Rational homotopy theory and differential forms [2ed.] 978-1-4614-8467-7, 1461484677, 978-1-4614-8468-4

Table of contents :
Front Matter....Pages i-xi
Introduction....Pages 1-3
Basic Concepts....Pages 5-20
CW Homology Theorem....Pages 21-25
The Whitehead Theorem and the Hurewicz Theorem....Pages 27-40
Spectral Sequence of a Fibration....Pages 41-52
Obstruction Theory....Pages 53-61
Eilenberg–MacLane Spaces, Cohomology, and Principal Fibrations....Pages 63-67
Postnikov Towers and Rational Homotopy Theory....Pages 69-81
deRham’s Theorem for Simplicial Complexes....Pages 83-93
Differential Graded Algebras....Pages 95-102
Homotopy Theory of DGAs....Pages 103-111
DGAs and Rational Homotopy Theory....Pages 113-118
The Fundamental Group....Pages 119-126
Examples and Computations....Pages 127-140
Functorality....Pages 141-149
The Hirsch Lemma....Pages 151-163
Quillen’s Work on Rational Homotopy Theory....Pages 165-176
A ∞ -Structures and C ∞ -Structures....Pages 177-185
Exercises....Pages 187-221
Back Matter....Pages 223-227

Citation preview

Progress in Mathematics 16

Phillip Griffiths John Morgan

Rational Homotopy Theory and Differential Forms Second Edition

Progress in Mathematics Volume 16

Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein

For further volumes: http://www.springer.com/series/4848

Phillip Griffiths • John Morgan

Rational Homotopy Theory and Differential Forms Second Edition

Phillip Griffiths Institute for Advanced Study Princeton University Princeton, NJ, USA

John Morgan Simons Center for Geometry and Physics Stony Brook University Stony Brook, NY, USA

ISBN 978-1-4614-8467-7 ISBN 978-1-4614-8468-4 (eBook) DOI 10.1007/978-1-4614-8468-4 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013946908 Mathematics Subject Classification (2010): 57R20, 57R57, 58A10, 58A14, 55R20, 55R10, 55R45, 55R40 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface to the First Edition

This monograph originated as a set of informal notes from a summer course taught by the present authors, together with Eric Friedlander, at the Istituto Matematico “Ulisse Dini” in Florence during the summer of 1972. Even though more formal expositions of Sullivan’s theory have since appeared, including the major original source [26], there has been a steady continuing demand for the old Florence notes. Moreover, one of us (J.M.) has become involved in the subject again through a series of lectures given at the University of Utah in January, 1980, together with joint work in progress with James Carlson and Herb Clemens on a new type of application of the theory to algebraic geometry. Since the Florence notes represented an approach and point of view that does not appear in the literature, we decided to publish the present revised and corrected version. The material in this monograph is outlined in the table of contents and is informally discussed in the introduction below. Here we should like to observe that the text roughly divides into two parts. The first seven chapters essentially constitute an introductory course in algebraic topology with emphasis on homotopy theory. The main prerequisite is some familiarity with simplicial homology, covering spaces, and CW complexes. Chapters 9–15 cover the main topic of differential forms and homotopy theory, with emphasis on the homotopy-theoretic and functorial properties of differential graded algebras and minimal models, a topic that does not appear explicitly in detail in the literature. An extensive set of exercises, frequently with copious hints, forms an essential complement to the material in the text. We would like to make several acknowledgements to colleagues whose help and advice have been invaluable. The first and foremost is to Dennis Sullivan. It was he who introduced us to the idea of relating homotopy theory and differential forms and who explained to us his theory around which these notes are built. The second is to Francesco Gherardelli who organized the original summer course and to the Istituto Matematico “Ulisse Dini” and the city of Florence, which together provided excellent mathematical and cultural conditions for the initial preparation of the notes. While in Florence we benefited from conversations with Ngo Van Que, Jim Carlson, and Mark Green. Finally, Moishe Breiner prepared v

vi

Preface to the First Edition

a beautifully handwritten set of notes that constituted the original version of this monograph. We would also like to thank the University of Utah and abovementioned coworkers of J.M. for providing support and motivation leading to the revision of the Florence notes. Finally we would like to point out two predecessors of the present theory. The first is Whitney’s book [27]. As explained to us by Sullivan, this book contains the genesis of the use of differential forms to solve the commutative cochain problem and thus get the homotopy type of the space. The main thing lacking at the time Whitney wrote the book was the Q-structure. Secondly the relationship between differential forms and homotopy theory was anticipated by Chen [2]. Many of the results we find from a general viewpoint were established, frequently in stronger form, by him using the method of iterated integrals.

Preface to the Second Edition

Thirty years have passed since the publication of the first edition, and we felt this monograph deserved updating. The essential structure and presentation remains the same, but several major additions seemed appropriate. We have included in an appendix a proof of the correspondence between rational minimal models and rational Postnikov towers different, and we feel more intuitive, from the one presented in the first edition. This proof relies on a set of polynomial forms with a filtration whose Serre spectral sequence agrees with the usual one for a fibration. We have also added a chapter describing Quillen’s approach to rational homotopy theory and comparing and contrasting it with Sullivan’s. Lastly, we have added a chapter on operads and A1 algebras and indicated briefly how the commutative version, C1 -algebras, gives another algebraic description of rational homotopy theory. The second author thanks Mohammed Abouzaid and Bruno Vallette for their help with the chapter on A1 -structures and Eric Malm for his help in preparing the diagrams and figures. Princeton, NJ, USA Stony Brook, NY, USA

Phillip A. Griffiths John W. Morgan

vii

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 CW Complexes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 First Notions from Homotopy Theory.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5 5 8 13 19

3

CW Homology Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21 21 22 24

4

The Whitehead Theorem and the Hurewicz Theorem . . . . . . . . . . . . . . . . . 4.1 Definitions and Elementary Properties of Homotopy Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Whitehead Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Completion of the Computation of  n .Sn / . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 The Hurewicz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Corollaries of the Hurewicz Theorem .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Homotopy Theory of a Fibration . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Applications of the Exact Homotopy Sequence . . . . . . . . . . . . . . . . . . . .

27

5

Spectral Sequence of a Fibration . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Fibrations over a Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Generalities on Spectral Sequences . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 The Leray–Serre Spectral Sequence of a Fibration .. . . . . . . . . . . . . . . . 5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41 41 42 43 45 48

6

Obstruction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Definition and Properties of the Obstruction Cocycle .. . . . . . . . . . . . .

53 53 54

27 29 31 33 34 38 39

ix

x

Contents

6.3 6.4 6.5

Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Obstruction to the Existence of a Section of a Fibration . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

57 58 58

7

Eilenberg–MacLane Spaces, Cohomology, and Principal Fibrations. 7.1 Relation of Cohomology and Eilenberg–MacLane Spaces . . . . . . . . 7.2 Principal K. ; n/-Fibrations .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63 63 64

8

Postnikov Towers and Rational Homotopy Theory .. . . . . . . . . . . . . . . . . . . . 8.1 Rational Homotopy Theory for Simply Connected Spaces . . . . . . . . 8.2 Construction of the Localization of a Space . . . .. . . . . . . . . . . . . . . . . . . .

69 73 79

9

deRham’s Theorem for Simplicial Complexes . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Piecewise Linear Forms .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Lemmas About Piecewise Linear Forms . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Naturality Under Subdivision .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Multiplicativity of the deRham Isomorphism .. .. . . . . . . . . . . . . . . . . . . . 9.5 Connection with the C1 deRham Theorem . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Generalizations of the Construction .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83 83 85 88 89 90 92

10 Differential Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 10.2 Hirsch Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 10.3 Relative Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 10.4 Construction of the Minimal Model .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 11 Homotopy Theory of DGAs . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Homotopies .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Obstruction Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Applications of Obstruction Theory .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Uniqueness of the Minimal Model . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 104 107 109

12 DGAs and Rational Homotopy Theory . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 Transgression in the Serre Spectral Sequence and the Duality .. . . . 12.2 Hirsch Extensions and Principal Fibrations .. . . .. . . . . . . . . . . . . . . . . . . . 12.3 Minimal Models and Postnikov Towers . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.4 The Minimal Model of the deRham Complex . .. . . . . . . . . . . . . . . . . . . .

113 113 114 115 117

13 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 1-Minimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2  1 ˝ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Functorality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

119 119 120 123 125

14 Examples and Computations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Spheres and Projective Spaces .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2 Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3 The Borromean Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.4 Symmetric Spaces and Formality.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

127 127 128 129 131

Contents

14.5 14.6 14.7 14.8 14.9

xi

The Third Homotopy Group of a Simply Connected Space . . . . . . . Homotopy Theory of Certain 4-Dimensional Complexes .. . . . . . . . . Q-Homotopy Type of BUn and Un . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Products .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Massey Products.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

132 134 135 137 138

15 Functorality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 The Functorial Correspondence . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Bijectivity of Homotopy Classes of Maps . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Equivalence of Categories . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

141 141 144 148

16 The Hirsch Lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.1 The Cubical Complex and Cubical Forms . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2 Hirsch Extensions and Spectral Sequences . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 Polynomial Forms for a Serre Fibration .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.4 Serre Spectral Sequence for Polynomial Forms . . . . . . . . . . . . . . . . . . . . 16.5 Proof of Theorem 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

151 151 154 156 159 163

17 Quillen’s Work on Rational Homotopy Theory .. . . .. . . . . . . . . . . . . . . . . . . . 17.1 Differential Graded Lie Algebras.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.2 Differential Graded Co-algebras.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.3 The Bar Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.4 Relationship Between Quillen’s Construction and Sullivan’s .. . . . . 17.5 Quillen’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

165 165 166 167 169 169

18 A1 -Structures and C1 -Structures.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1 Operads, Rooted Trees, and Stasheff’s Associahedron .. . . . . . . . . . . . 18.2 A1 -Algebras and A1 -Categories .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.3 C1 -Algebras and DGAs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

177 177 181 183

19 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 187 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 223

Chapter 1

Introduction

The purpose of this course is to relate the C1 -differential forms on a manifold to algebro-topological invariants. A model of results along these lines is deRham’s theorem, which says that the cohomology of the differential graded algebra (DGA) of C1 -forms is isomorphic to the singular cohomology with coefficients in R, i.e. HdR .M/ Š H .M; R/

.C1 deRham theorem/:

The main theorem of this course will be that from the DGA of C1 -forms, it is possible to calculate all of the real algebro-topological invariants of the manifold. More precisely, we shall be able to use the forms to obtain the (Postnikov tower) tensored with R of the manifold. In the next seven chapters of this book, we shall discuss the standard terminology, objects, and theorems of elementary homotopy theory, culminating in the description of the Postnikov tower of a space. We then define the localization of a CW complex at 0; this allows us to take a CW complex and replace it by one in which all torsion and divisibility phenomena have been removed (allowing one to focus on the Q-information in the original space). When we compare the Postnikov tower of the original space with that of its localization, we see that all the relevant information (homotopy and homology groups, k-invariants) has been tensored with Q. Once we have established these basic facts, we turn to the main theorem as shown to us by Sullivan. First, we define the rational p.l. forms on a simplicial complex K. They form a DGA defined over Q. By integration, these forms give Q-valued simplicial cochains on K, and this integration process induces an isomorphism of the cohomology of the rational p.l. forms to the usual (simplicial or singular) rational cohomology of the space: Hp:l: .K/ Š H .K; Q/ .p.l. deRham theorem/: There are two very important points here. The first is that we are working over Q rather than R, as we would be forced to do with C1 -forms. The second is that the P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__1, © Springer Science+Business Media New York 2013

