in the singly graded case (see also [18], Example 6, p. 159), we may extend ρ : (∧V, d1 ) → (∧V / ∧>m V, d1 ) to a bigraded Sullivan model ∼ =

χ : (∧V ⊗ ∧Z, d1 ) −→ (∧V / ∧>m V, d1 ) in which each Z p = ⊕k≥m Z k,p , d1 increases left degree by 1, and χ(Z) = 0. Because this is a Sullivan extension, Z is the increasing union of bigraded subspaces 0 = Z(0) ⊂ · · · ⊂ Z(r) ⊂ · · · such that d1 : 1 ⊗ Z p (r + 1) → ∧V ⊗ ∧(Z

m V, d) . proof: It follows from Step Four that ∧V ⊗(lk ⊕Z0 ) is preserved by d. Filter both (∧V, d)-models by F k = (−)≥k,∗ and observe that the morphism of spectral sequences induced by χ begins with the quasi-isomorphism '

χ : (∧V ⊗ (lk ⊕ Z0 ), d1 ) −→ (∧V / ∧>m V, d1 ) .  If (∧V, d) is a minimal Sullivan algebra and we set (∧V )m,p = (∧m V )p then the quadratic part of d, d1 , increases both bidegrees by 1. Thus, as in the proof of Theorem 9.4, the homology H(∧V, d1 ) inherits a bigrading H(∧V, d1 ) = { H m,p (∧V, d1 ) }. Proposition 9.5. Suppose (∧V, d) is a minimal Sullivan algebra and denote by d1 the quadratic part of the differential. (i) If H m+1,∗ (∧V, d1 ) = 0, then H j,∗ (∧V, d1 ) = 0 for j ≥ m + 1. (ii) cat (∧V, d1 ) = e(∧V, d1 ) ≤ m. (iii) cat (∧V, d) ≤ e(∧V, d1 ). proof: (i) Recall from Corollary 9.1 the (∧V, d)-semifree resolution χ : ' (∧V ⊗ (lk ⊕ Z0 ), d) → (∧V / ∧>m V, d). By construction, the differential in

page 257

January 8, 2015

258

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

∧V ⊗ (lk ⊕ Z0 ) preserves the decreasing filtration by left degree. Moreover, by Step Three in the proof of Proposition 9.4, '

χ : (∧V ⊗ (lk ⊕ Z0 ), d1 ) → (∧V / ∧>m V, d1 ) is a (∧V, d1 )-semifree resolution. Now if H m+1,∗ (∧V, d1 ) = 0, we shall construct a morphism ψ : (∧V ⊗ (lk ⊕ Z0 ), d1 ) → (∧V, d1 ) of bigraded (∧V, d1 )-modules such that ψ is the identity in ∧V ⊗ 1. For this recall the filtration Z(r) introduced at the start of the proof of Proposition 9.4. If ψ is already defined in some Z0 (r) and if m+1,∗ z is an element of Z0 (r + 1) then d1 (1 ⊗ z) ∈ [∧V ⊗ (lk ⊕ Z0 (r))] and m+1 ψ(d1 (1 ⊗ z)) is a d1 -cycle in ∧ V . By hypothesis, ψ(d1 (1 ⊗ z)) = d1 Φ for some Φ ∈ ∧m V and as in the proof of Proposition 9.4 we may extend ψ to Z0 (r + 1) and then by induction on r to all of ∧V ⊗ (lk ⊕ Z0 ). It follows that mcat(∧V, d1 ) ≤ m, and that H j,∗ (∧V, d1 ) = 0 for j ≥ m + 1. (ii) Recall from Proposition 9.3 that e(∧V, d1 ) ≤ mcat (∧V, d1 ). But if e(∧V, d1 ) = q then H q+1,∗ (∧V, d1 ) = 0 and so by the argument above, mcat (∧V, d1 ) ≤ q. Thus mcat(∧V, d1 ) = e(∧V, d1 ) and by Theorem 9.4 cat(∧V, d1 ) = e(∧V, d1 ). (iii) Suppose e(∧V, d1 ) = m and consider the retraction ψ : (∧V ⊗ (lk ⊕ Z0 ), d1 ) → (∧V, d1 ) constructed above. We perturb ψ to a retraction ϕ : (∧V ⊗ (lk ⊕ Z0 ), d) → (∧V, d) , establishing thereby that cat(∧V, d) = mcat(∧V, d) ≤ m. For this suppose ϕ has been constructed in some Z0 (r) such that ϕ(1 ⊗ z) − ψ(1 ⊗ z) ∈ ∧≥m+1 V . If z ∈ Z0 (r + 1) then d(1 ⊗ z) ∈ ∧V ⊗ (lk ⊕ Z0 (r)) and ϕ(d(1 ⊗ z)) is a d-cycle in ∧≥m+1 V , satisfying ϕd(1 ⊗ z) − d1 ϕ(1 ⊗ z) ∈ ∧≥m+2 V . Thus dψ(1⊗z)−ϕd(1⊗z) is a d-cycle in ∧≥m+2 V . Since H >m,∗ (∧V, d1 ) = 0, dψ(1 ⊗ z) − ϕd(1 ⊗ z) = d1 Ψ(0) + Φ(0) with Φ(0) ∈ ∧≥m+3 V . If follows that d(ψ(1 ⊗ z) − Ψ(0)) is a d-cycle in ∧≥m+3 V . Iterating this process yields an element Ψ such that ϕd(1 ⊗ z) = dΨ and Ψ − ψ(1 ⊗ z) ∈ ∧≥m+2 V . We may extend ϕ to 1 ⊗ z by setting ϕ(1 ⊗ z) = Ψ. The existence of ϕ then follows by a standard induction argument.  Corollary 9.2. If (∧W, d) is a Sullivan 1-algebra, then e(∧W, d) = cat(∧W, d) = max { i | H i (∧W, d) 6= 0 } . Moreover, H i (∧W, d) 6= 0 for i ≤ e(∧W, d).

page 258

January 8, 2015

14:59

9473-Rational Homotopy II

Lusternik-Schnirelmann category

RHT3

259

proof: This follows from Proposition 9.5 because for Sullivan 1-algebras, d = d1 , and the left degree coincides with the ordinary degree.  Proposition 9.6. Suppose (∧V, d) is a minimal Sullivan algebra for which dim V i < ∞ if i ≥ 2. If cat (∧V, d) = m < ∞ then cat (∧V 1 , d) < ∞ and for some N , H i (∧V 1 , d) = 0, i ≥ N . proof: Since each V i , i ≥ 2, is finite dimensional there is a finite dimensional subspace U ⊂ V 1 such that d(1 ⊗ V i ) ⊂ ∧U ⊗ ∧V ≤i , 2 ≤ i ≤ m. Without loss of generality we may suppose d : U → ∧2 U , and thus obtain a Sullivan extension (∧U ⊗ ∧V [2,m] , d) → (∧V, d) → (∧W, d) in which W i = 0 for 2 ≤ i ≤ m. The Mapping Theorem 9.3 then asserts that cat(∧W, d) ≤ m. Since the differential in ∧W 1 is purely quadratic, Step Three in the proof of Proposition 9.4 gives a (∧W 1 , d)-semifree resolution ' χ : (∧W 1 ⊗ (lk ⊕ Z0 ), d) −→ (∧W 1 / ∧>m W 1 , d) in which Z0 is the increasing union of subspaces 0 = Z0 (0) ⊂ · · · ⊂ Z0 (r) ⊂ · · · with d : 1⊗Z0 (r) → ∧m+1 W 1 ⊕(W 1 ⊗Z0 (r−1)). It follows by induction on r that Z0 is concentrated in degree m (i.e. left degree coincides with the normal degree). ' Now let (∧W ⊗ (lk ⊕ M ), d) −→ (∧W/ ∧>m W, d) be a semifree (∧W, d) resolution restricting to the obvious surjection in ∧W ⊗ 1. Then χ lifts to a morphism of (∧W 1 , d)-modules, ϕ : (∧W 1 ⊗ (lk ⊕ Z0 ), d) → (∧W ⊗ (lk ⊕ M ), d) , extending the inclusion of ∧W 1 in ∧W . But since by the Mapping Theorem 9.3, cat (∧W, d) ≤ m, there is a retraction ψ : (∧W ⊗(lk ⊕M ), d) → (∧W, d) of (∧W, d)-modules extending the identity in (∧W, d). Now consider the morphism ψ ◦ ϕ : (∧W 1 ⊗ (lk ⊕ Z0 ), d) → (∧W, d) of (∧W 1 , d)-modules. It extends the identity in ∧W 1 and satisfies (ψ ◦ ϕ)(Z0 ) ⊂ (∧W )m = (∧W 1 )m . Thus ψ ◦ ϕ is a retraction (∧W 1 ⊗ (lk ⊕ Z0 ), d) → (∧W 1 , d). It follows from Hess’ Theorem 9.4, that cat (∧W 1 , d) ≤ m, and therefore by Proposition 9.5 that H i (∧W 1 , d) = 0, for i > m. Finally, we have a spectral sequence converging from ∧U 1 ⊗ H(∧W 1 , d) to H(∧V 1 , d) and it follows that H i (∧V 1 , d) = 0, i > dim U 1 + m. In particular, cat (∧V 1 , d) ≤ dim U 1 + m. 

page 259

January 8, 2015

14:59

9473-Rational Homotopy II

260

9.5

RHT3

Rational Homotopy Theory II

cat = e(−)#

The Toomer invariant was originally introduced in the hope that e(∧V, d) would provide ( [63]) an algebraic description of rational category. While this proved not to be the case ( [18], §29(b), Example 2), the next Theorem shows that the idea was close to being correct. Moreover, Corollary 9.4 shows that this idea is correct if H(∧V, d) is a Poincar´e duality algebra, or if d : V → ∧2 V is purely quadratic. Recall now that if (M, d) is a (∧V, d) module then M # is the (∧V, d)module defined by (a · f )(m) = (−1)deg a·deg f f (a · m)

and (df )(m) = −(−1)deg f f (dm) ,

for a ∈ ∧V, m ∈ M and f ∈ M # . Theorem 9.5. Suppose (∧V, d) is a minimal Sullivan algebra and H(∧V, d) has finite type. Then cat (∧V, d) = e((∧V )# ) . proof: Given Theorem 9.4, this is identical with the proof of Theorem 19.16 in [18]. While that Theorem states as hypotheses that (∧V, d) is simply connected and of finite type, the only use of these hypotheses is the fact that the natural morphism of (∧V, d)-modules σ : (∧V, d) → (((∧V )# )# , d) is a quasi-isomorphism. But this is true if H(∧V, d) has finite type. Now, if ' H(σ) is an isomorphism and ρ : (P, d) → ((∧V )# , d) is a semifree resolution, it follows that the composite (ρ)# ◦ σ : (∧V, d) → (P # , d) is a quasi-isomorphism. With this, the proof of Theorem 29.16 in [18] applies verbatim to prove Theorem 9.5.  Corollary 9.3. Suppose X and Y are path connected topological spaces, that (∧V, d) and (∧W, d) are minimal Sullivan algebras, and that H(X), H(Y ), H(∧V, d) and H(∧W, d) all have finite type. Then cat0 (X × Y ) = cat0 X + cat0 Y and cat((∧V, d) ⊗ (∧W, d)) = cat(∧V, d) + cat(∧W, d) .

page 260

January 8, 2015

14:59

9473-Rational Homotopy II

Lusternik-Schnirelmann category

RHT3

261

proof: There is an obvious morphism Hom(∧V, lk) ⊗ Hom(∧W, lk) → Hom(∧V ⊗ ∧W, lk) of (∧V, d) ⊗ (∧W, d)-modules, and since H(∧V, d) and H(∧W, d) have finite type, this is a quasi-isomorphism. Now the Corollary follows from the Theorem by an argument identical to the proof of Theorem 30.2 in [18].  Corollary 9.4. Suppose (∧V, d) is a minimal Sullivan algebra and that H(∧V, d) is a Poincar´e duality algebra. Then cat (∧V, d) = e (∧V, d) . proof: Let z ∈ (∧V )n be a cycle representing the fundamental class and let f : ∧V → lk be a linear map of degree −n such that f (z) = 1 and f ◦ d = 0. Then it follows in a straightforward way from the definitions that a morphism of (∧V, d)-modules ϕ : (∧V, d) → (∧V )# is defined by ϕ(Φ)(Ψ) = f (Φ ∧ Ψ). The fact that H(∧V, d) is a Poincar´e duality algebra implies that ϕ is a quasi-isomorphism, and the Corollary follows. 

9.6

Jessup’s Theorem

Suppose M is a graded vector space and f ∈ Homk (M, M ) for some k ≤ 0. Then f is locally conilpotent if ∩p f p (M ) = 0, and f is conilpotent if for all i there is some h(i) such that f h(i) (M )∩M i = 0. Clearly conilpotence implies local conilpotence, and if M has finite type then these two conditions are equivalent. Definition. An Engels derivation is a locally nilpotent derivation θ in (∧W, d) of degree ≤ 0 for which either (i) W has finite type and H(θ) is conilpotent, or (ii) θ is conilpotent. Now suppose θ is a derivation of degree ≤ 0 in a minimal Sullivan algebra (∧W, d) and define a linear map f : W → W by requiring that θ − f : W → ∧≥2 W . Lemma 9.1. If f is conilpotent, then so is θ.

page 261

January 8, 2015

14:59

9473-Rational Homotopy II

262

RHT3

Rational Homotopy Theory II

proof: Straightforward arguments show first that f extends to a conilpotent derivation θf in ∧W and then, because θ − θf : ∧k W → ∧≥k+1 W , that θ is conilpotent.  Lemma 9.2. Suppose θ is an Engels derivation in a minimal Sullivan algebra (∧W, d) and that (xn , zn )n≥0 is an infinite sequence of pairs of elements in ∧W satisfying dzn = 0

and

θzn+1 = zn + dxn ,

n ≥ 0.

Then there is an infinite sequence of elements yn ∈ ∧W such that dyn = zn

and

θyn+1 = yn + xn ,

n ≥ 0.

proof: If W has finite type and H(θ) is conilpotent the argument proving Lemma 31.14 in [18] applies verbatin to give the Lemma. Otherwise, θ itself is conilpotent. It follows that H(θ) is conilpotent and hence, if γi is an infinite sequence of classes in H(∧W, d) such that H(θ)γi+1 = γi then each γi = 0. In particular, in this case we may write zi = dwi . Now define, for each s, a sequence of elements yi (s), i ≤ s as follows: ys (s) = ws

and yi (s) = θyi+1 (s) − xi ,

i < s.