1

2

1 Introduction

p.l. forms are a differential, graded-commutative algebra—the simplicial or singular cochains over Q are not commutative. Thus, the p.l. forms have a good property of ordinary cochains (they are defined over Q) and a good property of C1 -forms (they are graded commutative). Both these properties are essential. Next, we turn to the homotopy theory of DGAs, which are always implicitly assumed to be associative and graded commutative. Given one such, A, we show how to extract a minimal model for it. This is a DGA, MA , which satisfies some internal condition, together with a map of DGAs: ¡A W MA ! A which induces an isomorphism on cohomology. In the case that H1 .A/ D 0, the internal properties that MA is required to satisfy are: 1. It is free as a graded-commutative algebra with generators in degrees  2 only. 2. For all x 2 MA , the element dx is decomposable. It turns out that, given A, these properties characterize MA up to isomorphism. We shall show, in addition, that when A is the algebra of p.l. forms on a simply connected, simplicial complex X, then MA is dual to the rational Postnikov tower of X. The duality between minimal models defined over Q and rational Postnikov towers is described in Chap. 12. Schematically we have a “commutative diagram”: fSimplicial Complexesg

p:l: forms

/ fDGAs= Qg

localization at 0

 fQ spacesg

Postnikov tower

minimal model

 fPostnikov towers=Qg o

 / fminimial DGAs=Qg

Given a C1 -manifold M, we can smoothly triangulate M. Let K be the simplicial complex of this triangulation. We have both the C1 -forms on M and p.l. forms on K. These are both included in Ap:C1 .M/ the DGA of “piecewise C1 -forms” on M, (i.e., the forms whose restriction to each simplex of the triangulation is smooth), and the inclusions AC 1 .M/

/ A 1 .M/ o p: C

Ap:l: .K/ ˝Q R

1 Introduction

3

induce isomorphisms in cohomology. From this comparison theorem, it follows that the minimal models satisfy M.AC1 .M// Š M.Ap:l: .M// ˝Q R: This is the precise statement that “the deRham complex contains all the real algebraic-topological information from the manifold M.” Schematically the theory is arranged as follows: C 1 forms

fManifoldsg O

/ fDGAs=Rg D

 / fDGAs=Rg O

p: C 1 forms

ftriangulatedg manifolds Q QQQ QQQ QQQ QQQ QQ(  p:l: forms / fDGAs=Qg f Simplicial g complexes  fQspacesg

 / fMinimal models=Qg 5 lll lll l l ll lll u ll l

˝R

˝R

/ fDGAs=Rg

 ~ / fMinimal modelsg over R

 fPostnikov towers=Qg

Though these notes concentrate mainly on the case of simply connected spaces, there are generalizations to the nonsimply connected case. In purely algebraic terms, part of the theory of the nonsimply connected case is similar to the simply connected one. When we try to make comparisons with homotopy, the results are much weaker. The available information from the algebra of forms which is most meaningful in classical terms deals with the fundamental group. Chapter 13 discusses this.

Chapter 2

Basic Concepts

Here we give a brief introduction to the basics of CW complexes, homotopy theory, homology, and the algebraic topology of manifolds. Here are some more references for more details on the material in this chapter. For a good introduction to CW complexes, homology, and cohomology, consult Greenberg’s book [7]. For a more encyclopedic treatise on algebraic topology which covers all the homotopy theory presented in this course, save localization, one should see Spanier’s book [23]. For another account of some of the topics presented later in this course, such as obstruction theory, one should see Hu’s book [9].

2.1 CW Complexes It will suffice for the purposes of this course (and for most other situations, also) to do homotopy theory for a restricted class of spaces. These are the spaces which are homotopy equivalent to CW complexes. All naturally encountered spaces have this property (e.g., manifolds, algebraic varieties, loop spaces on CW complexes, K. ; n/s). Moreover for these spaces, the Whitehead theorem which states that .fW X ! Y is a homotopy equivalence if and only if f is an isomorphism on homotopy groups—cf. Sect. 9.2 for a proof) is true. What this means is that the usual functors of homotopy theory are powerful enough to decide when two CW complexes are homotopically equivalent. We begin with the definition of a CW complex. Let Dn denote the unit n-disk, namely, Dn D fx D .x1 ; : : : ; xn / 2 Rn W jjxjj2  1g and let Sn1 denote the unit .n  1/ sphere, i.e., the boundary @Dn of Dn . (Note: In these notes, we have also used the notation en , for n-cell, as another symbol for Dn .) Given X and a continuous map fW Sn1 ! X, we form the adjunction space P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__2, © Springer Science+Business Media New York 2013

5

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2 Basic Concepts

X [f Dn which is the quotient space of the disjoint union of X and Dn where every a 2 @Dn is identified with f.a/ 2 X. (Note: Above we required that f be a continuous map; usually we shall omit mention of continuity, with the understanding that map means continuous function.) Geometrically, what we have done is attach an n-cell to X

To give a space X, the structure of a CW complex means intuitively that X is obtained from a point by successively attaching cells. More precisely we have subspaces X.i/ of X with  D X.1/  X.0/  X.1/  : : : ;

.i/ X D [1 iD0 X

such that (1) X.iC1/ is obtained from X.i/ by attaching .i C 1/-cells and (2) if X ¤ X.n/ for any n (thus X is infinite dimensional), then X has the weak topology with respect to the X.n/ ’s meaning that U  X is an open set if and only if U \ X.n/ is open for all n. We call X.n/ the n-skeleton of X. (Note: Infinite-dimensional CW complexes such as CP1 , the infinite Grassmannians, the infinite sphere, etc. are very useful in homotopy theory. The weak topology means that in all cases, “1” can be well approximated by “arbitrarily large n.” Thus, for example, a map of a compact space into the infinite CW complex X is simply given by a map into XN for large some N. Also, a map fW X ! Y from a CW complex to a space is continuous if and only if its restriction to each skeleton X.n/ is continuous.

Examples of CW Complexes 1. The n-sphere Sn D fpt:g [f Dn where fW @Dn ! fpt:g is a degenerate attaching map. 2. The complex projective space CPn is given a CW structure inductively by CPn D CPn1 [f D2n where fW S2n1 ! CPn1

2.1 CW Complexes

7

is the Hopf map. More precisely, if we think of CPn as the lines through the origin in CnC1 , then taking D2n to be the unit ball in Cn , the attaching map fW @D2n ! CPn1 assigns to each point on the unit sphere in Cn the complex line joining through the origin containing that point 5. 3. CP1 D limn!1 CPn is the infinite CW complex having one 2n-cell for each n 2 ZC and with the attaching maps given as above. 4. Any simplicial complex K has the natural structure of a CW complex. The n-cells of this CW structure are exactly the n-simplices. Conversely, if X is a CW complex, then there is a simplicial complex K and a homotopy equivalence from K to X (cf., Exercise (13)). 5. A CW pair .X; A/ is a pair of spaces A  X such that X is obtained from A by attaching cells. (It is not necessary that A itself be a CW complex.) If .X; A/ is a CW pair, then we denote by X.n/ [ A the union of A with all cells of dimension  n. Again, if X is obtained by attaching infinitely many cells to A, then X is given the limit (or weak) topology. If X is a CW complex and A  X is a subcomplex, then .X; A/ is a CW pair. CW complexes are constructed so that, almost by definition, one works inductively up through the skeleton. As an example of this, we prove the homotopy extension theorem for CW pairs. Theorem 2.1. Given a CW pair .Y; X/, a map fW Y ! Z, and a homotopy FW X  I ! Z with FjXf0g D f jX , then there is an extension GW Y  I ! Z of F such that G.y; 0/ D f.y/. Proof. Step I: Given fW Dn ! Z and FW Sn1  I ! Z with FjSn1 f0g D fj@Dn , find GW D  I ! Z extending F with GjDn f0g D f. In the picture of Dn  I

we are given a map G0 on the union of the “bottom” .Dn  f0g/ and the “side” .@Dn  I/. We want to extend to a map on all of Dn  I. This is done taking the projection p of Dn  I onto .Dn  f0g/ [ .Sn1  I/ from the point f.middle of Dn /  f2gg

8

2 Basic Concepts Dn × {2}

Dn × {1}

Dn × {0}

and defining G.y; t/ D G0 .p.y; t//. Note: In this argument, as throughout homotopy theory, 99% of the proof is to find the correct “picture.” If this is done properly, no geometric argument will be difficult (although some algebraic computations may be messy). Step II: Given fW Y ! Z and FW X  I ! Z, we shall inductively construct G.i/ W .Y  f0g/ [ Œ.X [ Y.i/ /  Ig ! Z. Given G.i1/ , consider any i-cell Di’ and attaching map @Dia ! Y.i1/ . Then we have G.i1/ ı f’  IW .Si1  I/ [ .Di’  f0g/ ! Z .i/

and we may use Step I to extend this map over Di’  1 to a map Ga . Doing this over each i-cell and taking the union of the maps give G.i/ . Let G D [i G.i/ (i.e., GjY.i/ D G.i/ /. By the definition of the weak topology, G is continuous and gives the required extension of F, by the construction. 

2.2 First Notions from Homotopy Theory In homotopy theory, one always considers CW complexes modulo an equivalence relation, that of homotopy equivalence. Two maps f0 ; f1 W X ! Y are homotopic if there exists FW X  I ! Y with F.x; 0/ D f0 .x/ and F.x; 1/ D f1 .x/. We may set ft .x/ D F.z; t/ and think of the ft as giving a continuous deformation of f0 into f1 . Maps fW X ! Y and gW Y ! X are homotopy inverses if g ı f  idX and f ı g  idY (here, the notation “” means “is homotopic to,” and idX W X ! X is the identity map). A map fW X ! Y is a homotopy equivalence if it has a homotopy inverse; X and Y are homotopy equivalent if there is a homotopy equivalence fW X ! Y. Homotopy equivalence is an equivalence relation on the collection of CW complexes. In homotopy theory, one considers spaces as equivalent if they are homotopy equivalent (in particular, topological dimension is not defined). An equivalence class of homotopy equivalent spaces is said to be a homotopy type. For the rest of this course, space will mean CW complex, with the only exception

2.2 First Notions from Homotopy Theory

9

that we shall speak of path spaces and loop spaces on CW complexes (definitions below), which are not CW complexes as they stand. However, they always have the homotopy type of a CW complex [17] and so may be unambiguously considered as “spaces.” While the category of CW complexes is quite flexible and easy to work with for many applications in homotopy theory, sometimes, for example, in defining an appropriate DGA of forms, it is better to work with simplicial complexes.Of course, as we have already remarked, any simplicial complex is a CW complex. The converse is true up to homotopy. Lemma 2.2. Let X be a CW complex. Then there is a simplicial complex K homotopy equivalent to X. Proof. The proof is by induction over the skeleta of X. Suppose inductively that for some n  0, we have a simplicial complex Kn and a homotopy equivalence ®n W Kn ! X.n/ . For each .n C 1/-cell e’ of X, we have the attaching map f’ W Sn ! Xn . There is a map g’ W @nC1 ! Kn with ®n ı g’ homotopic to f’ . By subdividing the standard triangulation of @nC1 to produce a triangulation  of @nC1 , we can arrange that g’ is a simplicial map (without subdividing Kn ). Consider a collar neighborhood C of @nC1 in nC1 with @nC1 corresponding to the 0-end. The product  I defines a linear cell structure on C. We form the quotient CN of C where we identify points in @nC1  C if they have the same image under g’ . N of the product of each simplex of  with I is a cell. Let B be Then the image in C nC1 N Now by induction on the simplices ¢ of the triangulation the image of @ in C. , we triangulate the image ¢N of ¢  I in CN without subdividing ¢N \ B. Suppose that N At the 1-end, take the cone to the barycenter of ¢  f1g we have done this for .@¢/. of the triangulation of @¢  f1g. This produces a triangulation of the boundary of the N a triangulation that does not subdivide the image of ¢ f0g in C. N We then cell ¢N in C, take the cone over this triangulation to v¢  f1=2g, where v¢ is the barycenter of ¢. Then we extend the triangulation on the 1-end of C to a triangulation over the rest of nC1 . The produces a simplicial complex K0 , containing K as a subcomplex, and there is an extension of the homotopy equivalence from K0 to X.n/ [f’ e’ extending ®n . Performing this construction simultaneously for every .n C 1/-cell of X produces a simplicial complex KnC1 containing Kn and a homotopy equivalence ®nC1 W KnC1 ! X.n/ extending ®n . Taking the limit (i.e., increasing union) over all n establishes the result.  In many problems in homotopy theory, we wish to make a construction relative to a map fW X ! Y: It is frequently easier to work with an inclusion rather than an arbitrary map. This is always possible up to homotopy equivalence. Theorem 2.3. Given fW X ! Y, there is a space Mf , the mapping cylinder of f, inclusions jW X ! Mf and iW Y ! Mf , and a map  W Mf ! Y