A quick calculation shows that dyi (s) = zi and θyi+1 (s) = yi (s) + xi . In particular, θ(yi (s + 1) − yi (s)) = yi−1 (s + 1) − yi−1 (s). Let n(i) be the integer such that θn(i) (∧W ) ∩ (∧W )deg yi = 0 . Then if s − i ≥ n(i) yi (s + 1) − yi (s) = θs−i (ys (s + 1) − ys (s)) = 0 . Define the sequence yi by setting yi = yi (s), any s > i + n(i).  Now consider a Sullivan extension (∧V ⊗ ∧Z, d) in which (∧V ⊗ ∧Z, d) is itself a minimal Sullivan algebra. Then the inclusion V ,→ V ⊕Z dualizes to a surjection L → E from the homotopy Lie algebra of (∧V ⊗ ∧Z, d) to the homotopy Lie algebra of (∧V, d). Recall also from §4.3 the holonomy representation θ of L in H(∧Z, d). We extend Jessup’s Theorem 31.10 of [18] to

page 262

January 8, 2015

14:59

9473-Rational Homotopy II

Lusternik-Schnirelmann category

RHT3

263

Theorem 9.6. With the hypotheses above let x1 , · · · , xr be linearly independent elements of Eeven . Assume that for each i, either (i) V ⊕ Z has finite type, adE (xi ) is nilpotent, and θ(xi ) is conilpotent, or (ii) xi is the image of yi ∈ L and adL yi is nilpotent. Then cat(∧V ⊗ ∧Z, d) ≥ cat(∧Z, d) + r . proof: In the case of hypothesis (i) the proof of Theorem 31.10 of [18] applies verbatim. In the case of hypothesis (ii) let the xi be ordered so that deg x1 = · · · = deg xs < deg xs+1 ≤ · · · ≤ deg xr . Divide by V ≤deg x1 and by a subspace of V 1+deg x1 to arrange that V = ⊕si=1 lkvi ⊕ V >deg x1 +1 , where hvi , yj i = δij . Then denote U = ⊕si=2 lkvi ⊕ V >deg x1 +1 and choose the direct summand, Z, of V in V ⊕ Z so that hZ, yj i = 0, 1 ≤ j ≤ r. By the Mapping Theorem 9.3 (p. 248), this does not increase the LS category of the Sullivan algebra. Next in ∧v1 ⊗ ∧U ⊗ ∧Z write d(1 ⊗ Φ) = v1 ⊗ θΦ + 1 ⊗ dΦ, Φ ∈ ∧U ⊗ ∧Z. Define χ : U ⊕ Z → U ⊕ Z by requiring θ − χ : U ⊕ Z → ∧≥2 (U ⊕ Z). Then a brief calculation (cf. §2.1) gives hw, [y1 , y]i = hχ(w), yi for w ∈ U ⊕ Z and y ∈ L. Since adL y1 is nilpotent it follows that χ is conilpotent. By Lemma 9.1, θ is conilpotent and hence θ is an Engels derivation in ∧(U ⊕ Z). Now, in view of Lemma 9.2, the proof of Proposition 31.13 in [18] applies verbatim to show that cat (∧V ⊗ ∧Z, d) ≥ cat(∧U ⊗ ∧Z, d) + 1 . Finally, by construction, y2 , · · · , yr are in the image of the homotopy Lie algebra of (∧(U ⊕Z), d). It follows by induction that cat(∧U ⊗∧Z, d) ≥ cat(∧Z, d) + r − 1, which establishes the Theorem.  As a corollary we deduce the following application. e Proposition 9.7. Let X be a Sullivan CW complex with universal cover X. Suppose r linearly independent elements xi in the fundamental Lie algebra, L0 , of X are such that each adL0 xi is nilpotent. Then e . cat0 (X) ≥ r + cat0 (X) In particular, if L0 is finite dimensional then e + dim L0 . cat0 (X) ≥ cat0 (X)

page 263

January 8, 2015

14:59

264

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

proof: Let (∧U, d) be the minimal Sullivan model of X. We apply Theorem 9.6 to the Sullivan extension ∧U 1 ⊗∧U ≥2 , noting that (Theorem 7.2, p. 197) e  (∧U ≥2 , d) is the minimal Sullivan model of X. Finally, we prove a variant of Theorem 9.6. Theorem 9.7. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) such that dim V 1 = ∞. If cat (∧V, d) < ∞ then there are elements x, y ∈ L0 for which ( ad x)k y 6= 0 ,

k ≥ 0.

proof: By Proposition 9.6 (p. 259), cat (∧V 1 , d) < ∞ and so we may assume without loss of generality that V = V 1 . Choose a surjection (∧V, d) → (∧W, d) such that dim W = ∞ and cat (∧W, d) is minimized. Then the homotopy Lie algebra of (∧W, d) is a sub Lie algebra of the homotopy Lie algebra of (∧V, d), and it is sufficient to prove the Theorem for (∧W, d). Denote the homotopy Lie algebra of (∧W, d) by L. Write (∧W, d) as a Sullivan extension (∧w ⊗ ∧Z, d) with dw = 0 and define x ∈ L by hw, xi = 1 and hZ, xi = 0. Let E ⊂ L be the homotopy Lie algebra of (∧Z, d). Then (cf. §2.2) the differential in ∧W is given by d(1 ⊗ z) = 1 ⊗ dz − w ⊗ θz, where θ is the derivation of degree 0 in (∧Z, d) given by hθz, yi = hz, [x, y]i ,

y ∈E.

It follows that ker ad x = [ Z/θ(Z) ]# . Now if ad x acts locally nilpotently in E then ker ad x is infinite dimensional and so Z/θ(Z) is infinite dimensional. Moreover, the ideal I generated by θ(Z) in ∧Z is clearly preserved by d and by θ and so division by I gives a quotient Sullivan algebra of the form (∧w ⊗∧(Z/θ(Z)), id⊗d0 ). It follows from Theorem 9.3 (p. 248) and Proposition 9.1(iii) (p. 246) that cat (∧W, d) ≥ cat (∧w ⊗ ∧(Z/θ(Z)), id ⊗ d0 ) = 1 + cat (∧(Z/θ(Z)), d0 ) , contradicting our hypothesis on (∧W, d). Thus ad x is not locally nilpotent, which establishes the Theorem. 

page 264

January 8, 2015

14:59

9473-Rational Homotopy II

Lusternik-Schnirelmann category

9.7

RHT3

265

Example

Let p : X → S 3 be the fibration associated with the projection of S 3 ∨ S 3 on the first factor, and let (A, d) ⊂ AP L (X) be a sub cochain algebra such ' that (A, d) → AP L (X) and A>3 = 0. Then a quasi-isomorphism '

(∧y ⊗ ∧(wi , i ≥ 1)/(wi wj ), d) −→ (A, d) is defined as follows: dy = 0 = dw1 , y and w1 are mapped respectively to representatives of the fundamental classes of the first and second factor S 3 , and dwi = ywi−1 , i ≥ 2. Thus deg y = 3 and deg wi = 1 + 2i. It follows that if (∧y ⊗ ∧Z, d) is a minimal Sullivan model for AP L (p), then ' (∧Z, d) → (∧(wi /(wi wj ), 0)). By Theorem 5.1 (p. 145) this implies that ∧wi /(wi wj ) is isomorphic to the cohomology algebra of the space ∨i≥1 S 2i+1 . This algebra has lk 1 ⊕ (⊕i≥1 lkwi ) as basis, and all products of elements in H + are zero. Thus the fibre, F , of p has the rational homotopy type of ∨i≥1 S 2i+1 . Next note that the homotopy Lie algebra, L, of (∧v, 0) is the abelian Lie algebra lka with degree a = 2. Thus Theorem 6.2 (p. 184) implies that H∗ (ΩS 2 : lk) is the polynomial algebra ∧a. Moreover, it follows from the quasi-isomorphism above that the holonomy representation of L in ⊕i≥1 lkwi is given by a · wi+1 = wi , i ≥ 1 and a · w1 = 0. Now applying Theorem 6.5 (p. 193) we find that the holonomy action F × ΩS 3 → ∨i≥1 S 2i+1 induces the map on homology given by wi · ak = wi+k ,

i ≥ 1, k ≥ 0 ;

where wi is the homology class dual to wi . Thus H+ (F ) is the free H∗ (ΩS 3 ; lk) module generated by w1 . On the other hand, cat0 S 3 = cat0 F = 1. Here Theorem 9.6 does not apply, reflecting the fact that the holonomy representation of L in H∗ (F ; lk) is not nilpotent. Consider now a continuous map ϕ : B = S 1 ×S 1 ×S 1 → S 3 that induces an isomorphism on H3 (−; lk), and let q : E → B be the pullback of ϕ and p. E

/X

ψ

q

 S1 × S1 × S1

ϕ



p

/ S3 .

A model of q is given by (∧(v1 , v2 , v3 ), 0) → (∧(v1 , v2 , v3 ) ⊗ ∧wi /(wi wj ), D)

page 265

January 8, 2015

14:59

9473-Rational Homotopy II

266

RHT3

Rational Homotopy Theory II

with Dw1 = 0, Dwi = v1 v2 v3 wi−1 for i ≥ 2, and deg v1 = deg v2 = deg v3 = 1. Here the homotopy Lie algebra of ∧(v1 , v2 , v3 ) is the abelian Lie algebra lkx1 ⊕ lkx2 ⊕ lkx3 , and the holonomy representation is trivial. Thus E is a Sullivan space. Since E has a model of nilpotency 4, cat0 (E) ≤ 4. Now by Proposition 9.7 4 ≥ cat0 (E) ≥ cat0 (S 1 × S 1 × S 1 ) + cat0 (F ) = 3 + 1 = 4 . Therefore, cat0 (E) = 4.

page 266

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Chapter 10

Depth of a Sullivan algebra and of a Sullivan Lie algebra

In this Chapter, from §10.3 on, we shall be considering minimal Sullivan algebras (∧V, d) and graded Lie algebras L, together with pairings h , i : V × sL → lk. Consistent with the conventions of Chapter 2 we shall often abuse notation and write these as pairings (v, x) 7→ hv, xi where v ∈ V and x ∈ L. 10.1

Ext, Tor and the Hochschild-Serre spectral sequence

Let A be a graded algebra. We recall the classical functors Ext and Tor in the graded context. Thus a free resolution P∗ → M of a right A-module M is an exact sequence of right A-modules of the form → Pp+1 → Pp → · · · → P0 → M → 0 , in which the Pn are free A-modules. If Q and N are respectively a left and a right A-module then: → Pp+1 ⊗A Q → Pp ⊗A Q → · · · → P0 ⊗A Q and ← HomA (Pp+1 , N ) ← HomA (Pp , N ) ← · · · ← HomA (P0 , N ) are complexes of graded vector spaces, with corresponding homology denoted by TorA (M, Q) = { TorA p (M, Q) }

and ExtA (M, N ) = { ExtpA (M, N ) }

p with each TorA p (−, −) and ExtA (−, −) itself a graded vector space. Here p is called the homological degree and we often write   r p p,r−p A TorA (M, N ) . p (M, N ) r = Torp,r−p (M, N ) and [ ExtA (M, N ) ] = ExtA

267

page 267

January 8, 2015

268

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

For any left A-module Q, Hom(Q, lk) is the right A-module given by (f · a)(x) = f (a · x), f ∈ Hom(Q, lk), a ∈ A and x ∈ Q. Thus if P∗ → M is an A-free resolution of M we have the identification HomA (Pp , Hom(Q, lk)) = Hom(Pp ⊗A Q, lk) . Recall that we denote Hom(−, lk) by (−)# . Thus the identifications above yield the important relation  # ExtA (M, Q# ) = TorA (M, Q) . (10.1) Definition. If A is a graded algebra then the grade, gradeA M , of a right A-module M is the least integer k such that ExtkA (M, A) 6= 0. If ExtA (M, A) = 0 we say gradeA M = ∞. The depth of an augmented graded algebra A, depth A, is the grade of the trivial A-module lk. The global dimension of A, gldim A, is the greatest integer k (or ∞) such that ExtkA (lk, −) 6= 0. Next, suppose given a graded Lie algebra L. A representation of L in M is the same as a U L-module structure in M . Thus we may abuse language and refer to M as an L-module. If M and N are L-modules, then M ⊗ N becomes a L-module via the diagonal U L → U L ⊗ U L extending x 7→ x ⊗ 1 + 1 ⊗ x, x ∈ L. Similarly, Hom(M, N ) is endowed with the L-structure given by (x·f )(m) = x·f (m)+(−1)deg x·deg f f (x·m) , x ∈ L , m ∈ M, f ∈ Hom(M, N ) . Now recall from §2.1 the Cartan-Chevalley-Eilenberg construction (∧sL⊗M, ∂) associated with any left L-module M . If M = U L and the representation of L in U L is by left multiplication, then H(∧sL ⊗ U L, ∂) = lk, as follows from the Poincar´e-Birkhoff-Witt theorem ( [18], Proposition 22.3). Thus (∧sL ⊗ U L, ∂) is a resolution of lk by free right U L-modules. It follows that for any right L-module, M , (∧sL ⊗ U L, ∂) ⊗ M becomes, via the diagonal in U L, a complex of L-modules. Lemma 10.1. (∧sL ⊗ U L, ∂) ⊗ M is a U L-free resolution of M . '

proof: The augmentation (∧sL ⊗ U L) → lk defines a quasi-isomorphism ' (∧sL ⊗ U L, ∂) ⊗ M → M . It remains to show that ∧p sL ⊗ U L ⊗ M is U L free, and for this it is sufficient to show that U L ⊗ M is U L free. Consider the right U L-module M ⊗ U L and define a U L-linear map ϕ : M ⊗ U L →

page 268

January 8, 2015

14:59

9473-Rational Homotopy II

Depth of a Sullivan algebra and of a Sullivan Lie algebra

RHT3

269

U L ⊗ M by ϕ : Φ ⊗ a 7→ (1 ⊗ Φ) · a. Then recall ( [18], Proposition 21.2) ∼ = that we may identify ∧L with U L via the linear isomorphism ∧L → U L P given by x1 ∧ · · · ∧ xr 7→ (−1)σ xσ(1) · · · · · xσ(r) (Here the sum is over all permutations of {1, · · · , r}.) Since U L acts diagonally in U L ⊗ M it follows that for a ∈ ∧r L, ϕ(Φ ⊗ a) − (−1)deg Φ·deg a a ⊗ Φ ∈ ∧ 0.