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2 Basic Concepts

 / Mf X @@ O @@ @@ i @@ f  Y j

 

where   and i are homotopy inverses and   ı j D f. Thus, we may replace Y by a homotopy equivalent space in which X is included.) Proof. Define Mf D .X  I/ [f Y

where .x; 1/ is identified with f.x/ 2 Y. Then  W Mf ! Y is given by  .x; t/ D f.x/ for all x 2 X and  .y/ D y for all y 2 Y (this is consistent), and this gives a retraction of Mf onto Y.  Note: If X and Y are CW complexes and fW X ! Y is a cellular map (i.e., f.X.i/ /  Y.i/ /, it is easy to give Mf the structure of a CW complex. In Exercise (32), a proof of the fact that any f is homotopic to a cellular map f0 is outlined, so that Mf  Mf0 which is a CW complex (the notation A  B means that A and B are homotopy equivalent). Thus, we may consider any map as an inclusion without leaving the category of CW complexes. Above we discussed the homotopy extension property (h. e. p.) and proved that a subcomplex of a CW complex always has the h. e. p. Now there is a dual property to the h. e. p. called the homotopy lifting property or covering homotopy property. Given spaces E, B, we say that  W E ! B has the homotopy lifting property if given f

any space Y, a map Y ! E, and a homotopy gt of g D   ı f, there is a homotopy ft of f such that   ı ft D gt (thus, the homotopy ft “covers” or “lifts” the homotopy gt ). Here f is said to be a lifting of g, and covering homotopy says that if a map g can be lifted, then any homotopy gt of g can be lifted also. Not all maps have the homotopy lifting property; e.g., if B is connected, then   must be onto. If  W E ! B has the h. l. p. (D homotopy lifting property), then it is said to be a fibration. For any b 2 B, the fiber Fb D  l (b) is the preimage of the point. In a fibration, any two fibers are homotopy equivalent provided that the base is path connected (cf. Chap. 4). We let F be any space having the homotopy type of Fb (F is called a typical fiber). We write the fibration as

2.2 First Notions from Homotopy Theory

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F ! E ? ?  y B having in mind a picture like Fb ! ? ? y

E ? ? y

fbg ! B

Examples of Fibrations 1. Locally trivial fiber bundles, vector bundles, and the associated sphere bundles, covering spaces, are all examples of fibrations (for a discussion of these, cf. [25]). 2. Let X be a space with a base point x0 . Define the path space based at x0 2 X, P.X; x0 /, to be the set of all paths given by maps ¨W I ! X, ¨.0/ D x0 . The topology on P.X/ is the compact-open topology. Thus, a sub-basis for the open sets in P.X/ is given by taking K  I a compact subset and U  X an open set and letting < K; U > be all maps ¨W I ! X with ¨.K/  U. Define  W P.X/ ! X by  .¨/ D ¨.1/. This is a fibration. Homotopy Exact Sequence of a Fibration. Here is the statement: Theorem 2.4. Suppose that  W E ! B is a fibration with B path connected. Fix b 2 B, and let Fb be the fiber over b and iW Fb ! E the inclusion. Finally, fix e 2 Fb . Then we have a long exact sequence of homotopy groups: i#

 #

!  n .Fb ; e/ !  n .E; e/ ! n .B; b/ ! n1 .Fb ; e/ ! : Proof. Exactness at  n .E; e/. It is clear that  # ı i# D 0. Suppose a 2 n .E; e/ and  # .a/ D 0. Represent a by a map 'W .Sn ; p/ ! .E; e/, where p 2 Sn is the base point. Since  # .a/ D 0, there is a homotopy H from   ı 'W .Sn ; p/ ! .B; b/ to the constant map at b, a homotopy that is constant on fpg  I . Use the homotopy lifting Q Sn  I ! E property for the relative CW complex .Sn ; p/ to lift H to a homotopy HW n beginning at ' and sending fpg  I to e. The map HjSn f1g is a map .S ; p/ ! .Fb ; e/ representing a 2  n .E; e/. This proves that the image of i# contains the kernel of  # , completing the proof of exactness at  n .E; e/. Exactness at  n .B; b/. Let us first define the connecting homomorphism  n .B; b/ !  n1 .Fb ; e/. Given an element, aN 2  n .B; b/ represent it by a map W .Sn ; p/ ! .B; b/. Let pW .Dn ; @Dn / ! .Sn ; p/ be the map collapsing the boundary

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of the disk to the base point of the sphere. Use the fact that the disk is contractible and the homotopy lifting property to lift the composite of p followed by to a map Q W .Dn ; @Dn / ! .E; Fb / sending the base point (in @Dn ) to e. Then the restriction of Q j@Dn represents the image under the connecting homomorphism of a. We leave the proof that this is process determines well-defined homomorphism to the reader. It is clear from this construction that the composition of  # followed by the connecting homomorphism is zero since we can take the lift of aN D  # .a/ to be the composition the collapsing map pW .Dn ; @Dn / ! .Sn ; p/ followed by a. It follows that the image of the connecting homomorphism applied to  # .a/ D 0. Conversely, if the image of the connecting homomorphism applied to aN is zero, then the lifted map .Dn ; @Dn / ! .E; Fb / has the property that its restriction to the boundary is homotopic in .Fb ; e/ to a point map. Homotopy extension allows us to lift the original map of Sn to B to a map of Sn to E, showing that the image of  # contains the kernel of the connecting homomorphism. Exactness at  n .Fb ; e/. From the construction, any based sphere in .Fb ; e/ coming from the connecting homomorphism bounds a disk in .E; e/, showing that the composition of the connecting homomorphism followed by i# is zero. Conversely, if an element c 2  n .Fb ; e/ is trivial in  n .E; e/, then the sphere in .Fb ; e/ representing c bounds a disk in .E; e/ whose image under  # is a sphere of one higher dimension in .B; b/ whose image under the connecting homomorphism is c. 

The Loop Space Proposition 2.5. Let x0 2 X be given and let P.X; x0 / denote the space of paths in X beginning at x0 . Then  W P.X; x0 / ! X is a fibration. Proof. Given a path gW I ! X and an element gQ 0 2 P.X/ such that  .Qg0 / D g.0/ (i.e., given a path g in X and a path g0 beginning at x0 and ending at g.0/), we define gQ t 2 P.X; x0 / by ( gQ t .s/ D

gQ 0 .s.1 C t//

0s

g.s.1 C t/  1/

1 1Ct

1 1Ct

 s  1:

One sees easily that  .Qgt / D gt and that t ! gQ t is a continuous mapping of I into P.X; x0 /. This proves the homotopy lifting property for points. One checks that the construction varies continuously with the original data and hence gives the homotopy lifting property for all spaces.  Definition. The fiber  1 .x0 /  P.X; x0 / is denoted ˙.X; x0 / and is the loop space of X based at x0 .

2.3 Homology

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Lemma 2.6. P.X; x0 / is contractible (i.e., homotopy equivalent to a point). c

Proof. Define P.X; x0 /  I ! P.X; x0 / by c.¨; t/.s/ D ¨.ts/: Clearly, c.¨; 0/ is the constant path at x0 for any ¨ 2 P.X; x0 /, and c.¨; 1/ D ¨.  Corollary 2.7. Fix a point x0 2 X and let ¨0 be the constant loop at x0 . There is Š

an isomorphism  n1 .˙.X; x0 /; ¨0 / !  n .X; x0 /. Proof. We have the fibration P.X; x0 / ! X with fiber ˙.X; x0 / over the base point x0 2 X. The result now follows from Theorem 2.4 and the fact that P.X; x0 / is contractible and hence has trivial homotopy groups.  We have seen how to replace any map by an inclusion up to homotopy. It is also possible to replace any map by a fibration up to homotopy. Given fW X ! Y, form the set of pairs f.x; ¨/g where x 2 X and ¨ is a path in Y satisfying ¨.0/ D f.x/g. Q Q This is a subspace of X  P.Y/ where P.Y/ represents the space of all paths in Y. Q Q Define (Thus, P.Y/ D YI with the compact-open topology.) Call this subspace X. Q ! Y by  .x; ¨/ D ¨.1/. Define iW X ,! X Q by i.x/ D(x, constant path at f.x/).  W X Q is a homotopy equivalence, and that One checks easily that   ı i D f, that iW X ,! X Q ! Y is a fibration.  W X The freedom to convert maps up to homotopy into either inclusions or fibrations in homotopy category illustrates the elasticity of the notion of homotopy; to balance this flexibility, one has the perversity of nature that the homotopy groups of the simplest spaces, the spheres, have thus far been impossible to calculate.

2.3 Homology A chain complex C is a set of abelian groups Cn indexed by the integers with maps, called boundary maps, @n W Cn ! Cn1 satisfying @n1 ı @n D 0 for all n. The homology of a chain complex is a sequence of abelian groups defined by Hn .C / D Kernel @n =Image @nC1 : The kernel of @n is the group of cycles in degree n and the image of @nC1 is the group of boundaries in degree n so that the nth homology is the quotient of the group of cycles in degree n modulo the group of boundaries in degree n. Elements of Hn .C / are said to have degree n. Let X be a space. Here we shall introduce the construction of singular homology of X with Z coefficients and recall its basic properties. Let n be the standard n-simplex in RnC1 . This is the subset of RnC1 defined by n X ˇ n D f.t0 ; : : : ; tn /ˇti  0 for all i and ti D 1:g i D0

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Let Singn .X/ be the free abelian group generated by the singular n-simplices f

n ! X; which by definition are the continuous maps of n to X. The ith-face of n , denoted fi , is the subset where ti D 0. It is naturally identified with n1 by an affine linear isomorphism that preserves the order of the vertices (the extremal points).

These identifications of the faces of n with n1 induce a boundary map @n W Singn .X/ ! Singn1 .X/ defined by @n  D

n X .1/i jfi : iD0

This boundary maps satisfy @n1 ı @n D 0. This is the singular chain complex of a space: @nC1

@n

   ! SingnC1 .X/ ! Singn .X/ ! Singn1 .X/ !    ; The singular homology of X, denoted H .X/, is the homology of the singular chain complex, i.e., it is defined as the quotient of the singular cycles modulo the singular boundaries; i.e., Hn .X/ D Kernel@n =@nC1 SingnC1 .X/ Given a pair of spaces .X; A/, we define the relative singular chain complex Sing .X; A/ as the quotient Sing .X/=Sing .A/. This means the relative chain group indexed by n is the quotient Singn .X/=Singn .A/, and the boundary map is the one induced by the boundary map in the singular chain complex of X. The relative singular homology H .X; A/ is the homology of Sing .X; A/. Notice that there is a natural map Hn .X/ ! Hn .X; A/ induced by the quotient projection on the singular chain complexes. Also, there is a connecting homomorphism Hn .X; A/ ! Hn1 .A/.