Moreover the inclusion I ⊂ L induces a morphism of spectral sequences. UL I At the E 2 -level this exhibits the map TorU q (M, N ) → Torq (M, N ) as the composite UI I TorU q (M, N ) → lk ⊗U L/I Torq (M, N ) U L/I

= Tor0

(10.2)

UL I 2 ∞ (lk, TorU q (M, N )) = E0,q  E0,q  Torq (M, N ) .

For an ideal I ⊂ L there is also a convergent Hochschild-Serre spectral sequence ExtpU L/I (lk, ExtqU I (M, N )) =⇒ Extp+q U L (M, N ) . Here, the map ExtqU I (M, N ) ← ExtqU L (M, N ) decomposes as ExtqU I (M, N ) ← HomU L/I (lk, ExtqU I (M, N ))

(10.3)

0,q = Ext0U L/I (lk, ExtqU I (M, N )) = E20,q  E∞  ExtqU L (M, N ) .

Lemma 10.2. Suppose I is an ideal in a graded Lie algebra L and that L = lkx ⊕ I with deg x odd. Then for any right L-module M , Ext∗U I (M, U L) is a free U L/I-module.

page 270

January 8, 2015

14:59

9473-Rational Homotopy II

Depth of a Sullivan algebra and of a Sullivan Lie algebra

RHT3

271

proof: First recall that the right adjoint representation of L in C∗ (I), together with the (right) representation of L in M and right multiplication by U L in U L define a right representation of L in the complex Hom(C∗ (I) ⊗ M, U L) which induces the representation of L in ExtU I (M, U L) used to define ExtU L/I (lk, ExtU I (M, U L)). We denote this representation by α ⊗ y 7→ α ∧ y , y ∈ L, α ∈ ExtU I (M, U L) , noting that it depends only on the image y of y in L/I. Next note that left multiplication by U L in U L is a map of right U Lmodules and therefore makes ExtU I (M, U L) into a left U L-module, denoted by y ⊗ α 7→ y ∧ α , y ∈ L, α ∈ ExtU I (M, U L) . Finally, note that the right adjoint representation of L in U I, together with the right adjoint representation of L in C∗ (I) and the (right) representation of L in M define a right representation of L in the complex Hom(C∗ (I) ⊗ M, U I) which induces a right representation of L in ExtU I (M, U I). This we denote by β ⊗ y 7→ β • y , y ∈ L, β ∈ ExtU I (M, U I) . Thus, since the inclusion of U I in U L induces an inclusion of ExtU I (M, U I) in ExtU I (M, U L), a straightforward calculation gives β ∧ y = β • y + (−1)deg β·deg y y ∧ β , y ∈ L, β ∈ ExtU I (lk, U I) . Now observe that the Poincar´e Birkoff Witt Theorem implies that mul∼ = tiplication in U L is an isomorphism ∧x ⊗ U I → U L of right U I-modules. Since dim ∧x < ∞ this gives an isomorphism Hom(C∗ (I) ⊗ M, U L) = ∧x ⊗ Hom(C∗ (I) ⊗ M, U I) and hence shows that the map x ⊗ β 7→ x ∧ β induces an isomorphism ∼ = ∧x ⊗ ExtU I (M, U I) −→ ExtU I (M, U L) . The observations above then yield β ∧ x = β • x + (−1)deg β x ∧ β so that ExtU I (M, U L) = ExtU I (M, U I) ⊕ ExtU I (M, U I) ∧ x . But the map − ∧ x depends only on x ∈ L/I and, in U L/I, x2 = 0. In other words, ExtU I (M, U L) is a free U L/I-module.  Lemma 10.3. Suppose M and N are U L-modules, and I is an ideal in L. If N = N≥0 has finite type and I acts trivially in N then the E2 -term of the Hochschild-Serre spectral sequence for ExtU L (M, N ) is given by
I E2p,q = ExtpU L/I TorU q (M, lk), N .

page 271

January 8, 2015

14:59

9473-Rational Homotopy II

272

RHT3

Rational Homotopy Theory II

proof: We have to show that
I ExtpU L/I (lk, ExtqU I (M, N )) = ExtpU L/I TorU q (M, lk), N . Because of our hypotheses on N , it follows from (10.1) that
I # # ExtqU I (M, N ) = TorU q (M, N ) and so h i# L/I I # ExtpU L/I (lk, ExtqU I (M, N )) = TorU (lk, TorU . p q (M, N )) On the other hand, because N (and thus N # ) are trivial U I-modules, I UI # # we have TorU as U L/I-modules. Thus q (M, N ) = Torq (M, lk) ⊗ N

L/I I L/I I # # TorU lk, TorU = TorU TorU . p q (M, lk) ⊗ N p q (M, lk), N The Lemma follows from a second application of (10.1).  10.2

The depth of a minimal Sullivan algebra

Let (∧V, d) be a minimal Sullivan algebra and recall (§2.1) that the associated quadratic Sullivan algebra (∧V, d1 ) is defined by d1 : V → ∧2 V and d − d1 : V → ∧≥3 V . Then by Corollary 3.4 (p. 105) the acyclic closure of (∧V, d1 ) can be written in the form (∧V ⊗ ∧U, d1 ) in which d1 : ∧U → V ⊗ ∧U . Using the canonical linear isomorphism Hom∧V (∧V ⊗ M, −) = Hom(M, −) we let Homp∧V (∧V ⊗ ∧U, ∧V ) ⊂ Hom∧V (∧V ⊗ ∧U, ∧V ) correspond to the subspace Hom(∧U, ∧p V ). Then a chain complex of graded vector spaces Hom0∧V (∧V ⊗ ∧U, ∧V ) → · · · → Homp∧V (∧V ⊗ ∧U, ∧V ) → Homp+1 ∧V (∧V ⊗ ∧U, ∧V ) → · · · is defined by setting d1 (f ) = d1 ◦ f − (−1)deg f f ◦ d1 . Now (∧V ⊗ ∧U, d1 ) is a (∧V, d1 )-semifree resolution of lk and the homology of this complex is independent of the choice of bigraded semifree resolution. Thus we write H p (Hom∗∧V (∧V ⊗ ∧U, ∧V ), d1 ) = Extp(∧V,d1 ) (lk, (∧V, d1 )) and make the Definition. The depth of the minimal Sullivan algebra (∧V, d) is the least p (or ∞) such that Extp(∧V,d1 ) (lk, (∧V, d1 )) 6= 0.

page 272

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Depth of a Sullivan algebra and of a Sullivan Lie algebra

273

Similarly, the wedge degree in ∧V defines a grading in H(∧V, d1 ), which we denote by H p,∗ (∧V, d1 ) and we make the Definition. The global dimension of (∧V, d) is the largest p (or ∞) such that H p,∗ (∧V, d1 ) 6= 0. The next Theorem generalizes Theorem 35.13 in [18]. Theorem 10.1. For any minimal Sullivan algebra (∧V, d), depth (∧V, d) ≤ cat (∧V, d) ≤ gldim (∧V, d) . Moreover, if depth (∧V, d) = cat (∧V, d) then cat (∧V, d) = gldim (∧V, d). proof: As above let (∧V, d1 ) denote the quadratic Sullivan algebra for (∧V, d). Suppose cat (∧V, d) = m. In Corollary 9.1 (p. 257) we constructed a (∧V, d)-semifree resolution '

(∧V ⊗ (lk ⊕ Z0m,∗ ), d) −→ (∧V / ∧>m V, d) which after filtering by the wedge degrees gives a bigraded quasiisomorphism '

(∧V ⊗ (lk ⊕ Z0m,∗ ), d1 ) −→ (∧V / ∧>m V, d1 ) . It follows exactly as in the proof of Theorem 35.13 in [18] that it is sufficient to prove that if depth (∧V, d) ≥ m then H p,∗ (Hom∧V (∧V ⊗ (Z0m,∗ ⊕ lk), ∧V ), d1 ) = 0

if p < 0 .

Following the idea in §35 of [18] we let ∼ =

(∧V ⊗ ∧V ⊗ ∧U, d1 ) −→ (∧V, d1 ) be a Sullivan model for the multiplication (∧V, d1 ) ⊗ (∧V, d1 ) → (∧V, d1 ). Since ∧V is not required to be simply connected, we need to rely on Proposition 3.9 (p. 103). According to Corollary 3.4 (p. 105) we may assume that d1 : ∧U → (V ⊗ 1 ⊕ 1 ⊗ V ) ⊗ ∧U . Applying the augmentation ε : (∧V, d1 ) → lk to the middle term of (∧V ⊗ ∧V ⊗ ∧U, d1 ) and to (∧V, d1 ) ' gives a quasi-isomorphism (∧V ⊗ ∧U, d1 ) → lk, exhibiting this cdga as an acyclic closure of (∧V, d1 ). In the same way, dividing by ∧>k V yields quasi-isomorphisms
' ∧V ⊗ (∧V / ∧>k V ) ⊗ ∧U, d1 −→ (∧V / ∧>k V, d1 ) which for k = m lifts to a quasi-isomorphism of bigraded (∧V, d1 )-modules,
' ∧V ⊗ (∧V / ∧>m V ) ⊗ ∧U, d1 −→ (∧V ⊗ (lk ⊕ Z0m,∗ ), d1 ) .

page 273

January 8, 2015

14:59

9473-Rational Homotopy II

274

RHT3

Rational Homotopy Theory II

Thus we only have to prove that if depth (∧V, d) ≥ m, then for p < 0, H p,∗ (Hom∧V (∧V ⊗ ∧V / ∧>m V ⊗ ∧U, ∧V ), d1 ) = 0 .

(10.4)

Denote (Hom∧V (∧V ⊗ (∧V / ∧≥k V ) ⊗ ∧U, ∧V ), d1 ) by C(∧V / ∧≥k V ), and identify ∧≥k V /∧≥k+1 V as the trivial ∧V -module ∧k V . Then it follows by induction on k from the short exact sequences 0 → C(∧V / ∧≥k V ) → C(∧V / ∧≥k+1 V ) → C(∧k V ) → 0 that H p,∗ (C(∧V / ∧>k V )) = 0, for p < m − k. This proves (10.4).



Recall from Theorem 9.3 (p. 248) that if (∧V, d) → (∧Z, d) is a surjective morphism of minimal Sullivan algebra then cat (∧V, d) ≥ cat (∧Z, d). Here we establish the analogous Proposition 10.1. Suppose (∧W ⊗ ∧Z, d) is a decomposition of a minimal Sullivan algebra as a Sullivan extension (∧W ⊗∧Z, d) of a minimal Sullivan algebra (∧W, d). Then the fibre, (∧Z, d) satisfies depth (∧W ⊗ ∧Z, d) ≥ depth (∧Z, d) . proof: The associated quadratic Sullivan algebra of (∧W ⊗ ∧Z, d) decomposes as a Sullivan extension of quadratic Sullivan algebras (∧W, d1 ) → (∧W ⊗ ∧Z, d1 ) → (∧Z, d1 ). Thus according to Proposition 3.13 (p. 112), the acyclic closure of (∧W ⊗ ∧Z, d1 ) may be decomposed as a Λ-extension (∧W ⊗ ∧UW , d1 ) → (∧W ⊗ ∧UW ⊗ ∧Z ⊗ ∧UZ , d1 ) in which (∧W ⊗ ∧UW , d1 ) is the acyclic closure of (∧W, d1 ), the fibre (∧Z ⊗ ∧UZ , d1 ) is the acyclic closure of (∧Z, d1 ) and d1 : ∧UZ → (W ⊕ Z) ⊗ ∧UW ⊗ ∧UZ . Thus we may write Ext(∧W ⊗∧Z,d1 ) (lk, (∧W ⊗ ∧Z, d1 )) = H Hom∗∧W ⊗∧Z (∧W ⊗ ∧Z ⊗ ∧UW ⊗ ∧UZ , ∧W ⊗ ∧Z)




= H ( Hom∗∧W (∧W ⊗ Hom∗∧Z (∧Z ⊗ ∧UW ⊗ ∧UZ , ∧W ⊗ ∧Z)) ) . Filtering by the wedge degree of ∧W in the target space produces a first quadrant spectral sequence E∗p,q , for which   E1p,q = Homp∧W ∧W ⊗ ∧UW , Extq(∧Z,d ) (lk, (∧W, 0) ⊗ (∧Z, d1 )) . 1

If

Extq(∧Z,d ) (lk, (∧Z, d1 )) 1

= 0, q < m then it follows that

Extr(∧W ⊗∧Z,d1 ) (lk, (∧W ⊗ ∧Z, d1 )) = 0 ,

r < m.



page 274

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Depth of a Sullivan algebra and of a Sullivan Lie algebra

275

Example 1. The depth of a minimal Sullivan algebra (∧V, d) is equal to 0 if and only if V is finite dimensional and is concentrated in even degrees. Suppose first that V is finite dimensional and concentrated in even degrees. In that case ∧U is an exterior algebra, ∧U = ∧(u1 , · · · , ur ). Then a cycle f ∈ Hom0∧V ((∧V ⊗ ∧U, d), (∧V, d)) is defined by f (u1 · · · ur ) = 1 and f (α) = 0 if α ∈ ∧q is concentrated in odd degrees and is abelian. It follows that L>q is solvable and finite dimensional (Proposition 11.5). Thus L is finite dimensional and so L = rad L. In particular, by Theorem 11.1, dim Leven = depth L.  Now we describe briefly how the Propositions and Theorem above follow from the results of §10. The key fact is that a connected graded Lie algebra L of finite type is the homotopy Lie algebra of its associated quadratic Sullivan algebra (∧V, d1 ) = C ∗ (L), as shown in Proposition 2.1 (p. 55); here C ∗ (L) is the Cartan-Chevalley-Eilenberg construction defined in §2.1. Let (∧V ⊗ ∧U, d1 ) denote the acyclic closure of (∧V, d1 ). Then, since L ∼ = is connected, Theorem 6.1 (p. 180) provides an isomorphism U L → (∧U )# which converts right multiplication in U L to the dual of the holonomy representation of L in ∧U . Thus this isomorphism dualizes to an isomorphism ∼ =

∧U −→ (U L)# of U L-modules, where L acts in ∧U by the holonomy representation and in (U L)# by the dual of right multiplication in U L. With these identifications, Proposition 11.1 (i) follows immediately from the isomorphism above, and (ii) restates the definitions, p. 268. For

page 303

January 8, 2015

14:59

304

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

assertion (iii) recall that by definition the quadratic part, (∧V, d1 ) of (∧V, d) is C ∗ (L), and so Theorem 10.2 (p. 278) gives depth L = depth(∧V, d1 ) = depth(∧V, d) and gldim L = gldim(∧V, d1 ) = gldim (∧V, d) . Thus assertion (iii) restates Theorem 10.1 (p. 273). p For Proposition 11.2 note that since (U L)# ∼ = ∧U , ExtU E (lk, U L) =  # UE Torp (lk, ∧U ) . Thus assertion (i) translates to depth E = depthL E, which is Proposition 10.15(ii) (p. 300). With this, assertions (ii) and (iii) follow from Proposition 10.6 (p. 290). For Proposition 11.3 observe first that by Proposition 10.7 (p. 290), dim L < ∞. Then it follows from Theorem 10.5 (p. 294) that L is concentrated in odd degrees, and hence is an abelian Lie algebra. Thus L ∧x TorU ∗ (lk, U L) is the finite tensor product of spaces of the form Tor (lk, ∧x) UL with deg x odd, and a straightforward calculation gives Tor≥1 (lk, U L) = 0. In view of Proposition 11.3, Proposition 11.4(i) is the dual to Proposition 10.8 (p. 291) while Proposition 11.4(ii) is dual to Proposition 10.9 (p. 291). For Proposition 11.4(iii), recall from Proposition 10.4(iv) (p. 282) that the acyclic closure of C ∗ (L/I) has the form (C ∗ (L/I) ⊗ ∧S, d1 ) in which ∧S is the L/I-module of Proposition 10.10 (p. 292). Thus assertion (iii) follows by dualizing Proposition 10.10. Next, in Proposition 11.5, assertion (i) is Proposition 10.8 (p. 291) and (ii) is a special case of Theorem 10.6 (p. 295). Finally, Theorem 11.1 is just Theorem 10.6.

11.2

Modules over an abelian Lie algebra

Proposition 11.6. Suppose a connected graded Lie algebra, L, of finite type is abelian. If for some L-module, M , and some m, Extm U L (M, U L) 6= 0, then for some x ∈ M and some n the map U (L≥n ) → M , a 7→ a · x, is injective. Lemma 11.1. If F is a connected graded Lie algebra of finite type and concentrated in odd degrees, and if N is any F -module, then ExtpU F (N, U F ) = 0, p ≥ 1. proof: Since F is concentrated in odd degrees it is abelian. In view of Proposition 11.4(ii), it is sufficient to prove the Lemma when F is finitely

page 304

January 8, 2015

14:59

9473-Rational Homotopy II

Depth of a connected graded Lie algebra of finite type

RHT3

305

generated (and therefore finite dimensional). Let v1 , · · · , vr be a basis of F . The short exact sequence 0 → v1 N → N → N/v1 N → 0 makes it sufficient to prove the Lemma for v1 N and N/v1 N . But v12 = 12 [v1 , v1 ] = 0 in U F and so v1 · v1 N = 0. Thus we are reduced to the case of F -modules N for which v1 N = 0. Iterating this argument for v2 , · · · , vr reduces us to the case F · N = 0. Thus N = ⊕lk ei is the direct sum of one dimensional F -modules. But in this case Y ExtU F (⊕i lkei , U F ) = ExtU F (lkei , U F ) i

and it follows from Proposition 11.3 that Ext∗U F (lk, U F ) = Ext0U F (lk, U F ).  proof of Proposition 11.6: First note that because L is abelian all subspaces of L are ideals. Now if m = 0 then, because Ext0U L = HomU L , there is an x ∈ M and an f ∈ HomU L (M, U L) such that f (x) 6= 0. Choose n such that f (x) ∈ (U L) 0. Moreover, if, in addition, log index H(∧V, d) < ∞ then for some fixed R,   log dim Lk = log index L < ∞ . lim max n→∞ n≤k≤n+R k Theorem 12.10. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) for which V = V ≥2 has finite type and infinite dimension. If cat (∧V, d) < ∞ and log index H(∧V, d) < log index L, then there are constants R, β, γ > 0 with the following property: for some K and all n ≥ K,   β log dim Lk log n ≤ max ≤ log index L + . log index L − γ n≤k≤n+R n k n Theorem 12.11. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) for which V = V ≥2 has finite type and infinite dimension. If H >N (∧V, d) = 0 then there are constants β, γ > 0, depending only on H(∧V, d), and with the following property: for some K and all n ≥ K,   log n log dim Lk β ≤ max log index L − γ ≤ log index L + . n+2≤k≤n+N n k n 12.4

The gap theorem

Theorem 12.3. Suppose X is an N -dimensional rationally hyperbolic connected CW complex. Then for each n ≥ 0 there is some k ∈ [n + 2, n + N ] such that πk (X) ⊗ Q 6= 0. e is also an N -dimensional rationally hyperbolic proof: The covering space X e = H ≤N (X) e connected CW complex. Thus by Lemma 12.1 (p. 316) H(X) e is finite dimensional. Thus the minimal Sullivan model, (∧V, d), for X

page 318

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Trichotomy

319

satisfies H(∧V, d) = H ≤N (∧V, d) is finite dimensional and so V = V ≥2 has finite type. The proof of Theorem 33.3 in [18] now shows that for each n there is some k ∈ [n + 2, n + N ] such that V k 6= 0. This gives Theorem 12.3.  12.5

Rationally infinite spaces of finite category

Theorem 12.6 Suppose (∧V, d) is the minimal Sullivan model of a rationally infinite simply connected CW complex, X. If cat X < ∞ then for some K and R, and for all n ≥ K, dim V k = ∞ ,

some k ∈ [n, n + R] .

proof: We proceed in three steps and prove successively: Step One: dim V k = ∞, some k. Step Two: dim V k = ∞ for infinitely many k. Step Three: proof of the Theorem. Step One: Suppose dim V k < ∞ for all k. Then H(X) = H(∧V, d) has finite type and so Theorem 1.6 (p. 43) shows that πk (X) ⊗ Q ∼ = V has finite type, contradicting our hypothesis. Thus dim V k = ∞ for some k. Step Two: Suppose dim V k = ∞ for only finitely many k. Then for some N we have dim V N = ∞

and dim V k < ∞ ,

k>N.

Divide the Sullivan algebra by V N , d) in which dim W N is finite, but may be arbitrary large. If cat X = m < ∞ then Theorem 9.2 (p. 247), and the Mapping Theorem 9.3 (p. 248) give cat(∧W N ⊗ ∧V >N , d) ≤ cat(∧V, d) ≤ m . Since d(W N ) = 0 it follows by Proposition 9.3 (p. 250) that ∧m+1 W N ⊂ Im d . Now for v ∈ V >N write dv = ϕ(v) + ψ(v) with ϕ(v) ∈ ∧≥2 W N and ψ(v) ∈ ∧W N ⊗ ∧+ V >N . Since ∧m+1 W N ⊂ Im d it follows that ∧m+1 W N

page 319

January 8, 2015

14:59

9473-Rational Homotopy II

320

RHT3

Rational Homotopy Theory II

is contained in the ideal generated by ϕ(V [N +1,(m+1)N −1] ). Thus setting p = dim V [N +1,(m+1)N −1] we have dim ∧m+1 W N ≤ p dim ∧m−1 W N . But as dim W N → ∞ so does dim ∧m+1 W N /dim ∧m−1 W N , and we have again arrived at a contradiction. Thus dim V k = ∞ for infinitely many k. Step Three: Here we suppose the Theorem itself is false, and deduce a contradiction. Again denote cat(∧V, d) by m. Since the Theorem is false, we may find an infinite sequence N1 < N2 < · · · with the following two properties: dim V Ni = ∞

and dim V j < ∞ ,

N i + 1 ≤ j ≤ Ni + i + 1 .

We now construct a quotient Sullivan algebra ϕ : (∧V, d) → (∧W, d) such that W has finite type, and such that  ∼  s = s   if dim V s < ∞ . ϕ : V → W , NiX +i+1 (12.4) j  Ni  dim W ≥ (m + 1)i dim V , i ≥ 1 .   j=Ni +1

The construction is by an obvious induction: If ϕ is constructed in ∧V ≤s it defines a surjection (∧V, d) → (∧W ≤s ⊗ ∧V >s , d). If dim V s+1 < ∞ then extend ϕ by the identity in V s+1 . If dim V s+1 is ∞ then, since (∧W ≤s ) has finite type, there are subspaces U s+1 ⊂ V s+1 of finite codimension for which dU s+1 = 0. Extend ϕ by dividing by some U s+1 , where U s+1 is chosen so that W s+1 satisfies (12.4) if s + 1 = Ni , some i. Then by the Mapping Theorem 9.3 (p. 248), cat (∧W, d) ≤ m . The rest of the proof proceeds by an analysis of the homotopy Lie algebra, L, of (∧W, d). Since W has finite type, so does L. Thus setting depth L = q we have from Theorem 10.1 (p. 273) and Theorem 10.2 (p. 278) that q = depth L ≤ cat(∧W, d) ≤ m . Then, from (12.4), we obtain dim LNi −1 ≥ (q + 1)i

N i +i X j=Ni

dim Lj ,

i ≥ 1.

(12.5)

page 320

January 8, 2015

14:59

9473-Rational Homotopy II

Trichotomy

RHT3

321

Next observe that for any sub Lie algebra F ⊂ L there is an isomorphism U L = U F ⊗ C of U F -modules. We shall use without further reference the consequence: F UF TorU k (Q, U F ) 6= 0 ⇔ Tork (Q, U L) 6= 0 .

In particular, recall from Proposition 10.7 (p. 290) that there is a sub Lie algebra E ⊂ L generated by finitely many elements x1 , · · · , xt such that the inclusion induces a non-zero map E UL TorU q (Q, U E) → Torq (Q, U L) .

Thus depth E ≤ q. Moreover, if depth E = 0 then by Proposition 11.3 E (p. 302) TorU k (Q, U L) = 0, k ≥ 1. Thus it would follow that depth L = 0 and so, by that Proposition, that dim L < ∞. Since by Step Two this is not the case, we have 0 < depth E ≤ q . Now choose any i > t t M

Pt

j=1

deg xj . Then

adL (xj ) : LNi −1 −→

j=1

t M

LNi −1+deg xj

j=1

PNi +i maps LNi −1 into a vector space of dimension at most t j=N dim Lj . It i follows from (12.5) that for Z = ∩tj=1 ker adL (xj ) we have the inequality dim(Z ∩ LNi −1 ) ≥ qi

N i +i X

dim Lj .

(12.6)

j=Ni

Moreover, since Z commutes with E, the vector space Z ∩ E is a central ideal in E. Therefore Theorem 11.1 (p. 303) gives dim Z ∩ E < ∞, and so for some M , Z ∩ E≥M = 0. Denote L(1) = L>M ∩ Z. On the one hand, the composite L E U (E⊕L(1)) (Q, U L) → TorU TorU q (Q, U L) q (Q, U L) → Torq

is non-zero, and so depth(E ⊕ L(1)) ≤ q. Moreover, U (E ⊕ L(1)) is the tensor product of the algebras U E and U L(1), and thus M U (E⊕L(1)) U L(1) E Tork (Q, U (E⊕L(1))) = TorU (Q, U L(1)) . j (Q, U E)⊗Tor` j+`=k

Thus q ≥ depth(E ⊕ L(1)) = depth E + depth L(1). Since depth E > 0 this yields depth L(1) ≤ q − 1 .

page 321

January 8, 2015

322

t

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

On the other hand, for i sufficiently large (Ni > M and i ≥ Pt j=1 deg xj ), (12.6) yields dimL(1)Ni −1 ≥ qi

N i +i X

dimL(1)j .

j=Ni

Since depth L(1) < q we may now iterate the process above to construct an infinite sequence of infinite dimensional Lie algebras L = L(0 ⊃ L(1) ⊃ · · · with strictly decreasing depths. This is obviously impossible, and so the assumption that the Theorem fails produces a contradiction. This completes the proof.  Theorem 12.7. Suppose X is a rationally infinite connected CW complex. If cat X < ∞ then for some K and R, and for all n ≥ K, dim πk (X) ⊗ Q = ∞ ,

some k ∈ [n, n + R] .

The Theorem will follow via the Postnikov-type decomposition Lemma below from a simple adaptation of the proof of Theorem 12.5. Lemma 12.2. Let Y be a simply connected topological space such that for some n, dim πi (Y ) ⊗ Q < ∞ ,

i dim πni (Fi ) ⊗ Q ≥ (m + 1)i

niX +i+1

dim πj (X) ⊗ Q .

j=ni +1

Now according to Theorem 9.7 in [18] each Fi admits a rational homology equivalence into a rational space (Fi )Q . The maps Fi → Fi−1 induce maps (Fi )Q → (Fi−1 )Q , which we may also convert to fibrations. Thus we may suppose that the Fi are themselves rational spaces, and that the groups π∗ (Fi ) = π∗ (Fi ) ⊗ Q satisfy the condition above. ∼ = Set F = lim Fi . By construction, πj (Fi ) −→ πj (Fi−1 ) if j < ni , and ←− ∼ = so πj (F ) = lim πj (Fi ) → πj (Fi ) if j < ni . Moreover, it follows from the ←− conditions above that π∗ (F ) = π∗ (F ) ⊗ Q has finite type. Thus, by Lemma 12.1 and Theorem 12.5, the minimal Sullivan model (∧V, d) of F satisfies dim W ni ≥ (m + 1)i

niX +i+1

dim W j ,

i ≥ 1.

j=ni +1

It also follows that π∗ (F ) ⊗ Q → π∗ (X) ⊗ Q is injective, and so again by Theorem 25.6 in [18], cat (∧W, d) = cat0 (F ) ≤ cat0 X ≤ m . Thus the homotopy Lie algebra of (∧W, d) satisfies (12.5) (p. 320) and the proof of Theorem 12.5 shows that this is impossible. This completes the proof of Theorem 12.6. 

page 324

January 8, 2015

14:59

9473-Rational Homotopy II

Trichotomy

12.6

RHT3

325

Rationally infinite CW complexes of finite dimension

Theorem 12.8. Suppose X is an N -dimensional rationally infinite connected CW complex. Then for each n ≥ 0 there is some k ∈ [n + 2, n + N ] such that dim πk (X) ⊗ Q = ∞ .

proof: All spaces in this proof will be based with base points denoted by ∗, and all maps and homotopies will be base point-preserving; for simplicity of notation a based space (Y, ∗) will usually be denoted simply by Y . Moreover, in view of Lemma 12.1 it is sufficient to prove the Theorem for e Thus we may assume X is simply connected. the universal cover X. The proof of the Theorem will be in three steps Step One: The groups Gq (S, Y ), q ≥ 2. Step Two: The exact sequence Gq (S ∪h Dk+1 , Y ) → Gq (S, Y ) → πq+k (Y ) . Step Three: Completion of the proof. Step One: The groups Gq (S, Y ), q ≥ 2. Here we generalize the subgroups Gq (−) ⊂ πq (−) introduced by Gottlieb in [24]. For this, fix a continuous map σ:S→Y from a connected CW complex pointed by a 0-cell. Then a relative Gottlieb map is a pair (ϕ, f ) : (S q ×S, S q ×{∗}) → Y such that ϕ restricts to f ∨σ in S q ∨ S. We say (ϕ1 , f1 ) ∼ (ϕ2 , f2 ) if there is a homotopy of pairs restricting to the constant homotopy in ∗ × S, and denote the set of homotopy classes by Gq (S, Y ). Addition in Gq (S, Y ) is defined as follows. Denote the equator sphere in S q by S q−1 and let ∆ : S q → S q /S q−1 = S q ∨ S q be the quotient map. Using the equality (S q ∨ S q ) × S = (S q × S) ∪∗×S (S q × S) we suppose (ϕ1 , f1 ) and (ϕ2 , f2 ) are relative Gottlieb maps and set ϕ1 + ϕ2 = (ϕ1 ∪ ϕ2 ) ◦ (∆ × idS ) : S q × S → Y . Then ϕ1 + ϕ2 restricts to f1 + f2 in S q and we set (ϕ1 , f1 ) + (ϕ2 , f2 ) = (ϕ1 + ϕ2 , f1 + f2 ). This construction clearly passes to homotopy classes,

page 325

January 8, 2015

326

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

and thereby makes Gq (S, Y ) into an abelian group. By construction, the map (ϕ, f ) 7→ f defines a homomorphism, α : Gq (S, Y ) → πq (Y ) .