2.3 Homology

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The basic properties of homology are: (i) A map fW X ! Y induces a map on homology f W H .X/ ! H .Y/ which depends only on the homotopy class of f. (ii) An orientation for Sn determines an isomorphism Hn .Sn / Š! Z. If we reverse the orientation, then this isomorphism is multiplied by 1. A representative cycle for the class corresponding to 1 2 Z is determined by a homeomorphism @nC1 ! Sn which is orientation-preserving. (iii) Given Y  X, there is an exact homology sequence @

: : : Hn .Y/ ! Hn .X/ ! Hn .X; Y/ ! Hn1 .Y/ ! : : : (iv) If U  Y  X is such that U  interior .Y/; then we have the excision property H .X  U; Y  U/ Š H .X; Y/: (v) Any homology theory, that is, a functor from the homotopy category to abelian groups which satisfies (ii)–(iv), is naturally identified with singular homology [cf. Exercise (40)]. The most interesting of the axioms is probably excision, which is essentially the same as Mayer–Vietoris. We define singular homology with coefficients in an abelian group G by forming the chain complex @n ˝1

   ! Singn .X/ ˝ G ! Singn1 .X/ ˝ G !    and taking its homology. These groups are denoted Hn .X; G/. When no coefficients are specified, Z coefficients are understood. We also have cohomology. A cochain complex is a set of abelian groups fCn g indexed by the integers with coboundary maps ı n W Cn ! CnC1 satisfying ı nC1 ı ı n D 0. The cohomology Hn .C / is the quotient of kernel ı n (the cocycles) modulo the image of ı n1 (the coboundaries). The singular cohomology of X is the cohomology of the singular cochain complex •nC1

 Hom.Singn .X/; G/

•n

 Hom.Singn1 .X/; G/

 :::

where •n is the map dual to @nC1 . These groups are denoted Hn .X; G/. Again if no G is made explicit, Z coefficients are understood.

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Products in Homology and Cohomology. Singular cohomology Hn .X/ has the structure of a graded-commutative ring. That is to say, there is a multiplication Hp .X/ ˝ Hq .X/ ! HpCq .X/ with the property that ’ [ “ D .1/deg’deg“ “ [ ’. We begin with an explicit formula for the cup product in cohomology, the Whitney formula. Suppose that ’ is a singular cochain of degree p for X and “ is a singular cochain of degree q for X. We define the cup product ’ [ “ by defining its value on any singular simplex W pCq ! X by h’ [ “; i D h’; jf rp ih“; jbkq i; where f rp is the p-dimensional face of pCq , the face spanned by the first p C 1 vertices, and bkq is the q-dimensional face of pCq , the face spanned by the last q C 1 vertices. It is easy to see that this product is associative and satisfies •.’ [ “/ D •.’/ [ “ C .1/p ’ [ •.“/: It follows easily that the cochain cup product induces an associative product on cohomology. The cup product has no symmetry properties on the cochain level; nevertheless, it turns out for classes a 2 Hp .X/ and b 2 Hq .X/, we have a [ b D .1/pq b [ a: That is to say, the induced product makes cohomology into an associative, gradedcommutative ring with unit. There are related algebraic structures: (1) Homology has a co-associative, graded-commutative co-multiplication W Hk .X/ ! ˚iCjDk Hi .X/ ˝ Hj .X /: (2) There is a cap product: Hp .X/ ˝ Hq .X / ! Hqp .X/: The first pairing is induced by the following map on the chain level: For a singular simplex  of degree k,  7!

k X

D jf ri ˝ jbkki :

iD0

The second pairing is induced by the following map on the chain level: Let  be a singular simplex of degree q and let “ be a singular cochain of degree p. Then  \ “ D h“; f rp ./ibkqp ./:

2.3 Homology

17

Since singular homology and cohomology are defined using the same underlying chain complex, they are closely related, as expressed in the universal coefficient theorem. First, notice that singular n-cochains can be evaluated on singular n-chains that this leads to a pairing Hn .X/ ˝ Hn .X/ ! Z; and hence a homomorphism Hn .X/ ! Hom.Hn .X/; Z/: This homomorphism appears in the universal coefficient theorem. First, we need to introduce Tor and Ext in the category of abelian groups. Suppose we are given two abelian groups A and B. There is a short free resolution of A, i.e., a short exact sequence f0g ! F1 ! F0 ! A ! f0g with F0 and F1 being free abelian groups. We define Tor.A; B/ to be the kernel of the map F1 ˝ B ! F0 ˝ B, and we define Ext.A; rmB/ to be the cokernel of the map Hom.F0 ; B/ ! Hom.F1 ; B/. Up to canonical isomorphism, these definitions are independent of the choice of the free resolution of A. Theorem 2.8 (Universal Coefficient Theorem). 1. There is a short exact sequence: 0 ! Ext.Hn1 .X/; Z/ ! Hn .X/ ! Hom.Hn .X/; Z/ ! 0: 2. For any abelian group G, there are short exact sequences: f0g ! Hn .X/ ˝ G ! Hn .XI G/ ! Tor.Hn1 .X/; G/ ! f0g; 0 ! Ext.Hn1 .X/; G/ ! Hn .XI G/ ! Hom.Hn .X/; G/ ! 0: When the homology of X is finitely generated in each degree, this implies: 1. Hn .X/=Torsion and Hn .X/=Torsion are dual free abelian groups. 2. Torsion.Hn1 .X// and Torsion.Hn .X// are Pontrjagen dual finite abelian groups. Properties of the Homology of Closed, Oriented Manifolds. The homology and cohomology groups of a closed, oriented manifold satisfy a strong relationship, Poincaré duality. Theorem 2.9. Let M be a closed, oriented manifold of dimension n. Then there is a fundamental class ŒM 2 Hn .M/ and cap product with this class induces an isomorphism \ŒM W Hq .M/ ! Hnq .M/:

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2 Basic Concepts

While this theorem holds for topological manifolds, it has nice descriptions in the case when M is smooth. For a smooth manifold, the fundamental class ŒM 2 Hn .M/ can be described as follows. Take a smooth triangulation of M and choose a total ordering for the vertices of this triangulation. This induces an ordering of the vertices of each n-dimensional simplex and hence an identification of each n-dimensional simplex with n . For each n-dimensional simplex  of the triangulation, we define a sign  by comparing the orientation on  induced P from that of M with the standard orientation of the n-simplex. Then the sum c D  , where the sum is taken over all the n-dimensional simplices of the triangulation, is a singular n-chain in M. Each .n  1/-dimensional simplex in the triangulation occurs as in the boundary of exactly two n-simplices, and it is easy to see that the signs cancel out so that c is a cycle. It represents the fundamental class of M. Putting this together with the universal coefficient theorem, we see that Poincaré duality implies that the rank of Hk .M/ is equal to the rank of Hnk .M/ and the torsion in Hk .M/ is isomorphic to the torsion in Hnk1 .M/. Actually, there are much stronger statements than just abstract isomorphisms: There are dual pairings which we will now describe. Given homology classes a 2 Hk .M/ and b 2 Hnk .M/, we can represent them by singular cycles so that each singular simplex maps smoothly into M and furthermore so that the simplices in the representative for a and those for b meet transversely. This means that the intersection is a finite set of points and at each intersection point the tangent spaces of the two simplices are complementary subspaces in the tangent space of M. Since M and each of the simplices is oriented, we can associate a sign, ˙1, to each intersection point and the sum of these signs over all intersection points is an integer which depends only on the homology classes and not on the cycle representatives. This defines a pairing Hk .M/ ˝ Hnk .M/ ! Z: In fact, its adjoint is the composition of Poincaré duality followed by the usual evaluation map Hk .M/ ! Hnk .M/ ! Hom.Hnk .M/; Z/; so that the pairing is perfect on the quotients of these homology groups by their torsion subgroups. This is called the intersection pairing. The symmetry of cup product means that this pairing is also graded-commutative a  b D .1/deg.a/deg.b/ b  a: There is an analogous pairing for the torsion groups: Given torsion classes a 2 Hk .M/ and b 2 Hnk1 .M/, we choose representative cycles, ’ and “, that are disjoint. Then some multiple N ’ is the boundary of a .k C 1/-chain  that we can make transverse to “. Counting signed intersections as before, we form .  “/=N 2 Q=Z. This defines a pairing

2.4 Categories and Functors

19

Torsion Hk .M/ ˝ Torsion Hnk1 .M/ ! Q=Z which is a perfect pairing in the sense that its adjoint is an isomorphism. This pairing is called the linking pairing. This construction is most pleasing when the manifold is smooth and the cycle representatives are themselves closed, oriented smooth submanifolds of M. Notice that if Pk is a closed, oriented submanifold of M, then it represents a homology class ŒP 2 Hk .M/. One way to see this is to take the image of the fundamental class of P under the inclusion P ,! M. Another is to take a smooth triangulation of P, choose a total ordering of the vertices of this triangulation, and then take the signed sum of the top-dimensional simplices forming a k cycle in M. Given closed, oriented submanifolds Pk and Qnk in M meeting transversally, we form the homological intersection P  Q. This is the sum over the (finite) set of points of P \ Q of a sign for each point, a sign that compares on one hand the sum of the orientation for P at the intersection point followed by that of Q and on the other the orientation of M at that point. This homological intersection is exactly the value of the intersection pairing, as defined above using Poincaré duality, of the homology classes represented by P and Q.

2.4 Categories and Functors Now we will introduce the language of categories and functors which so permeates mathematics today. There are two reasons for doing this. One is that it is encountered when reading reference articles, and the second is that it gives a convenient formalism for stating and remembering many of the results contained in these notes. In the notes, we do not always bother to give the category-theoretic reformulation of the results we prove. These will sometimes be left to the reader as exercises. A category, C, consists of a collection of objects, Obj.C/, and for any pair of objects A and B of the category a set of morphisms from A to B. This set is denoted by Hom.A; B/ or HomC .A; B/. Several axioms are required to be satisfied: 1. Hom.A; A/ always contains a distinguished element IA , the identity morphism of A. 2. For objects A, B, C of C, there is a composition Hom.A; B/  Hom.B; C/ ! Hom.A; C/ which is associative .fg/h D f.gh/ and for which IA is a unit; i.e., f ı IA D f and IB ı f D f for any f 2 Hom.A; B/. Note: The objects of C do not necessarily have to have underlying sets and a morphism from A to B does not have to be a set map. It is just considered as an arrow. Examples. 1. Abelian groups. The objects are abelian groups; the morphisms are group homomorphisms.