(12.7)

Step Two: The exact sequence Gq (S ∪h Dk+1 , Y ) → Gq (S, Y ) → πq+k (Y ) . Suppose now that the map σ of Step One is the restriction of a map σ : S ∪h Dk+1 → Y , where S ∪h Dk+1 is the CW complex obtained by attaching a (k + 1)-cell to S via a map h : S k → S. (As usual, for any r ≥ 1, Dr denotes the disk with boundary S r−1 .) Denote S ∪h Dk+1 by T . Then a relative Gottlieb map (ϕ, f ) for S and Y defines the map ϕ ∪ σ : (S q × S) ∪∗×S T → Y . On the other hand, recall the classical identification (Dq+k+1 , S q+k ) = (Dq × Dk+1 , Dq × S k ∪S q−1 ×S k S q−1 × Dk+1 ). Let ρ : Dq → Dq /S q−1 = S q be the quotient map. Thus we may define ψ = (ϕ ◦ (ρ × h) ∪ (∗ × σ)) : S q+k → Y , where ∗ denotes the constant map of S q−1 on the base point of S. The homotopy class of this map depends only on the homotopy class of (ϕ, f ). Thus the correspondence ϕ 7→ ψ gives a set map Gq (S, Y ) → πq+k (Y ) . q

(12.8) q

On the other hand, restriction from S × T to S × S defines a homomorphism Gq (T, Y ) → Gq (S, Y ) and we complete this section by showing that (12.8) is a homomorphism with kernel the image of Gq (T, Y ). To show that (12.8) is a homomorphism we define the additive structure in πq+k (T ) by the map S q+k → S q+k /S q+k−1 = S q+k ∨ S q+k with equator sphere S q+k−1 given by S q+k−1 = Dq−1 × S k ∪S q−2 ×S k S q−2 × Dk+1 . That (12.8) is a homomorphism is then a straightforward verification from the construction of ψ. Finally, ψ is null homotopic precisely when it extends to Dq × Dk+1 or, equivalently, when ϕ ◦ σ extends to ((S q × S) ∪∗×S T ) ∪ψ (Dq × S k+1 ). But a second trivial verification gives ((S q × S) ∪∗×S T ) ∪ψ (Dq × Dk+1 ) = S q × T .

page 326

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Trichotomy

327

It follows that the kernel of (12.8) is the image of Gq (T, Y ). Step Three. Completion of the proof of Theorem 12.8. As observed at the start of the proof we may assume X is a simply connected N -dimensional CW complex, pointed by a zero cell. We establish Lemma 12.3. Suppose for some n ≥ 0 that dim πk (X) ⊗ Q < ∞ ,

k ∈ [n + 2, n + N ] .

Then dim πn+1 (X) ⊗ Q < ∞. proof: We suppose the Lemma fails for some n, and deduce a contradiction. For this we construct by induction an increasing sequence of finite sub complexes of X, ∗ = S0 ⊂ S1 ⊂ · · · ⊂ X and maps ϕi : S n+1 × Si → Si+1 such that the maps fi = ϕi |S n+1 ×∗ represent linearly independent elements of πn+1 (X) ⊗ Q. By hypothesis dim πn+1 (X) ⊗ Q = ∞ and so we may find f0 : S n+1 → X. Suppose ϕi , Si are constructed, and let Si+1 ⊂ X be any finite sub complex containing ϕi (S n+1 × Si ). Then (cf. (12.7)) we have the commutative diagram ⊂

Gn+1 (Si+1 , X)

Gn+1 (Si , X)

' w πn+1 (X) . It follows from Step Two by induction on the number of relative cells in (Si+1 , Si ) that Gn+1 (Si+1 , X) ⊗ Q has finite codimension in Gn+1 (Si , X). This will also be true for each Gn+1 (Sj , X)⊗Q ⊂ Gn+1 (Sj−1 , X)⊗Q, j ≤ i. In particular, since Gn+1 (∗, X)⊗Q = πn+1 (X)⊗Q, Gn+1 (Si+1 , X)⊗Q has finite codimension in πn+1 (X) ⊗ Q. Choose (ϕi+1 , fi+1 ) ∈ Gn+1 (Si+1 , X) so that [fi+1 ], [fi ], · · · , [f0 ] are linearly independent in πn+1 (X) ⊗ Q. Finally, define γ : S n+1 × · · · × S n+1 (N + 1 factors) → X by composing id×fN

γ : S n+1 × · · · × S n+1 −→ S n+1 × · · · × S n+1 ×SN −1 | {z } | {z } N +1 factors

id×ϕN −1

−→

N factors

S n−1 × · · · × S n−1 ×S N −2 → · · · → X . | {z } N −1 factors

page 327

January 8, 2015

328

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

By construction, πn+1 (γ) ⊗ Q is injective. Moreover, as described at the start of the proof of Lemma 12.2, there is a fibration F →Y →B ∼ =

in which π≥n+1 (B) = 0, πj (Y ) → πj (B) for j < n + 1, and Y ' X. Since S n+1 × · · · × S n+1 is n-connected the map γ 0 : S n+1 × · · · × S n+1 → Y corresponding to γ is homotopic to a map γ 00 : S n+1 × · · · × S n+1 → F for which πn+1 (γ 00 ) ⊗ Q is injective. ∼ = But π∗ (F ) → π∗ (Y ) → π∗ (X) is injective, and so Theorem 25.6 in [18] and Theorem 9.2 (p. 247) give cat0 F ≤ cat0 X ≤ N . Now the Hurewicz homomorphism identifies πn+1 (γ 00 ) ⊗ Q with Hn+1 (γ 00 ) and so Hn+1 (γ 00 ) is injective. Denote by βi ∈ Hn+1 (S n+1 × · · · × S n+1 ) the image of [S n+1 ] under the inclusion of the ith sphere. Since Hn+1 (γ 00 ) is injective there are classes ωi ∈ H n+1 (F ) such that hH n+1 (γ 00 )ωi , βj i = δij . In particular ω0 • · · · • ωN 6= 0, contradicting the fact that cat0 F ≤ N . This completes the proof of the Lemma.  It follows from Lemma 12.3 that if dim πk (X)⊗Q < ∞, k ∈ [n+2, n+N ] then dim πk (X) ⊗ Q < ∞ for 2 ≤ k ≤ N . As described in the proof of Lemma 12.2 (p. 322) we may, by adjoining cells of dimension ≥ N + 2 to X, construct a CW complex Z such that π≤N (Z) = π≤N (X) and π>N (Z) = 0. Convert the map X → Z to a fibration and use a Serre spectral argument ∼ = to see that H≤N (X) −→ H≤N (Z). Since π∗ (Z) has finite type so, by Theorem 1.6 (p. 43) and Lemma 12.2 (p. 322), does H∗ (Z). Thus H≤N (X) has finite type. But X is an N -dimensional CW complex. Thus H∗ (X) = H≤N (X) has finite type and Theorem 1.6 (p. 43) asserts that π∗ (X)⊗Q has finite type. This contradicts the hypothesis of Theorem 12.7 and thereby completes the proof. 

page 328

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Chapter 13

Exponential growth

Recall from §12.1 that for any graded vector spaces, S = {Sk }k≥0 and V = {V k }k≥1 , we write log index S = lim sup k

log dim Sk k

and log index V = lim sup k

log dim V k . k

In this Chapter we consider minimal Sullivan algebra (∧V, d) with homotopy Lie algebra L, in which V = V ≥2 has finite type but is infinite dimensional. Our purpose is to prove (Proposition 12.2) log index H(∧V, d) ≤ log index L , and log index H(∧V, d) < ∞ ⇔ log index L < ∞ , and the following three Theorems stated in §12.3. Proposition 12.2 follows from Proposition 13.2 immediately below and the rest of the Chapter is devoted to the proofs of the Theorems. As explained in Chapter 12, these establish a very strict form of exponential growth for the sequence (dim Ln ) and thus for the ranks of the homotopy groups of CW complexes. Theorem 12.9. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) for which V = V ≥2 has finite type and infinite dimension. If cat(∧V, d) < ∞ then log index L > 0. Moreover, if, in addition, log index H(∧V, d) < ∞ then for some fixed R,   log dim Lk lim max = log index L < ∞ . n→∞ n≤k≤n+R k

329

page 329

January 8, 2015

330

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Theorem 12.10. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) for which V = V ≥2 has finite type and infinite dimension. If cat (∧V, d) < ∞ and log index H(∧V, d) < log index L, then there are constants R, β, γ > 0 with the following property: for some K and all n ≥ K,   log dim Lk β log n ≤ max ≤ log index L + . log index L − γ n≤k≤n+R n k n Theorem 12.11. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) for which V = V ≥2 has finite type and infinite dimension. If H >N (∧V, d) = 0 then there are constants β, γ > 0, depending only on H(∧V, d), and with the following property: for some K and all n ≥ K,   log dim Lk β log n ≤ max ≤ log index L + . log index L − γ n+2≤k≤n+N n k n All three Theorems establish exponential growth detected in intervals of a fixed length. Essentially the only difference is that in the first one the error is just some positive “ε” decreasing to zero, while in the second and third it can be established as β/n and γ/ log n. Additionally in the third Theorem, where H >N (∧V, d) = 0, the length of the interval is N − 2 and β and γ depend only on H(∧V, d). Consequently the proofs of the three Theorems can and will be presented as a single proof except that the technique for estimating the error will vary, depending on which set of hypotheses are being considered. The strategy for the proof is then as follows: §13.1 contains the proof of Proposition 12.2. §13.2 establishes essential technical lemmas about the growth of a universal enveloping algebra. §13.3 detects exponential growth in intervals of the form [k, k+(m+1)k] where m = cat (∧V, d). §13.4 establishes the basis for the error estimates β/n and γ/ log n in the second and third Theorems. §13.5 detects exponential growth in intervals whose length grows, but in a more constrained way. §13.6 detects exponential growth in intervals of constant length, and shows that if H >N (∧V, d) = 0 then the exponential growth occurs in intervals of length N − 2.

page 330

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Exponential growth

13.1

331

The invariant log index

When log index V > 0 we say that V grows at most exponentially. Recall also (§1.1) that the Hilbert series of a graded vector space V of finite type P is the formal power series k dim Vk z k . Remarks. For graded vector spaces, V , of finite type: (i) log index V = −∞ if and only if dim V < ∞. Otherwise log index V ≥ 0. (ii) log index V = α > 0 means that : (a) For each ε > 0, there is some K such that for k ≥ K, dim V k ≤ ek(α+ε) (b) there is an infinite sequence ki such that log (dim V ki )1/ki → α. (iii) In particular, α = log index V < ∞ if and only if e−α is the radius of P convergence of the Hilbert series dimV k z k . We begin by recalling the well-known result of Babenko: Proposition 13.1. ([3]) Let L = L≥1 be a graded Lie algebra of finite type. If log index L > 0, then L and U L have the same log index. proof: Since L ⊂ U L, log index L ≤ log index U L. Thus we need only consider the case log index L < ∞. Set an = dim Ln . Then the Hilbert series of U L is given by Q 2n+1 a2n+1 X ) p n (1 + z . bp z = Q 2n )a2n (1 − z n p P Denote by R the radius of convergence of the series n an z n . It follows from Remark (iii) that R < 1. Then for z < R, by the mean value theorem, we have z z ≤ log(1 + z) ≤ z ≤ log(1 − z)−1 ≤ . 1+R 1−R Therefore ∞ X X X ak z k X zk log( bp z p ) ≤ ak z k + ≤ ak . 1−R 1−R p k odd

k even

k=1

This shows that the radius of convergence of U L is greater than or equal to R, and since U L contains L, they are equal. 

page 331

January 8, 2015

14:59

9473-Rational Homotopy II

332

RHT3

Rational Homotopy Theory II

Babenko’s result has the following important consequence: Proposition 13.2. Let L be the homotopy Lie algebra of a minimal Sullivan algebra (∧V, d) in which V = V ≥2 has finite type. Then log index H(∧V, d) ≤ log index L ≤ 2 sup i≥2

log dim H i (∧V, d) . i−1

In particular, log index H(∧V, d) < ∞

⇔

log index L < ∞ .

proof: Since ∧V is the universal enveloping algebra of V , regarded as an abelian Lie algebra, it follows from Proposition 13.1 (p. 331) that log index ∧V = log index V = log index L = log index U L. The first inequality is immediate. For the second, we may suppose log index H(∧V, d) < ∞, in which case log dim H i (∧V, d) < ∞. i−1 i≥2
1 Thus ω = supi≥2 dim H i (∧V, d) i−1 < ∞. We show now that sup

dim (U L)k ≤ (2ω)k ,

k ≥ 1.