20

2 Basic Concepts

2. Topological spaces. The objects are the usual topological spaces; the morphisms are continuous maps. 3. Homotopy category. The objects are spaces homotopy equivalent to CW complexes, and the morphisms are homotopy classes of maps. 4. Analogous to 1, we have the categories of groups, rings, fields, vector spaces over a field, and modules over a ring. An isomorphism in a category is a morphism with an inverse; i.e., fW A ! B is an isomorphism if there exists gW B ! A with g ı f D IA and f ı g D IB . The isomorphisms in 1, 2, and 3 are, respectively, group isomorphism, homeomorphism, and homotopy equivalence. A (covariant) functor between two categories, F W C ! C 0 , is an assignment of objects F .A/ in C 0 to objects A in C and morphisms F .f/ in HomC 0 .F .A/; F .B// to morphisms f in HomC .A; B/ such that identities and compositions are preserved. Clearly a functor sends isomorphic objects in C to isomorphic objects in C 0 . For any category C, there is an opposite category C opp . These two categories have the same objects and by definition HomC opp .A; B/ D HomC .B; A/. A contravariant functor from C to C 0 is by definition a covariant functor from C opp to C 0 . A homotopy functor is a functor from the homotopy category to another category. Algebraic topology is the study of homotopy functors into algebraic categories— i.e., groups, rings, fields, chain complexes, and vector spaces. Under a homotopy functor, homotopy equivalent spaces have assigned to them isomorphic objects. This is why, in homotopy theory, it is permissible to identify homotopy equivalent objects. As an example of a functor of algebraic topology, consider homology. The homology functor assigns to each space X a graded group H .X/ and to a continuous map fW X ! Y an induced map on homology f W H .X/ ! H .Y/. If f  g, then f D g so that homology is actually a homotopy functor. (See the next chapter for more details on singular homology.)

Chapter 3

CW Homology Theorem

3.1 The Statement Suppose that X is a CW complex and let X.n/ denote its n-skeleton. Then the .nC1/skeleton X.nC1/ is obtained by attaching .n C 1/-cells to X.n/ by maps f

@enC1 ! X.n/ : Q n .X/ the free abelian group generated by the oriented n-cells of X. Denote by C Since each n-cell has two orientations, there are two generators in CQ n .X/ for each n-cell. Introduce the relation that the two generators corresponding to the opposite orientations of a cell are negatives of one another. Denote the quotient group by Cn .X/. It is a free abelian group of rank equal to the number of n-cells. If we are given an orientation for an n-cell en of X, then choosing an orientation preserving isomorphism n ! en determines an element in Hn .X.n/ ; X.nC1/ /. The resulting element depends only on the orientation of en and changes sign if we reverse the orientation. Thus, the construction determines a map ®n W Cn .X/ ! Hn .X.n/ ; X.n1/ /. As we shall see below, an easy application of the excision theorem for homology shows that this mapping is an isomorphism. Let us assume for the moment that ®n is an isomorphism. Define the boundary map @W Cn .X/ ! Cn1 .X/ to be the composition: ®n

i

@

Cn .X/ ! Hn .X.n/ ; X.n1/ / ! Hn1 .X.n1/ / ! Hn1 .X.n1/ ; X.n2/ / ®1 n1

! Cn1 .X/: Theorem 3.1 (CW Homology Theorem). fC .X/; @g is a chain complex. Furthermore, there is a natural identification of its homology with the singular homology of X.

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__3, © Springer Science+Business Media New York 2013

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3 CW Homology Theorem

Remarks. (1) In case X has only finitely many cells in each dimension, the chain groups Cn .X/ are finitely generated, free abelian groups. Hence, the above theorem gives a fairly effective way of calculating homology. (2) If .X; A/ is a CW pair, i.e., if X is built by inductively attaching cells to A, then one has an analogously defined chain complex fCn .X; A/; @g which computes the relative singular homology H .X; A/.

3.2 The Proof Let us begin the proof of Theorem 2.1 with a couple of simple lemmas. Lemma 3.2. Let X D Y[f em . Then 

Hi .X; Y/ D 0 i ¤ m; Hm .X; Y/ Š Z:

Proof. Let N be a neighborhood of @em in en and U D Y[f N.

Then there is a deformation retraction of U onto Y (cf. the picture). The excision and homotopy axioms give N Š Hi .int.em /; int.en / \ @N/ Hi .X; Y/ Š Hi .X; U/ Š Hi .em ; @em /; where the first and third isomorphisms come from the fact that homotopy equivalent pairs have isomorphic homology and the second comes from excision. But Hi .em /D0 for i > 0, and so the exact homology sequence and computation of the homology of Sm1 D @em give (for i 6D 1)  Hi .e ; @e / Š Hi1 .@e / D m

m

m

Z for i D m 0 otherwise:



Q  .X/ to be the kernel We define the reduced homology of a space X, denoted H of the map H .X/ ! H .pt/ induced by the collapsing map X ! pt.

3.2 The Proof

23

Remark. A similar proof gives the following useful result: Proposition 3.3. Let (X, A) be a CW pair. Let X=A denote the CW complex Q i .X=A/. obtained by shrinking A to a point. Then Hi .X; A/ Š H The proof is by induction on the dimension of the cells and proceeds similarly to the proof of Lemma 2.2. Details are left to the reader. It is excision which makes homology more computable than other homotopy functors such as the homotopy groups. Lemma 3.4. Let X be a CW complex. Then Hi .X; X.n1/ / D 0

for i  n  1

Hi .X.n/ ; X.n1/ / D 0

for i ¤ n:

The map ®n W Cn .X/ ! Hn .X.n/ ; X.n1/ / is an isomorphism. Proof. These statements are all proved from Lemma 2.2 using induction on the number of cells (plus a direct limit argument if X has infinitely many cells).  Proof (Proof of Theorem 3.1). We have an isomorphism ®n W Cn .X/ ! Hn .X.n/ ; X.n1/ / .n/ .n1/ Q Q / ! and @W Cn .X/ ! Cn1 .X/ is ®1 n1 ı @ ı ®n where @W Hn .X ; X .n1/ .n2/ Hn1 .X ;X / is the boundary in the homology long exact sequence for the triple X.n2/  X.n1/  X.n/ . Since @Q n1 ı @Q n D 0, it follows that @  @W Cn .X/ ! Cn2 .X/ is zero. Thus, fC .X/; @g is a chain complex. Setting Z D X.n2/ and applying Lemma 2.2, we have an exact sequence

0 ! Hn .X.n/ ; Z/ ! Hn .X.n/ ; X.n1/ / ! H.n1/ .X.n1/ ; Z/ @Q ? ? ? ?D ?D ?D y y y 0 !

Hn .X.n/ /

!

Cn .X/

! @

Cn1 .X/:

Thus, Hn .X.n/ / is identified with the cycles in Cn .X/. Similarly, there is a commutative diagram: D

CnC1 .X/

/ HnC1 .X.nC1/ ; X.n/ /

@

/ Hn .X.n/ / m6 m m mmm m m m  mmm @ .n/ HnC1 .X; X /  0

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3 CW Homology Theorem

This allows us to identify @CnC1 .X/  Zn .C .X// with @.HnC1 .X; X.n/ //  Hn .X.n/ /. Applying 2.2 and 2.4 once again, we find the following is exact: @

HnC1 .X; X.n/ / ! Hn .X.n/ / ! Hn .X/ ! 0: Thus, Zn .C .X//=@CnC1 .X/ D Hn .X.n/ /[email protected] .X; X.n/ // D Hn .X/. If fW X ! Y is a continuous mapping between CW complexes, then f is homotopic to a cellular map. If we deform f to be cellular, then it induces maps on all the relative groups used in establishing the equivalent of H .C .X// with H .X/. Hence, this equivalence is functorial.  is Remark. Choose an orientation for each n-cell and each .n C 1/-cell of X. If enC1 ’ n an .n C 1/-cell, then @enC1 D † c e for some integer coefficients c . This means “ ’“ “ ’“ ’ ! X.n/ is the attaching map for enC1 that if f’ W @enC1 ’ ’ , then there is a singular chain c in X.n1/ so that @enC1  †c’“ en“  c is a singular boundary in X.n/ . ’ If the attaching maps are generic, then there is an appealing geometric description of @W Cn .X/ ! Cn1 .X/. Choose an oriented n-cell .en ; Oe / and an oriented .n  1/cell .¢ n1 ; O¢ /. To calculate the coefficient of .¢ n1 ; O¢ / in @.en ; Oe /, one simply ®e considers the attaching map @en ! X.n1/ . Generically this map will be transverse to some point p¢ in ¢ n1 . Comparison of the orientations allows us to assign an algebraic intersection number ˙1 to each point in ®1 e .p¢ /. The sum of these is the sought-after coefficient.

3.3 Examples 1. RP2 D S1 [z2 D2 where D2 D fz 2 CW jzj  1g. The boundary map sends ŒD2 ; OD2 ! 2Œe1 ; Oe1 . 2. The 2-torus is obtained by attaching D2 to S1 _ S1 by aba1 b1

®1 .Pa / D 2 points with opposite sign ®1 .Pb / D 2 points with opposite sign. Hence, @ŒD2 ; OD2 D 0. 3. The n-sphere Sn D x0 [f en . Thus, Hi .Sn / D 0 for i ¤ 0; n and Hn .Sn / D Z. 4. Complex projective space CPn is obtained from CPn1 by attaching a cell of dimension 2n. Inductively, one argues that CPn has a cell decomposition with exactly one cell occurring in dimensions 0; 2; 4; : : : ; 2n. Thus, in C .X/, all boundary maps must be zero. Hence, H2i .CPn / D Z for 0  i  n H .CPn / D 0 for all other :

3.3 Examples

25

5. Consider the map fn W S1 ! S1 given by ei™ ! ein™ S1 [fn e2 . Then

for n 2 Z. Let Xn D

H1 .Xn / D Z=n Z: The reason is that @e2 D nS1 (cf. example 1 for n D 2). 6. As we saw in Chap. 2, a simplicial complex has a natural CW complex structure. The CW homology theorem gives a purely combinatorial way to calculate H .jKj/. (Here jKj is the topological space associated to K.) The complex fC .jKj/; @g is exactly the simplicial chain complex. 7. If .X; A/ is a CW pair, then there is an analogously defined complex fCn .X; A/; @g. The group Cn .X; A/ is a free abelian group with one generator for each n-cell in X–A. Arguments similar to the ones above show that the homology of this complex is naturally identified with H .X; A/.

Chapter 4

The Whitehead Theorem and the Hurewicz Theorem

4.1 Definitions and Elementary Properties of Homotopy Groups For spaces X and Y, we denote by ŒX; Y the set of homotopy classes of maps from X to Y. For pairs of topological spaces .X; A/ and .Y; B/, we denote by Œ.X; A/; .Y; B/ the set of homotopy classes of maps of the pair .X; A/ to the pair .Y; B/. If H is such a homotopy, then for each t, the map Ht is required to map A to B. If .X; x0 / and .Y; y0 / are based spaces, then we denote Œ.X; x0 /; .Y; y0 / by ŒX; Y 0 . There is a similar notation homotopy classes of maps of triples Œ.X; A; A0 /; .Y; B; B 0 / where A0  A  X and B 0  B  Y Recall that the set ŒS1 ; X 0 has a group structure induced by composing loops. This group is the fundamental group and is denoted  1 .X; x0 / (or  1 .X/ if the base point plays no important role). We define sets  n .X; x0 / D ŒSn ; X 0 . Then  0 .X/ is exactly the set of path components of X. We show how to make  n .X/ an abelian group for n  2. To define the group structure on  n .X/; n  2 and show that it is abelian, it is convenient to give a different description. Let In denote the n-cube. Then  n .X/ D Œ .In ; @In /I .X; x0 / . The composition is

This is abelian as the following sequence of pictures indicates (the unmarked areas are sent to the base point).

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__4, © Springer Science+Business Media New York 2013

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4 The Whitehead Theorem and the Hurewicz Theorem

For n  2, define  n .X; A/ to be Œ.In ; @In ; @In  In1  f1g/; .X; A; a0 / : The group law is given as before. An argument similar to the one above shows that for n  3;  n .X; A/ is an abelian group. Given f 2 Œ.X; A/; .Y; B/ , there is an induced map f W   .X; A/ !   .Y; B/: This means that the homotopy groups form a covariant functor from the homotopy category (of pairs) of spaces to the category of groups. Remarks. (i) The homotopy groups are easier to define but much harder to calculate than the homology groups. One of the main points of this course will be to give an effective way to calculate   ˝ Q from the DGA of differential forms. (ii) We show that  k .Sn / D 0 if k < n and  3 .S2 / ¤ 0. To see the first, let fW Sk ! Sn be given with k < n. Deform f until it is cellular. Since the k-skeleton of Sn can be taken to be a point, this deformation carries f to a constant map.  (In general, this argument shows that  k .X.m/ / !  k .X/ for k  m  1 and that  k .X.k/ / !  k .X/ is onto.) To show that  3 .S2 / ¤ 0, consider CP2 as an adjunction space CP2 D .CP1 /[f e4 where CP1 D S2 . The homotopy type of an adjunction space depends only on the homotopy class of the attaching map. Thus, if  3 .S2 / D 0, then CP2 would be homotopy equivalent to S2 [constant e4 D S2 _ S4 . This would imply that if g generates H2 .CP2 /, then g [ g D 0 in H4 .CP2 /. But we know (say by Poincaré duality) that g [ g ¤ 0. There is an analogous argument showing that  4n1 .S2n / ¤ 0. If fW Sn ! X, then there is induced f W Hn .Sn / ! Hn .X/. But Hn .Sn / is identified, once and for all, with Z. Thus, we have f .1/ 2 Hn .X/. This determines a natural transformation: HW  n .X/ ! Hn .X/ Œf ! f .1/ called the Hurewicz homomorphism. (It is an easy exercise to show that H is indeed a homomorphism.) Applying this in the special case where X D Sn , we have HW  n .Sn / ! Hn .Sn / D Z: H.f/ is also called the degree of f and sometimes denoted by deg.f/. Theorem 4.1 (Brouwer). The map HW  n .Sn / ! Z is an isomorphism.

4.2 The Whitehead Theorem

29

Since H.IdSn / D 1, it is obvious that H is onto. It is harder to show that H is one-to-one, i.e., that if deg.f/ D deg.g/, then f ' g. We shall prove this later in this chapter. If x0 2 A  X, then there is an exact homotopy sequence i

@

j

: : : !  nC1 .X; A/ !  n .A/ !  n .X/ !  n .X; A/ ! : : : As one consequence of this and the calculations above, we see that excision is false for the homotopy groups. @



The easiest example of this is that  3 .D2 ; S1 / !  2 .S1 / is zero whereas  3 .S2 ; D2 / Š  3 .S2 / ¤ 0. Of course, where excision to hold for homotopy groups, then they would satisfy all the axioms for homology. If that were true, then the argument in Chap. 3 which identified singular homology with H .C .X; A// could be used to identify the homotopy groups with H .C .X; A//. It would follow that the homology and homotopy groups were isomorphic.

4.2 The Whitehead Theorem Theorem 4.2. Let X and Y be CW complexes with base points, x0 and y0 , being 0-cells. Let fW .X; x0 / ! .Y; y0 / be a map inducing isomorphisms Š

f W  n .X; x0 / !  n .Y; y0 / for all n  0 Suppose that Y is connected. Then fW X ! Y is a homotopy equivalence. Let us begin with a special case. Suppose dim X < 1 and  n .X/ D 0 for all n  0. We shall show that X is contractible. (This is a special case of the theorem for x0 ,! X:/ Since  0 .X/ is the one point set, the zero skeleton can be deformed to x0 2 X .

Use the homotopy extension property (Theorem 2.1) to obtain a continuous family: ft W X ! X such that f0 D IdX and f1 .X.0/ / D x0 .

0t1

30

4 The Whitehead Theorem and the Hurewicz Theorem

Now consider X.1/ . The image under f1 of any 1-cell e1 is a loop based at x0 . Since  1 .X/ D 0 we can deform f1 .e1 / through loops based at x0 to the constant loop. Doing this for each l-cell defines a homotopy from f1 jX.1/ to the constant map f2 W X.1/ ! x0 . It is a homotopy relative to X.0/ . Using homotopy extension, we can find a homotopy ft W X ! X; 1  t  2, with f2 jX.1/ D constant. Continue in this fashion using the fact that  n .X/ D 0 for all n. In the end, we have a homotopy from IdX to the constant map of X to x0 2 X. Remarks. (i) The same argument shows that if  n .X/ D 0 for n < N, then there is a homotopy ft W X ! XI

0t1

with f0 D Id and f1 .X.N1/ / D x0 : Q k .X/ D 0 for k < N. As a consequence, H (ii) In case X is infinite dimensional, we make the first homotopy last for 0  t  1=2, the second last for 1=2  t  3=4, etc. and then define f1 .X/ D x0 . Using the fact that the topology on X is the weak (or limit) topology, it is an easy exercise to show that the proposed homotopy is indeed continuous. (iii) There is a relative version of the theorem which states that if .X; A/ is a CW pair and if  n .X; A/ D 0 for all n, then there is a deformation retraction f1 W X ! A; i.e., there is a homotopy ft W X ! XI 0  t  1, such that f0 D identity; f1 .X/  A, and ft jA D identityA . The proof is essentially the same as in the absolute case. General Case: Given fW X ! Y, choose a cellular map f0 homotopic to f. The pair .Mf0 ; Y / is a relative CW complex. Since Mf0 retracts to Y, it follows that  k .M0f / D  k .Y/ for all k. If f0 W  k .X/ !  k .Y/ is an isomorphism for all k, i

then so is  k .X/ !  k .Mf0 /. Hence,  k .Mf0 ; X/ D 0 for all k. Thus, remark (iii) r implies that there is a retraction rW Mf0  X. The composition Y  Mf0 ! X is a homotopy inverse for f0 and hence for f. This completes the proof of the Whitehead theorem. Example. Think of Sk as Ik =f@Ik g. There is a product map I2 =f@I2 g  I2 =f@I2 g ! I4 =f@I4 g:

4.3 Completion of the Computation of  n .Sn /

31

This is a map fW S2  S2 ! S4 . Since  k .S2  S2 / Š  k .S2 /   k .S2 /, one sees easily that f W  k .S2  S2 / !  k .S4 / is trivial for all k. But f is not homotopic to a constant map. The easiest way to see this is to note that f W H4 .S2  S2 / ! H4 .S4 / is nonzero (in fact, it is an isomorphism). In dealing with spaces that are not CW complexes, a map fW X ! Y is not necessarily a homotopy equivalence if it induces an isomorphism in all homotopy groups. When it does, we say that f is a weak homotopy equivalence. This is not an equivalence relation, but it generates an equivalence relation weak homotopy equivalence which when restricted to CW complexes is homotopy equivalence by Whitehead’s theorem. In general if A is a CW complex and fW X ! Y is a weak homotopy equivalence, then f# W ŒA; X ! ŒA; Y is a bijection.

4.3 Completion of the Computation of  n .Sn / Theorem 4.3. The Hurewicz homomorphism  n .Sn / ! Hn .Sn / is an isomorphism. 

Proof. Since we have seen that HW  n .Sn / ! Z is onto, it suffices to show that if f .1/ D 0 in Hn .Sn /, then f  constant. First, deform f until it is a C1 mapping and p 2 Sn is a regular value for f. This means that f is a local diffeomorphism in a neighborhood of each xi 2 f1 .p/.

At each point xi 2 f1 .p/, there is a local degree of f. This is C1 if f is orientation-preserving near xi and 1 if it is orientation reversing there. We claim that deg.f/ D †xi 2f1 .p/ –f .xi / (where –f .xi / is the local degree of f at xi ). The easiest way to see this is to choose a C1 -form ¨ on sn of degree n supported in a small ball Dn  Sn containing p whose R  n integral over D is 1. Clearly, Sn f .¨/ D deg.f/. On the other hand, if we choose Dn sufficiently Small, there will be one component of f1 .Dn / for each xi 2 f1 .p/. If Ci is the component of f1 .D/ containing xi , then Z



f .¨/ D .deg.f/ at xi /  Ci

Z Dn¨

D deg.f/ at xi :

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4 The Whitehead Theorem and the Hurewicz Theorem

The result follows immediately. This argument assumes deRham’s theorem from Chap. 9. Alternatively one may use Hn .Sn / Š Hn .Sn ; p/ Š Hn .Sn ; Dn / Š Hn .Sn  Dn ; @Dn / Š Hn1 .Sn1 / together with Hn .Sn / Š Hn .Sn ; f1 .p// Š Hn .Sn ; f1 .Dn // Š Hn .Sn  f1 .Dn /; f1 .@Dn // ,! ˚i Hn1 .@Ci / to show that deg f D †i deg.fj@Ci /: Thus, we must show that if †xi2f1 .p/ .deg .f/ at xi / D 0, then f is homotopic to constant. We do this for n  2 (the case n D 1 will be proved afterwards). Assume first that f1 .p/ D fxg[fyg, that deg.f/ D 1 at x, and that deg.f/ D 1 at y. In Sn I, choose an embedded arc A connecting x and y and meeting Sn  f0g transversally at x and y

The disks around x and y [i.e. f1 .Dn /] extend to a tubular neighborhood N of A. In fact, N Š en  I

Note that f is defined on en  f0g and en  f1g and, since the local degree of f at x and y has opposite signs, the degrees of f on Sn1 f0g and Sn1  f1g are the same. By the induction assumption, f extends to a map FW Sn1  I ! @Dn . We may extend F to a map FW en  I ! Dn (this is a little exercise). Thus, f extends to a map QfW Sn  f0g [ N ! Sn such that QfW @N ! @Dn  Sn . Since Sn n int Dn is contractible, we can extend Qf to a map on the rest of Sn  I, sending the complement of N to Dn . This map will have p as a regular value with preimage the arc A. The resulting map on Sn  f1g will miss p and hence is homotopic to a constant.

4.4 The Hurewicz Theorem

33

If f1 .p/ has more than two points and n  2, then this argument allows us to deform f by a homotopy to “cancel” two of the points with opposite local degree. Continuing in this manner, we can deform f until the preimage of p is empty if the deg.f/ is 0. Once we have accomplished this, the result is clearly homotopic to a constant map. If n = 1, then we must take more care. The point is that the arc A between two points of opposite sign must be disjoint from xj  I for all other points xj 2 f1 .p/. If n  2, then general position ensures that this is always possible. If n D 1, then we must find a pair of points x and y 2 f1 .p/ of opposite local degree so that there are no other points of f1 .p/ “in between them” (i.e., so that one of the two arcs in S1 connecting them contains no other points of f1 .p//. Since this is always possible if deg.f/ D 0, we can still cancel a pair of points in this case.  There is a generalization of this result that we shall need. Theorem 4.4. For n > 1, the map HW  n ._i Sn / ! Hn ._i Sn / Š ˚i Z is an isomorphism. Proof. The proof is analogous to the one given above. One takes fW Sn ! _i Sni and deforms it until it is regular with respect to a point pi 2 Sni . (The point pi is different from the point at which all the Sni are joined together.) The cancelling argument shows that if the sum of the local degrees at xij 2 f1 .pi / is zero, then f is homotopic to a map missing pi . (This requires that n  2.) If H.f/ D 0, then all the local degrees †j .deg.f/ at xij / D 0 for all i. Thus, if H.f/ D 0, then f is homotopic to a map missing a point pi 2 Si for every i. This map is in turn homotopic to the constant map since _i .Si  pi / is contractible. 