(13.1)

Let (∧V ⊗∧U, d) be the acyclic closure (§3.6) of (∧V, d). By Proposition 3.12 (p. 111) dim U i = dim V i+1 = dim Li . By the Poincar´e Birkhoff Witt theorem, U L ∼ = ∧U as graded vector spaces and so dim(U L)k = dim (∧U )k , k ≥ 0. Thus it suffices to prove (13.1) with (U L)k replaced by (∧U )k . Consider the short exact sequence 0 → ∧+ V ⊗ ∧U → ∧V ⊗ ∧U → ∧U → 0 . Since H(∧V ⊗ ∧U, d) = lk and the quotient differential in ∧U is zero, it follows that d induces an injection of degree one: ∧U → H(∧+ V ⊗ ∧U, d). From the spectral sequence H + (∧V, d) ⊗ ∧U ⇒ H(∧+ V ⊗ ∧U, d) we then obtain k+1 X dim(∧U )k ≤ dim H i (∧V, d) dim (∧U )k+1−i , k ≥ 1. i=2

But when k = 1, dim (∧U )1 = dim U 1 = dim H 2 (∧V, d) ≤ ω . Thus, supposing by induction that (13.1) holds for k < `, we find `

dim (∧U ) ≤

`+1 X i=2

ω i−1 (2ω)`+1−i ≤ (2ω)` .

page 332

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Exponential growth

333

This establishes (13.1). By definition, log ω = supi≥2 so log index L = log index U L ≤ 2 log ω ≤ 2 sup i≥2

log dim H i (∧V,d) , i−1

and

log dim H i (∧V, d) . i−1

The final assertion of the Proposition is immediate because for any graded vector space S = S ≥1 of finite type, if log index S < ∞ so is sup i≥2

log dim Si . i−1 

13.2

Growth of graded Lie algebras

Proposition 13.3. Let L = L≥1 be a graded Lie algebra of finite type. Suppose an infinite sequence of integers 0 < r0 < r1 < · · · , and an infinite sequence of real numbers (λi )i≥0 , satisfy dim Lri ≥ eλi ri , i ≥ 0 ,

and λi+1 ≥ λi + 2

log(ri + 1) log 3 + , ri ri+1

i ≥ 0.

Then L contains a sub Lie algebra E, generated by subspaces in degrees ri , i ≥ 0, such that 1 λ i ri e ≤ dim (E/[E, E])ri ≤ eλi ri , 2

i ≥ 0.

Before undertaking the proof of this Proposition we establish three necessary Lemmas. Lemma 13.1. For any integer s ≥ 1 the coefficients in the power series expansion ∞

X 1−z = ak z k s 1−z−z k=0

satisfy a0 = 1, ak = 0 for 1 < k < s and ak ≤ (s + 1)k/s , k ≥ s. proof: Note that ∞

X 1 1−z = = 1 + z s` 1 − z − zs 1 − z s /(1 − z) `=1



1 1−z

`

page 333

January 8, 2015

14:59

9473-Rational Homotopy II

334

RHT3

Rational Homotopy Theory II

=1+

∞ X `=1

z

s`

 ∞  X `+j−1 j=0

`−1

zj .

In particular, a0 = 1 and ak = 0, 1 ≤ k < s. Fix k ≥ s and let q ≥ 1 and i ∈ [0, s − 1] be the unique integers such that k = qs + i. Then  q  X ` + (q − `)s + i − 1 ak = . `−1 `=1

Write m = ` − 1, so that ak =

 q−1  X m + (q − m − 1)s + i . m

m=0

But m + (q − m − 1)s + i ≤ m + (q − m − 1)s + s − 1 = (q − m)s + (m − 1) . Thus,   [(q − m)s + (m − 1)] · · · [(q − m)s + r] · · · [(q − m)s] m+(q−m−1)s + i ≤ m m! (q − 1)(q − 2) · · · (q − m + r) · · · (q − m) m!   q − 1 =sm . m ≤sm

Hence for k ≥ s ak ≤

q−1 X

sm

m=0



q−1 m



= (1 + s)q−1 ≤ (1 + s)k/s . 

The next Lemma shows that the growth of a graded Lie algebra F and the growth of its space of indecomposable elements F/[F, F ] are very close when F is sufficiently connected. Lemma 13.2. Suppose for some integer s ≥ 1 and some β ≥ 0 that a graded Lie algebra F satisfies Fk = 0 , k < s

and dim (F/[F, F ])k ≤ eβk , k ≥ s .

Then 1

dim (U F )k ≤ e(β + s

log(s+1))k

,

k ≥ 0.

page 334

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Exponential growth

335

proof: Let W be a graded vector space satisfying Wk = 0, k < s, and suppose that dim Wk is the integral part of eβk . The tensor algebra, T W , is the universal enveloping algebra of the free graded Lie algebra, E, generated by W : T W = U E. The respective Hilbert series W (z) for W and U E(z) for U E satisfy ( K ,

some j ≤ 4N N +5 K .

Definition. The critical degree for a simply connected minimal Sullivan algebra (∧V, d) is the least integer σ such that dim V 1+σ > [2(m + 1)]m+1 . Corollary 13.1. The critical degree σ for (∧V, d) satisfies σ < (m + 1)[n0 + 2m+3 r02 (m + 1)2m+4 ] . If H >N (∧V, d) = 0 then σ < 2N +2 N 2N +5 . The next lemma is the key technical step in establishing a weak form of exponential growth for the integers dim Lj . Lemma 13.4. Suppose for integers 0 < q < p ≤ 2q that dim V [q+1,p] = s ≥ 2m, and set λi =

max

dim V j−1 .

i(q+1)≤j≤ip

Then m+1 X i=2

1 λi [i(p − q − 1) + 1] > . si 2(m + 1)m+1

page 338

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Exponential growth

339

proof: Division by the ideal generated by V ≤q defines a quotient Sullivan algebra (∧V ≥q+1 , d). Moreover, the Mapping Theorem 9.3 (p. 248) gives cat(∧V ≥q+1 , d) ≤ m . In particular, V [q+1,p] ⊂ ker d, and hence ∧m+1 V [q+1,p] ⊂ Im d . Choose linear maps of degree 1, θi : V >p → ∧i V [q+1,p] , i ≥ 2, such that P d − i θi : V >p → V >p ∧(∧V ≥q+1 ). Then ∧m+1 V [q+1,p] ⊂

m+1 X

  θi (V >p )∧ ∧m+1−i V [q+1,p] .

i=2 i

Since ∧ V

[q+1,p]

is concentrated in degrees [i(q + 1), ip] it follows that

∧m+1 V [q+1,p] ⊂

m+1 X

  θi (V [i(q+1)−1,ip−1] )∧ ∧m+1−i V [q+1,p] .

i=2

Next, since s ≥ m + 1,   s dim ∧m+1 V [q+1,p] ≥ m + 1

and dim ∧m+1−i V [q+1,p] ≤ sm+1−i .

It follows that m+1 X

s m+1

dim V [i(q+1)−1,ip−1] sm+1−i ≥



λi (i(p − q − 1) + 1)sm+1−i ≥





,

i=2

and so m+1 X

s m+1



.

i=2

2

On the other hand, induction on m gives the inequality (m + 1)! ≤ Thus, since s ≥ 2m, we have   (s/2)m+1 2m sm+1 s m + 1 > (m + 1)m+1 = 2(m + 1)m+1 .


m+1 m+1 . 2

The Lemma follows from these inequalities.  proof of Proposition 13.5: Set q = n0 + kr0 and p = n0 + 2kr0 − 1, with k ≥ 2m. Then by Proposition 13.4(iv), s = dim V [q+1,p] ≥ k ≥ 2m .

page 339

January 8, 2015

14:59

9473-Rational Homotopy II

340

RHT3

Rational Homotopy Theory II

Now set λ = max dim V j , 2q + 1 ≤ j ≤ (m + 1)p. Thus, in the notation of Lemma 13.4, λi ≤ λ, 2 ≤ i ≤ m + 1. Since 1/si ≤ 1/s2 , Lemma 13.4 gives m

1 λ [(m + 1)(p − q)] > . 2 s 2(m + 1)m+1

It follows that λ>

s2 2(p − q)m(m + 1)m+2

and so, since s ≥ k λ>

k . 2r0 (m + 1)m+3

(13.5)

But by definition, λ = dim V j for some j ≤ (m + 1)p, and thus (i) follows from (13.5). (ii) Suppose H >N (∧V, d) = 0. Thus (∧V, d) is quasi-isomorphic to a commutative cochain algebra (A, d) such that A+ = A[2,N ] . It follows that k (A+ ) = 0 if k > N/2 and so it follows from Proposition 9.1(i) (p. 246) that m ≤ N/2. Moreover, in Proposition 13.4(iv) we may, by Proposition 13.4(v), take n0 = 0 and r0 = N −1 in applying part (i) of this Proposition. When N ≥ 4, since m + 1 ≤ N (ii) is immediate from (i). When N = 2 or 3 we have r0 = 1 or 2 and m = 1, and so (ii) follows from (i) by a direct calculation.  Proposition 13.6. (Weak exponential growth) Suppose for some q (e.g. q = critical degree) that dim V q+1 > [2(m + 1)]m+1 . Then q extends to an infinite sequence q = q0 < q1 < · · · such that for each i ≥ 0, qi+1 + 1 = `i (qi + 1) − 1 with 2 ≤ `i ≤ m + 1, and such that  m+1
`i −1 1 dim Lqi > [2(m + 1)]m+1 , dim Lqi ≥ dim Lqi−1 , 2(m + 1) and  dim Lqi ≥

1 2(m + 1)

i +1 ! qq+1

m+1 dim Lq

.

proof: By hypotheses, dim V q0 +1 > [2(m + 1)]m+1 . Now suppose the qi are constructed for i ≤ k and set dim V qk +1 = s. By construction, s ≥

page 340

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Exponential growth

341

[2(m+1)]m+1 ≥ 2m, and so we may apply Lemma 13.4 (p. 338) with q = qk and p = qk + 1. Since then p − q − 1 = 0 we obtain m+1 X λi 1 > , i s 2(m + 1)m+1 i=2 where λi = dim V i(qk +1)−1 . sj In particular, some λj must satisfy λj > 2(m+1) m+2 . We set `k = j and qk+1 = `k (qk + 1) − 2. Then  m+1
`k 1 (dim V qk +1 )`k qk+1 +1 ≥ dim V > dim V qk +1 2(m + 1)m+2 2(m + 1)  m+1 1 2(m+1) [2(m + 1)] , ≥ 2(m + 1) which proves the first two inequalities. Now set  m+1 1 a= , 2(m + 1) and n = 1 + `k + · · · + `2 · · · `k . Then it follows from the second inequality that dim V qk+1 +1 ≥ an · (dim V q+1 )`1 `2 ···`k . Since each `i ≥ 2, n/(`1 `2 · · · `k ) ≤ 21k + · · · + 12 < 1. Thus !`1 `2 ···`k  m+1 1 qk+1 +1 q+1 dim V ≥ dim V . 2(m + 1) But

qk+1 +1 q+1

≤ `1 `2 · · · `k and this proves the third inequality. 

Corollary 13.2. Given a sequence (qi ) as in the Proposition then for all i > j ≥ 0, log dim Lqj log dim Lqi (m + 1) log 2(m + 1) ≥ − . qi + 1 qj + 1 qj + 1 In particular, if s > q0 then log dim Li log dim Lq0 − (m + 1) log 2(m + 1) max ≥ . i+1 q0 + 1 s+1≤i 0. proof: Since qi+1 = `i (qi + 1) − 2 ≤ (m + 1)(qi + 1) − 2 ≤ (m + 2)qi , 1 Proposition 13.6 implies that dim Lqi > C qi , where C = (a dim Lq ) q+1 . It follows that log index L > 0.  Corollary 13.5. There is some K > 1 and some integer n such that k X

dim Li ≥ K k ,

k ≥ n.

i=0

proof: Set n = q, so that dim Lq > 1 1 (a dim Lq ) q+1 , and K = C m+2 . Then

1 a

= [2(m + 1)]m+1 , and set C =

dim Lqi ≥ C qi +1 . For an integer k ≥ n choose i so that qi ≤ k < qi+1 . Then k  1 qi+1 +2 X ≥ Kk . dim Li ≥ dim Lqi ≥ C qi +1 = C `i i=1

 proof of Theorem 13.1: We construct the sequence ri as follows. Set r0 = σ and then, given ri , let ri+1 ∈ (ri , (m + 1)(ri + 1)) be chosen so that for r ∈ (ri , (m + 1)(ri + 1)) log dim Lri+1 − (m + 1) log(2(m + 1)) log dim Lr − (m + 1) log(2(m + 1)) ≤ . r ri+1 It follows from Proposition 13.6 that for all i, dim Lri > (2(m + 1))m+1 . On the other hand, for any ε > 0 there are infinitely many integers log dim Lp > log index L − ε. Each such p must belong to an p such that p interval (ri−1 , (m + 1)(ri−1 + 1)), and thus log dim Lp − (m + 1) log 2(m + 1) log dim Lri − (m + 1) log 2(m + 1) ≤ . p ri It follows that   log dim Lri 1 1 > log index L − ε − (m + 1) log 2(m + 1) − . ri ri−1 ri For p (and therefore ri ) sufficiently large,

(m+1) log 2(m+1) ri−1

log dim Lri > log index L − 2ε . ri

< ε.

page 342

January 8, 2015

14:59

9473-Rational Homotopy II

Exponential growth

RHT3

343

On the other hand, Corollary 13.2 extends ri to a sequence of integers ri = q0 < q1 < · · · such that 2qj ≤ qj+1 < (m + 1)(qj + 1) and log dim Lqj log dim Lri (m + 1) log 2(m + 1) ≥ − . qj + 1 ri + 1 ri + 1 For any k ≥ i, (rk , (m + 1)(rk + 1)) must contain some qj and it follows as above that   log dim Lrk+1 log dim Lri 1 1 1 ≥ +(m+1) log 2(m+1) − − rk+1 ri + 1 rk+1 qj + 1 ri + 1 ≥ log index L − 4ε . Lk Thus since log index L = lim sup log dim it follows that k log dim Lri −→ log index L . ri



13.4

Approximation of log index L

In this section (∧V, d) is a minimal Sullivan algebra satisfying the following conditions:  P i V = V ≥2 has finite type and i≥2 dim V = ∞ , (13.6) cat (∧V, d) = m < ∞ , and we fix the following notation:    L is the homotopy Lie algebra of (∧V, d)   σ = critical degree of (∧V, d)     α = log index L. L