4.4 The Hurewicz Theorem We begin with the statement of this result. Theorem 4.5. Let X be a CW complex. If  k .X/ D 0 for k < n, then Q k .X/ D 0 for k < n (recall that H Q is the reduced homology), and (i) H (ii) HW  n .X/ ! Hn .X/ is an isomorphism provided n > 1. Q k .X/ D 0 for Proof. We have already seen that if  k .X/ D 0 for k < n, then H k < n. It remains to show that under this hypothesis, we have 

HW  n .X/ ! Hn .X/: Step I: Let us show that H is onto. Since  i .X/ D 0 for i < n, the argument given in the proof of the Whitehead theorem shows that there is a map fW X ! X such that (1) f is homotopic to the identity and (2) f.X.n1/ / D x0 . Let a 2 Hn .X/, and let †ncells a’ .en’ ; O’ / be a cycle representative for a in Cn .X/. (Here, we choose arbitrarily an orientation O’ for each en’ ). Clearly fjen’ W .en’ ; @en’ / ! .X; x0 /. Thus,

34

4 The Whitehead Theorem and the Hurewicz Theorem

fjen’ represents an element in  n .X; x0 /. Since f is homotopic to the identity, ’ D f .’/ D class of †’ f.en’ /. Since each f.en’ / is represented by a sphere, this latter class is in the image of H. We wish to show that HW  n .X/ ! Hn .X/ is injective, provided that n > 1 and  i .X/ D 0 for i < n. Let fW Sn ! X be a map such that f ŒSn D 0 in Hn .X/. We can deform f until f.Sn /  X.n/ and until f is transverse regular to a point pi in the interior of each n-cell eni . Let œi D †x2f1 .pi / .local deg at x/: The chain † œi Œeni is a cycle in Cn .X/ which represents f ŒSn 2 Hn .X/. Since f ŒSn D 0, this cycle is a boundary; i.e., there exists j such that †œi Œeni D @† j enC1 . Adding to fW Sn ! Xn the linear combination † j @enC1 makes each j j œi D 0, and this does not change the homotopy class of fW Sn ! X. There is a map §W X ! X such that (a) § is homotopic to idX and (b) §jX.n1/ is constant. This means that §jX.n/ factors through a wedge of spheres:

X.n/

§jX.n/

DD DD DD DD p "

_Sn

/ X > | || | || || g

Since fW Sn ! X.n/ has each œi D 0, the composition p ı fW Sn ! _Sn is homologous to zero. By Lemma 3.2, this means that p ı fW Sn ! VSn is homotopic to zero. Hence, g ı p ı f D § ı f is homotopically trivial. Since § Š idX , this means that f itself is homotopically trivial.  Note: The Hurewicz theorem is valid for all spaces and not just CW complexes. To establish this, one works with the cycles themselves rather than with the spaces into which they are mapped. For a proof of the general result, see [23].

4.5 Corollaries of the Hurewicz Theorem To begin with, there is (of course) a relative form. Let A  X be a CW pair and assume that  1 .A/ D 0 and  i .X; A/ D 0;

i < n .n  2/:

4.5 Corollaries of the Hurewicz Theorem

35

Then it follows that Hi .X; A/ D 0

for i < n

and the relative Hurewicz map  n .X; A/ ! Hn .X; A/ H

is an isomorphism (the proof that Hi .X; A/ D 0 for i < n and that H is surjective follows from the proof of the Whitehead theorem, just as in the absolute case). We now come to the corollaries of Hurewicz theorem. f

Corollary 4.6. If X and Y are simply connected CW complexes and X ! Y induces an isomorphism on homology, then f is a homotopy equivalence. Proof. Let Mf be the mapping cylinder for f. Then Y ,! Mf is a deformation retract, and using this, we make the identifications  i .Y/ Š  i .Mf / Hi .Y/ Š Hi .Mf /: The inclusion X ,! Mf gives exact sequences f

   !  i .X/ !  i .Y/ !  i .Mf ; X/ !  i1 .X/    ? ? ? ? ? ? ? ? y y y y f

   ! Hi .X/ ! Hi .Y/ ! Hi .Mf ; X/ ! Hi1 .X/    Using that f is an isomorphism on homology gives Hi .Mf ; X/ D 0 for i  0. The relative Hurewicz theorem now gives  i .Mf ; X/ D 0

for i  0:

It follows then that Mf deformation retracts onto X (cf. the proof of the Whitehead theorem).  Corollary 4.7. If X has the homotopy type of an n-dimensional CW complex and if  i .X/ D 0 for i  n, then X is contractible. Proof. Since  i .X/ D 0 for i  n; Hi .X/ D 0 for i  n. On the other hand, Q  .X/ D 0 and so   .X/ D 0 by the Hi .X/ D 0 for i > n by dimension. Thus, H Hurewicz theorem. Hence, by the Whitehead theorem, X is contractible.  Corollary 4.8. If X has the homotopy type of an n-dimensional CW complex and if  i .X/ D 0 for i  n  1, then

36

4 The Whitehead Theorem and the Hurewicz Theorem

X  _S n : In particular, if Hn .X/ Š Z, then X has the homotopy type of Sn . (Thus, a simply connected homology sphere (a homology sphere is a space with the same homology as a sphere) is homotopy equivalent to a sphere.) Proof. This follows from the proof of the Hurewicz theorem in the following way: We may assume that X D X.n/ . As before, there is a map W X ! X homotopic to the identity with .X.n1/ . Thus, factors g

f

X ! _Sn ! X: f

It follows that Hn ._Sn / ! Hn .X/ is onto and we may choose a basis for Hn ._Sn / Š ˚Hn .Sn / (little exercise) and Hn .X/ such that f is given by a matrix with c columns and b rows, where b is the rank of Hn .X/, and c is the rank of Hn ._Sn /, and each ai is ˙1: 0 1 a1 0    0    0 B 0 a 0   0    0C B C 2 B C B 0 0  0 0 0    0C : B C @ 0  0  0 0    0A 0   0 ab 0    0 Since  n ._Sn / Š Hn ._Sn /, we may find a map h Sn _    _ Sn JJ JJ j JJ h JJ JJ  J% n / X _S f

such that h has matrix with b columns and c rows: 0 a1 B0 B B B0 B B0 B @0 0

0 a2 0   0

 0  0  

1    C C C 0 0C C  0C C 0 ab A  0

n : : _ S…n /. using the obvious basis for Hn .S „ _ :ƒ‚ b

It follows that j is an isomorphism on homology.



4.5 Corollaries of the Hurewicz Theorem

37

Corollary 4.9. If X is simply connected and Hi .X/ D 0 for i > n, then X  Y.nC1/ ; i.e., X has the homotopy type of an .n C1/-dimensional CW complex. If, in addition, Hn .X/ is free, then X  Y.n/ . Proof. We have from the CW homology theorem Š

Hi .X.n/ / ! Hi .X/

i¤n

Hi .X / ! Hn .X/ ! 0 .n/

Using the exact homology sequence Hi .X.n/ / ! Hi .X/ ! Hi .X; X.n/ / ! Hi1 .X.n/ / ! Hi1 .X/ it follows that Hi .X; X.n/ / D 0 for i ¤ n C 1 and that i

0 ! HnC1 .X; X.n/ / ! Hn .X.n/ / ! Hn .X/ ! 0 is exact. Since Hn .X.n/ / is the kernel of the map from Cn .X.n/ ! Cn1 .X.n/ , it follows that Hn .X.n/ / is free abelian. Using the relative Hurewicz theorem, we see that  nC1 .X; X.n/ / Š HnC1 .X; X.n/ /: Thus, we may attach .n C 1/-cells to X.n/ to in such a way that the attaching maps form a basis for the kernel of i . For this new CW complex Y.nC1/ , Hi .Y.nC1/ / Š Hi .X/ by construction for all i. Hence, Y.nC1/ / and X are homotopy equivalent. The other statement is an exercise.  Corollary 4.10. Let X be a simply connected topological n-manifold. Then X has the homotopy type of an n-dimensional CW complex. Remark. It is pretty clear that X has the homotopy type of a simplicial complex [cf. Exercise (13)]. Thus, in the homotopy category, X can be triangulated. If X is smooth, then X can be smoothly triangulated (see pp. 124–125 of [27]). If X is a topological manifold, then there is exactly one obstruction to it being triangulated in a homogeneous fashion, an obstruction in H4 .XI Z=2Z/; see [12]. (Here, “homogeneous fashion” means that every point has a neighborhood whose closure is a subcomplex that is combinatorially isomorphic to a subcomplex of a rectilinear triangulation of Euclidean space of the same dimension.) There are topological manifolds admitting no triangulation, homogeneous or not; see [15, 22].

38

4 The Whitehead Theorem and the Hurewicz Theorem

4.6 Homotopy Theory of a Fibration Let  W E ! B be a fibration (i.e.,   has the homotopy lifting property). Let ”W I ! B be a path from b0 to b1 . We have a commutative diagram 

 1 .b0 /  f0g    .b0 /  I 1

/ E

p2

/ I

”

 / B

Applying homotopy lifting gives a map  1 .b0 /  I ! E covering ” ı p2 . It is easy to show that  1 .b0 /  f1g !  1 .b1 / is a homotopy equivalence. Thus, we see that if B is path connected, then all fibers are homotopy equivalent. If ” is a loop based at b0 , then the resulting homotopy equivalence  1 .b0 /  f1g !  1 .b0 / is a homotopy automorphism of  1 .b0 /. Its homotopy class depends only on the class of ” in  1 .B; b0 /. Thus, we have a representation  1 .B; b0 / ! Auto. 1 .b0 // where Auto . 1 .b0 // is the group of homotopy classes of homotopy equivalences of  1 .b0 /. This representation is the action of  1 .B; b0 / as homotopy classes of homotopy equivalences on the fiber. There are induced actions on the homology and cohomology of the fiber. Theorem 4.11. Let  W E ! B be a fibration, and let F D  1 .b0 /. There is an exact sequence: i

@

 

!  nC1 .B; b0 / !  n .F; e0 / !  n .E; e0 / !  n .B; b0 / ! where iW F ,! E is the inclusion. Proof. We have the long exact sequence of the pair F ,! E. @

: : : !  nC1 .E; F/ !  n .F/ !  n .E/ !  n .E; F/ ! : : : Also, we have   W  nC1 .E; F/ !  nC1 .B; b0 /. We claim that this map is an isomorphism. Once we know this, the exact sequence as claimed is derived immediately from the above exact sequence of the pair. We define an inverse to   W  nC1 .E; F/ !  nC1 .B; b0 /. Given fW .In ; @In / ! .B; b0 /, there is a lifting E ?         f / B In g

4.7 Applications of the Exact Homotopy Sequence

39

The reason is that, since In is contractible, the map f is homotopic to the constant map In ! b0 . (The homotopy must in general deform @In off of b0 .) By the homotopy lifting property, any map, homotopic to a map which lifts, itself lifts. Clearly, gW .In ; @In / ! .E; F/ determines an element in  n .E; F/ which projects via   to f. This proves that   is onto. To prove that it is one-to-one, we show that if g0 ; g1 W .In ; @In / ! .E; F/ are such that   ı g0 is homotopic to   ı g1 as maps (In ; @In / ! .B; b0 /, then g0 is homotopic to g1 . By homotopy lifting, we can assume that   ı g0 D   ı g1 . The result now follows from the next lemma by letting X D In1 .  Q 2 W X  I ! E which Q 1 and H Lemma 4.12. Given HW X  I ! B and two liftings H Q 1 and H Q 2 all of agree in X  f0g, there is a homotopy of lifting of H connecting H which agree on X  f0g. Q  f0g. We have a commutative diagram Proof. Let hQ D HjX

where the map JW XII ! B is projection onto XI followed by H. Since XII Q 1 [ hQ  I [ H Q2 deforms onto X  f0g  I [ X  I  f0g [ X  f1g  I, we can extend H to a lifting of J. 