Theorem 13.2. Let L be the homotopy Lie algebra of a minimal Sullivan algebra, (∧V, d), satisfying (13.6). Suppose H >N (∧V, d) = 0 and dim H i (∧V, d) ≤ h, i ≤ N . Then there are positive constants B0 and C0 , depending only on N and h, and such that if log r ≥ max{4, σ, m + 1} 2 log N h then log dim Li B0 log dim Li C0 max − ≤ αL ≤ max + . i r i log r r≤i r and so [x, rad L] ⊂ rad L. Thus rad L is a solvable ideal in L0 and so rad L ⊂ rad L0 . On the other hand, let I ⊂ L0 be a solvable ideal and let I(p) be the image of I ∩ Lp0 in L(p). If x ∈ I(p) and y ∈ L(q) then [y, x] ⊂ I(p + q), and

page 396

January 8, 2015

14:59

9473-Rational Homotopy II

Weight decompositions of a Sullivan Lie algebra

RHT3

397

it follows that ⊕p I(p) is an ideal in L. Moreover, if x ∈ I(p) and y ∈ I(q) such that x + x0 and y + y 0 and y 0 ∈ Lq+1 then there are elements x0 ∈ Lp+1 0 0 are in I. Thus if I is abelian it follows that [x, y] = 0 and so ⊕p I(p) is an abelian ideal in L and hence contained in rad L. P Now set J = I + rad L and for x ∈ J write x = xp with xp ∈ L(p). If P x1 , · · · , xq ∈ rad L then i>q xi ∈ J, and so xq+1 ∈ I(q + 1) + (rad L)(q + 1) ⊂ rad L. Thus each xi ∈ rad L, and since dim rad L < ∞ it follows that for some q, xi = 0, i > q. It follows that x ∈ rad L and I ⊂ rad L. An easy induction now shows that any solvable ideal in L0 is contained in rad L and hence rad L0 = rad L. ' (iv) It follows from Proposition 15.1(ii) that (∧V 1 , d) → C ∗ (L). Then by (i) dim H(C∗ (L))) < ∞, and (iv) follows from Theorem 15.2.  Corollary 15.1. If G is a discrete group whose classifying space BG is a 1-formal Sullivan space for which cat (BG) < ∞, then the Lie algebra L(G) = ⊕p Gp /Gp+1 of Example 1 (p. 389) satisfies (i) depth (L(G) ⊗ Q) ≤ cat (BG), (ii) dim rad (L(G) ⊗ Q) ≤ depth (L(G) ⊗ Q) ≤ cat (BG). (iii) There are constants b, c, αG and K such that 0 < αG < ∞ and
1 nαG e ≤ dim Gn /Gn+1 ⊗ Q ≤ eb enαG , k ≥K. c n proof: By Theorem 7.2 (p. 197), BG has a minimal Sullivan model of the form (∧V 1 , d) and H(∧V 1 , d) has finite type. Moreover, by Theorem 9.2 (p. 247), cat (∧V 1 , d) ≤ cat (BG). Thus the Corollary follows from Theorem 15.3.  Next, again consider a 1-formal minimal Sullivan algebra (∧V, d), and suppose dim H 1 (∧V 1 , d) < ∞. Denote by L the weighted Lie algebra defined in (15.5). Then it follows from Theorem 2.1 (p. 50) that L(1) ∼ = L/[L, L] is finite dimensional, and that Ln = ⊕i≥n L(i). In particular, L is a Sullivan Lie algebra, (∧V 1 , d) is the associated quadratic Sullivan algebra, and L is the associated weighted Lie algebra (cf. Example 1) for the fundamental Lie algebra of (∧V, d). Denote by L the free Lie algebra on L(1). Then the identity of L(1) extends to a surjective morphism of weighted Lie algebras ρ : L → L,

page 397

January 8, 2015

398

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

which is an isomorphism in weight 1. Definition. ([11]) The Chen Lie algebra of (∧V, d) is the quotient Lie algebra h(∧V, d) := L/K , where K is the ideal generated by ker ρ(2) : L(2) → L(2). If (∧V, d) is the minimal Sullivan model of a path connected topological space X then the Chen Lie algebra is called the Chen Lie algebra of X and is denoted by h(X). Theorem 15.4. Suppose (∧V, d) is a 1-formal minimal Sullivan algebra and that dim H 1 (∧V, d) < ∞. Then, with the notation above, the morphism ∼ = ρ0 : L/K −→ L induced by ρ is an isomorphism. proof: Denote L/K by E and by (∧Z, d) the quadratic Sullivan 1-algebra associated to the Sullivan Lie algebra E. The morphism ρ0 induces a morphism of Sullivan algebras ϕ : (∧Z, d) ← (∧V 1 , d) , with the restriction of ϕ to V 1 the dual of ρ0 . Since ρ0 is by definition surjective, ϕ : V 1 → Z is injective. In particular, ϕ defines a Sullivan extension (∧Z, d) = (∧V 1 ⊗ ∧W, d) . Now suppose W 6= 0. Then for some non-zero element w ∈ W , dw is a cycle in ∧2 V 1 . But since (∧V 1 , d) is 1-formal, by Proposition 2.7 (p. 88) every homology class in H 2 (∧V 1 , d) is represented by a cycle in ∧2 V01 . Thus we may assume dw ∈ ∧2 V01 , and so w ∈ Z1 . On the other hand, Z1 = (L/K)(2)# ⊕ (L/K)(1)# and by construction ∼ ∼ = = ρ0 : (L/K)(2) ⊕ (L/K)(1) −→ L(2) ⊕ L(1). Thus ϕ : V1 → Z1 and it follows ∼ = that w ∈ V1 , which is a contradiction. Thus ϕ : ∧V −→ ∧Z and ρ0 is an isomorphism.  Example. Let (∧W, d) be the minimal Sullivan model of an oriented Riemann surface of genus ≥ 2. According to Proposition 8.5 (p. 240) (∧W, d) is a formal Sullivan 1-algebra. Thus as observed above, its homotopy Lie algebra is the completion of its associated weighted Lie algebra L = ⊕n≥1 L(n). Moreover it follows from Theorem 15.4 and §8.5 that X L = L(a1 , b1 , · · · , ag , bg )/ [ai , bi ] ,

page 398

January 8, 2015

14:59

9473-Rational Homotopy II

Weight decompositions of a Sullivan Lie algebra

RHT3

399

a result which first appears in Sullivan [62].  Next, let E be the associated weighted Lie algebra for L. The gradation E = ⊕n≥1 E(n) extends to a gradation of the enveloping algebra U E and we denote by U E(t) the Hilbert series ∞ X U E(t) = dim (U E)(n)tn . n=0

Proposition 15.2. With the above notation, denote bq = dim H q (∧V, d). Then 1 . U E(t) = P q b tq (−1) q q proof: Since (∧V, d) is formal, we have a quasi-isomorphism ϕ : (∧V, d) → (H, 0), and V is equipped with a lower gradation V = ⊕p≥0 V(p) such that V(p) = (E(p + 1))# . This induces a lower gradation ∧V = ⊕p (∧V )(p) and d : V(p) → (∧2 V )(p−1) . Denote by (∧V ⊗ ∧U, d) the acyclic closure of (∧V, d). The quasi-isomorphism ϕ induces a quasi-isomorphism ϕ ⊗ 1 : (∧V ⊗ ∧U, d) → (H ⊗ ∧U, d) . It follows from the construction of the acyclic closure (p. 111) that the lower gradation of V induces one in ∧U for which d : U(p) → (∧V ⊗∧U )(p−1) and such that the linear part of the differential induces an isomorphism U(p) → V(p−1) . Therefore we have isomorphisms of graded vector spaces P (∧U )(p) ∼ = U E(p) and U E(t) = p≥0 dim (∧U )(p) tp . Because (∧V, d) is formal, it follows from Proposition 2.7 (p. 88) that q ∧V(0) generates H. Thus H q = H(0) , (H ⊗ ∧U )q(p) = H q ⊗ (∧U )(p) and d(U(p) ) ⊂ H 1 ⊗ (∧U )(p−1) . Thus for each n ≥ 0 the direct sum M H q ⊗ (∧U )(p) p+q=n

is a subcomplex. Now since H n (∧V ⊗ ∧U, d) = 0 if n > 0, we have  X 1, if n = 0 q (−1)q dim (H ⊗ ∧U )(p) = 0, if n > 0 . p+q=n Now remark that ! ! X X X q q p (−1) bq t · dim(∧U )(p) t = (−1)q dim(H⊗∧U )q(p) tp+q = 1 . q

p

p,q



page 399

January 8, 2015

400

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Example. There are many examples of 1-formal Sullivan 1-algebras, and a detailed study of 1-formality is given in the survey of Papadima and Suciu ( [54]). For instance, when G is an Artin group, then K(G, 1) is 1-formal. In [37], Kohno proves that the complement X in Cn of complex projective hypersurfaces is 1-formal. More precisely, if X is the complement of a family {H1 , H2 , · · · , Hr } of hyperplanes in Cn the Chen Lie algebra is generated by the {x1 , x2 , · · · , xr } with relations   xi , xj1 + xj2 + · · · + xjs = 0 , where j1 < j2 < · · · < js are such that codim Hj1 ∩ Hj2 ∩ · · · Hjs = 2. 

page 400

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Chapter 16

Problems

Problem 1. Cohomology of the spatial realization. If (∧V, d) is the minimal Sullivan model of a path connected space X and if | ∧ V, d| is a Sullivan space, then Theorem 7.8 shows that the natural map µX : X → | ∧ V, d)| is a rational homology equivalence. Is the converse true? In this case must X be a Sullivan space? Problem 2. Model and 1-model. Let (∧V, d) be a minimal Sullivan algebra for which dim H 1 (∧V, d) < ∞ and dim V i < ∞, i ≥ 2. If cat (∧V, d) < ∞ then Proposition 9.6 and Theorem 9.3 assert that cat (∧V 1 , d) < ∞ and cat (∧V ≥2 , d) < ∞. Is the converse true? More generally, for any minimal Sullivan extension (∧W ⊗ ∧Z, d), if cat (∧W, d) < ∞ and cat (∧Z, d) < ∞, is cat (∧W ⊗ ∧Z, d) < ∞? Problem 3. Exponential growth in homotopy. Chapters 12 and 13 establish exponential growth of the ranks of the homotopy groups of a simply connected rationally hyperbolic space. Moreover, Chapter 15 shows that, under some hypotheses, for the fundamental Lie algebra L0 of a minimal Sullivan algebra, the sequence dimLn0 /Ln+1 grows exponentially. 0 • Suppose L is the rational homotopy Lie algebra of a simply connected rationally hyperbolic space and that dim L/[L, L] is finite dimensional. Does the sequence dim Ln /Ln+1 grow exponentially? • Suppose for a group G that BG is a Sullivan space with infinite dimensional fundamental Lie algebra L0 . Does the sequence dim Ln0 /Ln+1 0 grow exponentially? • Let L be a graded Lie algebra with depth L < ∞ and dim L/[L, L] < ∞. Does the sequence Ln /Ln+1 grow faster than any polynomial? 401

page 401

January 8, 2015

14:59

402

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Problem 4. Structure of the homotopy Lie algebra. Chapter 14 establishes general structure theorems for the lattice of L-equivalence classes of ideals in a graded Lie algebra L = L≥1 of finite depth. This suggests the questions: • Is there an analogous lattice for a Lie algebra L0 of finite depth, with analogous structure theorems? • Are there analogous results for Q[π1 (X)] ⊗ U L≥1 in general or in the case of Sullivan spaces? • Let L be the homotopy Lie algebra of a simply connected rationally hyperbolic space. If depth L < ∞, does it follows that depth Leven < ∞? • In Chapter 14 it is shown that log index [L, L] = log index L. What conditions imply that log index L/[L, L] < log index L? Is this always the case if L is the homotopy Lie algebra of a simply connected rationally hyperbolic finite CW complex? • Suppose E is an abelian sub Lie algebra of the homotopy Lie algebra L of a simply connected space of finite category. If dim L = ∞, is it always true that log index E < log index L?

Problem 5. Sullivan spaces. • Determine a large class of Sullivan spaces of the form BG. • If X is a connected finite CW complex such that πi (X) = 0 if i ≥ 2, is X a Sullivan space? • If X has a single 2-cell and no cell of higher degree, must X be 1-formal? • If G is the quotient of a free group by one relation, is BG a Sullivan space?

Problem 6. Depth and category. According to Chapter 10, cat(∧V, d) ≥ depth (∧V, d1 ). Moreover if µ : (∧V, d1 ) → C ∗ (L) is a quasiisomorphism, then depth(∧V, d1 ) = depth L. Are there other conditions which imply depth L = depth(∧V, d1 )? Problem 7. Rationalization. Compare the rationalization XQ = |∧V, d| with the Bousfield-Kan construction Q∞ (X) ( [7]). Problem 8. Determine some algebraic conditions on a pronilpotent Lie

page 402

January 8, 2015

14:59

9473-Rational Homotopy II

Problems

RHT3

403

algebra L which imply an isomorphism ExtU L (lk, lk) ∼ = Extlk[GL ] (lk, lk) . Problem 9. • Determine some algebraic conditions on a graded Lie algebra that imply that the natural morphism lim C ∗ (L/Ln ) → C ∗ (L) −→ n is a quasi-isomorphism. • Determine conditions on a group G which imply that lim C ∗ ((G/Gn ); Q) → C ∗ (G; Q) −→ n is a quasi-isomorphism. Problem 10. Suppose L is a Sullivan Lie algebra concentrated in degree zero. • When is there a subspace S ⊂ L mapping isomorphically to L/[L, L] and generating a sub Lie algebra E ⊂ L isomorphic to the associated weighted Lie algebra of Chapter 15? • Is there a weaker condition on S for which '

lim C ∗ (E/E n ) → C ∗ (E)? −→ n

Problem 11. If L is a graded Lie algebra then each Lk is filtered by the subspaces Lk (p) = { x ∈ Lk | [L0 , · · · , L0 , x] = 0 }. To what extent can the | {z } p+1

results about Sullivan Lie algebras be extended to Lie algebras in which ∩n Ln0 = 0 and L0 /[L0 , L0 ] and each Lk (p + 1)/Lk (p) are finite dimensional. Problem 12. Suppose L is an infinite dimensional Sullivan Lie algebra concentrated in degree zero. Must it be true that if depth L < ∞ then for some n dim Ln /[Ln , Ln ] = ∞?

page 403

January 8, 2015

14:59

404

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Problem 13. Local and weak local nilpotence • According to Theorem 10.3 (p. 288) if L is a Sullivan Lie algebra of depth m then the elements x ∈ Leven such that ad x is locally nilpotent span a vector space of dimension ≤ m. Do the elements x ∈ Lodd for which ad x is locally nilpotent span a finite dimensional vector space? • If L is a Sullivan Lie algebra an element x ∈ L is weakly locally nilpotent if for some sequence n(k), (ad x)k : L → Ln(k) and limk→∞ k/n(k) = 0. If depth L < ∞ can every x ∈ L be weakly locally nilpotent? Problem 14. It is known ([43]) that a minimal simply connected Sullivan algebra whose homology is a Poincar´e duality algebra is the model of a commutative cochain algebra which itself is a Poincar´e duality algebra. Does this remain true in the non-simply connected case?

page 404

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Bibliography

[1] J.F. Adams and P.J. Hilton, On the chain algebra of a loop space, Comment. Math. Helvetici 30 (1956), 305-330. [2] J. Amoros, On the Malcev completion of Kahler groups, Comment. Math. Helv. 71 (1996), 192-212 [3] I.K. Babenko, Analytical properties of Poincar´e series of a loop space, Mat. Zametski 27 (1980) 751-764. [4] H. Baues and J.M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), 219-242. [5] J. Birman and T. Brendle, Handbook of geometric topology, Elsevier (2005). [6] A.K. Bousfiled and V.K. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 179 (1976). [7] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972). [8] E. Brieskorn, Sur les groupes de tresses (d’apr`es Arnold), S´eminaire Bourbaki 1971/72, Lecture Notes in Mathematics, 317, Springer-Verlag (1973). [9] H. Cartan and S.Eilenberg, Homological algebra, Princeton University Press (1956). [10] B. Cenkl and R. Porter, Malcev’s completion of a group and differential forms, J. Differential Geometry 15 (1980), 531-542. [11] K.T. Chen, Extension of C ∞ -function algebra by integrals and Malcev completion of π1 , Advances in Math. 23 (1977), 181-210. [12] O. Cornea, G. Lupton, J. Oprea and D. Tanr´e, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs 103, AMS (2003). [13] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of K¨ ahler manifolds, Invent. Math. 29 (1975), 245-274. [14] G. Denham and A. Suciu, On the homotopy Lie algebra of an arrangement, Michigan Math. J. 54 (2006), 319-340. [15] M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988), 543-556. [16] M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77-88. [17] Y. F´elix, La dichotomie elliptique-hyperbolique en homotopie rationnelle,

405

page 405

January 8, 2015

406

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Ast´erisque 176, SMF (1989). [18] Y. F´elix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer-Verlag (2001). [19] Y. F´elix, S. Halperin and J.-C. Thomas, An asymptotic formula for the ranks of the homotopy groups of finite complex, Expositiones Math. 25 (2007), 6776. [20] Y. F´elix, S. Halperin and J.-C. Thomas, The ranks of the homotopy groups of a finite complex, Can. J. Math. 65 (2013), 82-119. [21] Y. F´elix, J. Oprea and D. Tanr´e, Algebraic models in geometry, Oxford Graduate Texts in Math. 17 (2008). [22] W. Fulton, Algebraic curves, Addison-Wesley (1969). [23] T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 (1967), 417-427. [24] D.H. Gotllieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756. [25] G. Gr¨ atzer, General lattice theory, Birkh¨ auser (1998). [26] P. Griffiths and J. Morgan, Rational homotopy theory and differential forms, Second Edition, Progress in Mathematics 19, Birkh¨ auser (2013). [27] P.-P. Grivel, Formes diff´erentielles et suites spectrales, Ann. Inst. Fourier 29 (1979), 17-37. [28] S. Halperin, Lectures on Minimal Models, M´emoires de la Soci´et´e Math´ematique de France, 230 (1983). [29] S. Halperin and J.-M. Lemaire, Suites inertes dans les alg`ebres de Lie gradu´ees, Math. Scand. 61 (1987), 39-67. [30] S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Adv. in Math. 32 (1979), 233-279. [31] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65-71. [32] A. Hatcher, Algebraic topology, Cambridge University Press (2002). [33] K. Hess, A proof of Ganea conjecture for rational spaces, Topology 30 (1991), 205-214. [34] K. Hess, Rational homotopy theory: a brief introduction, Interactions between homotopy theory and algebra, 175-202, Contemp. Math. 436, Amer. Math. Soc. (2007). [35] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland (1975). [36] I. James, On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348. [37] T. Kohno, On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, Nagoya Math. J. 92 (1983), 21-37. [38] T. Kohno, S´erie des Poincar´e-Koszul associ´ee aux groupes de tresses pures, Invent. Math. 82 (1985), 57-75. [39] J.-L. Koszul, Homologie et cohomologie des alg`ebres de Lie, Bull. Soc. Math. France 78 (1950), 65-127. [40] A.L. Kurosh, Theory of groups, II, Chelsea Publishing (1956).

page 406

January 8, 2015

14:59

9473-Rational Homotopy II

Bibliography

RHT3

407

[41] P. Lambrechts, Analytic properties of Poincar´e series of spaces, Topology 37 (1998), 1363-1370. [42] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Ec. Norm. Sup. 71 (1954), 101-190. [43] J.M. Lemaire and F. Sigrist, Sur les invariants d’homotopie rationnelle li´es a ` la LS cat´egorie, Comment. Math. Helv. 56 (1981), 103-122. [44] MacLane, Homology, Springer-Verlag (1975). [45] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience Publishers (1966). [46] A.L. Malcev, Nilpotent groups without torsion, Jzv. Akad. Nauk. SSSR, Math. 13 (1949), 201-212. [47] P. May, Simplicial objects in algebraic topology, Chicago Univ. Press (1967). [48] P. May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 155 (1975). [49] J. Milnor and J.C. Moore, On the structure of Hopf algebras, Annals Math. 81 (1965), 211-264. [50] J. Morgan,The algebraic topology of smooth algebraic varieties, Inst. Hautes Etudes Sci Publ. Math. 148 (1970), 137-204. [51] J. Neisendorfer and L. Taylor, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. 74 (1978), 429-460. [52] P. Orlik, Seifert Manifold, Lecture Notes in Mathematics, 231 SpringerVerlag (1972). [53] P.Orlik and L.Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189. [54] S. Papadima and A. Suciu, Geometric and algebraic aspects of 1-formality, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009), 355-375. [55] S. Papadima and A. Suciu, Chen Lie algebras, Internat. Math. Res. Notices 21 (2004), 1057-1086. [56] S. Papadima and A. Suciu, Homotopy Lie algebras, lower central series and the Koszul property, Geom. Topol. 8 (2004), 1079-1125. [57] S. Papadima and S. Yuzvinsky, On rational K(π, 1) spaces and Koszul algebras, J. Pure Appl. Alg. 144 (1999), 157-167. [58] D. Quillen, Rational homotopy theory, Ann. Math. 90 (1969), 205-295. [59] H.K. Schenck and A. Suciu, Lower central series and free resolutions of hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002), 3409-3433. [60] J.P. Serre, Lie algebras and Lie groups, Benjamin Inc. (1965). [61] D. Sullivan, Differential forms and the topology of manifolds, Proceedings of the International Conference on Manifolds and Related Topics in Topology, Tokyo, 1973, University of Tokyo Press (1975), 37-49. [62] D. Sullivan, Infinitesimal computations in topology, Publ. IHES 47 (1977), 269-331. [63] G.H Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z. 138 (1974), 123-143. [64] J.H. van Lint and R.M. Wilson, A course in combinatorics, Cambridge University Press, second edition (2001). [65] C. Weibel, An introduction to homological algebra, Cambridge University

page 407

January 8, 2015

408

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Press (1995). [66] G. Whitehead, Elements of homotopy theory, Graduate Text in Math. 61, Springer-Verlag (1978). [67] J.H.C. Whitehead, A certain exact sequence, Ann. Math. 52 (1951), 51-110.

page 408

January 8, 2015

14:59

9473-Rational Homotopy II

RHT3

Index

H HK , 16 , 16 X πn (∧V, d), 35 |K|, 11 cn , 9 hol : π1 (Y ) → GF , 119 hol : GL → G(∧Z,d) , 126 mh∧V,di : (∧V, d) → AP L (h∧V i), 26 m|∧V,d| : (∧V, d) → AP L |∧ V |, 28

(AP L )n , 14 AP L (K), 14 AP L (X), 14 GL , 59 Gq (S, Y ), 325 Hhol, 129, 132 PL , 59 TV , 3 Λ-extension, 95 fibre of, 96 homotopy of maps, 97 lifting lemma, 97 minimal, 95 d ηL : U L → (∧U )# , 177, 178 expL , 60 ∼ = d γL : (∧sV )# → U L, 71

Acyclic closure ∧U , 102, 111 canonical conjugation, 170 comultiplication, 173 Associated homotopy fibration, 93

ιn : πn (| ∧ V, d|) → (Qn (∧V ))# , 36 λL : L → Sing |L|, 12 λX : |Sing X| → X, 12 λh∧V,di : h∧V i → Sing | ∧ V |, 27 ∧V , 3 logL , 60 LV , 5 ν(L), 368, 383 νL (I), 383

Braid group, 237 Cartan-Chevalley-Eilenberg, 47 Chain coalgebra C∗ (L), 47 Circle construction, 147 Classifying space, 93, 123 Cochain algebra, 7 Λ-algebra, 18 Λ-extension, 95 of polynomial forms AP L , 14 409

page 409

January 8, 2015

14:59

410

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Critical degree, 338 Depth of a graded algebra, 268 of a graded Lie algebra, 277 of a min. Sullivan alg., 272, 299 depthL E, 287 Differential graded algebra, 6 Differential module, 6 homotopy of morphisms, 7 morphism of modules, 6 semifree module, 7 semifree resolution, 7 Engels derivation, 261 Exponential group, 59 ExtA (M, N ), 267 Fibration condition, 117, 145 Fundamental class of S n , 17 Fundamental cycle cn of S n , 17 Fundamental simplex of ∆[n], 9 Gap theorem, 318 Global dimension of a graded algebra, 268 of a graded Lie algebra, 277 of a min. Sullivan alg., 273, 299 gradeA M , 268 Graded algebra, 3 augmentation, 3 commutative algebra, 3 derivation of, 3 free commut. graded algebra, 4 interchange isomorphism, 3 tensor algebra, 3 tensor product, 3 Graded Hopf algebra, 5 group-like elements, 5

primitive elements, 5 Graded Lie algebra, 4 admissible sub-Lie algebra, 307 associated quadratic Sullivan algebra, 45, 55, 276 centralizer of a subalgebra, 308 Closed ideal, 280 derivation of, 4 exponential group GL , 59 free graded Lie algebra, 5 hyper radical, 370 locally solvable, 303 lower central series, 46 nilpotent Lie algebra, 46 pronilpotent Lie algebra, 46 representation of, 4 solv length, 295 solvable Lie algebra, 295 Sullivan closure of a sub algebra, 280 Sullivan Lie algebra, 54 universal enveloping algebra, 5 weak depth, 307 weakly locally finite module, 307 Graded vector space, 1 S-large, 368 V 6 W , 367 ∼L , 368 dual of U = U # , 2 finite type, 2 full sub vector space, 367 Hilbert series, 2 Hom, 2 interchange isomorphism, 2 suspension of V = sV , 2 tensor product, 2 Hochschild-Serre spect. seq., 269,

page 410

January 8, 2015

14:59

9473-Rational Homotopy II

Index

271 Holonomy representation of ΩY on X ×Y P 0 Y , 122 of π1 (∧V ) in πn (∧Z), 143 of π1 (Y ) in πn (Y ), 142 of π1 (Y ) in F , 119 d of U L in H(∧Z, d), 132 of GL in H(∧Z, d), 130 of L in H(∧Z, d), 132 of U L in H(∧Z, d), 132 of a Λ-extension, 126 of a fibration, 119 Homotopy based homotopy, 20 of cdga maps, 20 Homotopy fibre, 93 Homotopy groups of a space fundamental Lie algebra, 46 homotopy Lie algebra, 46 Interchange isomorphism, 2, 3 LCS formula, 90 Lie ideal L-equivalence classes, 368 S-minimal, 372 νL (I), 383 height of an ideal, 381 hyper radical, 370 radical, 295 spectrum, 368 weak complement, 377 Lie module S-light, 372 isotropy sub Lie algebra, 372 Lifting lemma, 19, 21, 97 log index, 314 Loop cohomology coalgebra, 169

RHT3

411

Loop cohomology Hopf algebra, 176 Lower central series of a graded Lie algebra, 46 of a group, 63 LS category of a commut. cochain algebra, 246 minimal Sullivan algebra, 246 space, 245 Malcev completion, 213 Malcev k-complete group, 213 Mapping Theorem, 248 Mapping torus, 157 Milnor realization functor, 10 Minimal Sullivan model, 102 of a Riemann surface, 232 of a Seifert manifold, 239 of a wedge of circles, 229 uniqueness, 107 Module LS category, 250 Moore path free Moore path, 92 Moore loop space, 92 Moore path space, 92 Pairing, 2 Polynomial differential forms, 14 Pronilpotent Lie algebra, 46 closed subspace, 64 convergent sequence, 64 Rational K(π, 1) space, 40 Rational completion, 40 Rational homology equiv., 29 Rational homology type, 29 Rational LS category, 246 Rationally elliptic space, 314 Rationally equivalent spaces, 29

page 411

January 8, 2015

412

14:59

9473-Rational Homotopy II

RHT3

Rational Homotopy Theory II

Rationally hyperbolic space, 314 Rationally infinite space, 314 Rationally nilpotent group, 227 Rationally nilpotent space, 227 Representation locally nilpotent, 120 Riemann surface, 232, 235

Sullivan extension, 95 Sullivan Lie algebra, 54 Sullivan model, 23 of BG, 205 of a Riemann surface, 235 of a Seifert manifold, 240 Sullivan representative, 24

Seifert manifold, 239, 240 Semidirect product, 231 Simplicial set, 8 ∆[n], 9 ∂∆[n], 9 homotopy of maps, 10 Kan simplicial set, 10 simplicial interval, 30 Sing X, 10 Singular simplex functor, 10, 11 Spectrum, 368 Sullivan 1-algebra, 48 canonical filtration, 49 formality, 87 Sullivan algebra, 18 k-model, 109 1-finite, 18 formality, 87 fundamental group, 75 fundamental Lie alg. of, 46, 75 homology Hopf algebra of, 82 homotopy groups, 35 homotopy Lie algebra of, 46 indecomposable elements, 22 lifting lemma, 19, 21 minimality, 18 model, 102 quadratic part of d, 45 relative Sullivan algebra, 95 simplicial realization, 25 Sullivan condition, 18, 95

Toomer invariant, 250 TorA (M, Q), 267 d Transformation map ηL : U L → # (∧U ) , 176 Universal cover, 93 Universal enveloping algebra, 5 augmentation ideal, 56 d canonical conjugation of U L, 178 completion of U L, 56 diagonal, 56 primitive Lie algebra, 56, 59 w-depth L, 307 Weighted Lie algebra, 217

page 412