4.7 Applications of the Exact Homotopy Sequence 1.  2 .CPn / Š Z and  i .CPn / D  i .S2nC1 / for i ¤ 2. This follows immediately from the homotopy long exact sequence of the Hopf fibration S1 ! S2nC1 ! CPn and the fact that  i .S1 / D 0 for i > 1. [To calculate the higher homotopy groups exp of S1 , recall that R1 ! S1 is the universal cover. In general, the unique path Q ! X implies that  i .X/ Q Š  i .X/ if i > 1 if X Q is a covering lifting property of X 1 1 1 space of X. Since R is contractible,  i .S / Š  i .R / D 0 for i > 1.]

40

4 The Whitehead Theorem and the Hurewicz Theorem

2.  3 .S2 / Š Z. This is the special case of n D 1 in the above example. 3. Let B be the loop space on B. It is the fiber of PB ! B where PB is the path space of B. Since PB is contractible, this gives  i1 . B/ Š  i .B/.

Chapter 5

Spectral Sequence of a Fibration

5.1 Introduction We begin with a fibration F ! E ? ?  y B In Chap. 4, we saw that the homotopy groups of F,E,B are related by an exact sequence: @

: : : !  n .F/ !  n .E/ !  n .B/ !  n1 .F/ ! : : : Our goal now is to understand how the cohomologies of F, E, B are related. It is to be expected that the relationship will be somewhat complicated, because even in the case of a product E D F  B, the Kunneth theorem giving H .E/ in terms of H .F/ and H .B/. H .F  BI Q/ D H .FI Q/ ˝ H .BI Q/: is more complicated than the simple formula  n .F  B/ Š  n .F/ ˚  n .B/ for the homotopy groups.

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__5, © Springer Science+Business Media New York 2013

41

42

5 Spectral Sequence of a Fibration

5.2 Fibrations over a Cell The total space of a fibration over a CW complex is filtered by the increasing sequence of subspaces lying over the various skeleta of the base. To calculate the cohomology of a pair of spaces, one has a long exact sequence relating the cohomology of the subspace, the cohomology of the pair, and the cohomology of the total space. For a more general filtration, there is a much more complicated algebraic formalism—a spectral sequence—which relates the cohomology of the successive pairs in the filtration and the cohomology of the space. Suppose that B is a connected CW complex with p-skeleton B.p/ , and let E.p/ D 1   .B.p/ /. Let b0 2 B be the base point and F D  1 .b0 /. Proposition 5.1. If B is path connected and if  1 .B; b0 / acts trivially on H .F/, then there are isomorphisms: Hn .E.p/ ; E.p1/ / Š

Y

Hn . 1 .ep /;  1 .@ep //

fpcells in Bg

Š Cp .BI Hnp .F//: Proof. Consider a

®

ep ! B

fpcells in Bg

to be the map of all p-cells into B. Form ® E ! qep the induced fibration.Since .qep ; q@ep / ! .B.p/ ; B.p1/ / is a relative homeomorphism, so is .® E; ® E q @ep / ! .E.p/ ; E.p1/ /. Thus, by excision, H .E.p/ ; E.p1/ / Š H .® E; ® Ej@ep / Y D H .® Ejep ; ® Ej@ep /: pcells

 1  

Consider a fibration E ! ep . Let F0 D  1 .0/. We have a diagram F0  f0g _  F0  eP



/ E1   p2

 / ep

5.3 Generalities on Spectral Sequences

43

Since ep is contractible, the map p2 lifts to a map pQ W F0  ep ! E1 , extending the inclusion of F0 into E1 . This gives a fiberwise map pQ W .F0  ep ; F0  @ep / ! .E1 ; E1 j@ep /. Comparing the homotopy long exact sequences, we see that pQ induces an isomorphism on the relative homotopy groups. Thus, it is a homotopy equivalence of pairs. Thus, in our case, we have H .® Ejep ; ® Ej@ ep / Š H .F0 / ˝ H .ep ; @ep /: If  1 .B; b0 / acts trivially on H .F/, then choosing a path from b0 to 0 2 ep  B.p/ gives an identification of H .F/ with H .F0 / which is independent of the path. Thus, we can identify H .F0 / ˝ H .ep ; @ep / with H .F/ ˝ H .ep ; @ep /. This gives H .E.p/ ; E.p1/ / Š

Y

H .ep ; @ep / ˝ H .F/

pcells

Š Cp .BI H .F//: Returning now to our proposition, we may think of H .E.p/ ; E.p1/ / as a first approximation to H .E.p/ /, of H .E.p/ I E.p2/ / as a second and somewhat better approximation, of H .E.p/ I E.p3/ / as a third approximation, etc. It is pretty clear that the successive approximations are related by exact sequences in which the missing terms are given by the previous proposition. Thus, we can expect to get a whole chain of exact sequences, each related to the previous one and each one giving us better and better approximation to H .E/. The algebra involved is formalized in the following.

5.3 Generalities on Spectral Sequences Definition. A (first quadrant) spectral sequence consists of abelian groups Er D p;q p;q pCr;qrC1 fEr gp;q0 .r  0/ and maps dr W Er ! Er with d2r D 0, such that the homology H.Er ; dr / D ErC1 . In detail, p;q

p;q

ErC1 D

pCr;qrC1

ker dr W Er ! Er

pr;qCr1

im dr W Er

p;q

! Er

:

Remark. The most important thing about spectral sequences is to have in mind the picture: For each term Er , we plot the first quadrant in the (p, q) plane and put in p;q the group Er at the lattice point (p, q). Then the differentials map according to the picture.

44

5 Spectral Sequence of a Fibration

.p; q/

MM MMM MMM MMM MMM MMM MMM .p C r; q  r C 1/ MMM dr MMM MMM MMM &

p;q

An element ’ 2 Er is said to live to infinity if dr ’ D 0 (and thus ’ defines an p;q p;q element in ErC1 /, drC1 ’ D 0; : : :, etc. An element “ 2 Er is said to be killed ps;qCs1 if dr “ D : : : D ds1 “ D 0, but “ D ds ” for some ” 2 E . The spectral sequence is said to degenerate at Er if Er Š ErC1 Š : : : etc.; it is said to be degenerate if E2 Š E3 Š E4 Š    Spectral sequences are a necessary and useful tool and should not be either over- or underestimated. As with evaluation of calculus integrals, they are best learned by practice. As an exercise in understanding pictures, we prove the following lemma: p;q

Lemma 5.2. In a first quadrant spectral sequence if N > p; q C 1, then EN Š p;q ENC1 . Proof. In the picture, we find

p;q

pN;qCN1

so that dN  0 on EN and EN p;q

p;q

D 0 so that the image of dN in EN D 0. 

p;q

p;q

Definition. We set E1 D EN for N > p; q C 1 and En1 D ˚pCqDn E1 . As another exercise in the definitions, we have the following: p;q

Lemma 5.3. Given a first quadrant spectral sequence fEr g, we have 1;0 E1;0 2 D E1 :

5.4 The Leray–Serre Spectral Sequence of a Fibration

45

We also have an exact sequence d2

2;0 2;0 f0g ! E0;1 2 ! E2 ! E1 ! f0g:

Proof. The picture is



5.4 The Leray–Serre Spectral Sequence of a Fibration The main result concerning spectral sequences which interests us is the following: Theorem 5.4 (Leray–Serre). Let F ! E ! B be a fibration in which B is a connected CW complex with  1 .B/acting trivially on the cohomology of the fiber. p;q Then there is a spectral sequence fEr g in which E1 D Cp .B; Hq .F// and d1 D coboundary map of C .B; Hq .F//I p;q p;q

E2 D Hp .B; Hq .F//I and p;q

EN D

kerfHpCq .E/ ! HpCq .E.p1/ /g ; N > p; q C 1: kerfHpCq .E/ ! HpCq .E.p/ /g

Remarks. If we define F p HpCq .E/ D kerfHpCq .E/ ! HpCq .E.p1/ /g then clearly Hn .E/ D F 0 Hn .E/ F 1 Hn .E/ : : : F n Hn .E/ F nC1 Hn .E/ D 0; so that the groups F p Hn .E/ give a decreasing filtration on Hn .E/ whose associated p;q graded abelian group is ˚pCqDn E1 . This is usually written as pCq Ep;q .E/ r H) H

and one says that the spectral sequence converges (or abuts) to H .E/. Proof. Let Sing .E/ be the singular cochains of E. We define a filtration Fp .Sing .E// D kerfSing .E/ ! Sing .E.p1/ /g:

46

5 Spectral Sequence of a Fibration

It is clear that •Fp  Fp . Thus, Sing .E/ is a cochain complex with a decreasing filtration preserved by •. Under this circumstance, it is a purely algebraic result that there is a spectral sequence which converges to the cohomology on Sing .E/. The terms are p;q p;q Ep;q r D Zr =Br p pCq Zp;q .E// such that r D fcochains in F .Sing

• 2 FpCr .SingpCqC1 .E//g p;q pC1 Bp;q C •FprC1 .SingpCq1 .E // \ Fp g: r D fZr \ F

It is a computational result that • induces a differential, denoted dr on Er , and p;q that H .Er / (computed using the differential dr ) is ErC1 . By definition E0 is pCq pCq pCq Fp .Sing .E//=FpC1 .Sing .E// Š Sing .E.p/ ; E.p1/ /. The differential d0 p;q is the usual singular coboundary map on the relative cochains. Thus, E1 D p;q HpCq .E.p/ ; E.p1/ /. Proposition 5.1 gives E1 Š Cp .BI Hq .F//. It is easy to see p;q that under this identification, d1 becomes •B ˝ 1. Thus, E2 D Hp .BI Hq .F//. This completes the proof of the theorem.  There is another version of the Leray–Serre spectral sequence for C1 -manifolds. Let  W E ! B be a C1 map between C1 manifolds which is locally (in B) differentiably equivalent to a projection FU ! U. (We call such a smoothly locally trivial fibration.) We define a filtration on the C1 -differential forms of E, denoted

 .E/. The condition that ¨ be in Fi .  .E// is a pointwise condition which is required to hold for each point. A form ¨` is contained in Fi if and only if for every p 2 E and every set of tangent vectors £1 .p; : : : ; £` .p we have < ¨` .p/; £1 .p/ ^ : : : ^ £` .p/ >D 0 whenever `  i C 1 of the tangent vectors £j .p/ are vertical, i.e., are in the kernel of D p . Explanation: ¨` is an `-form, and hence, ¨` .p/ 2 ^` TEp . Thus, if £1 .p/; : : : ; £` .p/ are tangent vectors to E at p, then < ¨` .p/; .£1 .p/ ^ : : : ^ £` .p// > is a real number. Claim. p;q

E0 D

Fp . pCq .E// FpC1 . pCq .E//

is identified with C1 p-forms on B with values in C1 q-forms on the fibers. Proof of Claim. Let ¨pCq 2 Fp . pCq .E//. Let £1 .b/; : : : ; £p .b/ be tangent vectors to B at b D  .p/. Let £Q i be a tangent vector to E at p which projects onto £i .b/. Let £Q pC1 ; : : : ; £Q pCq be tangent vectors to  1 .p/ D Fp  E. Then < ¨pCq .p/; £Q 1 ^

5.4 The Leray–Serre Spectral Sequence of a Fibration

47

: : : ^ £Q pCq > is independent of the liftings £Q 1 ; : : : £Q p of the tangent vectors in B. This defines Fp . pCq / to the space of p-forms on B with values in q-forms on the fibers. Clearly, this map factors through Fp . pCq /=FpC1 . pCq /. One checks easily that it induces an isomorphism Fp . pCq /=FpC1 . pCq / Š (p-forms on B with values in q-forms on the fibers: The differential d0 becomes differentiation in the fiber direction under this identification. Hence, p;q

E1 D p .BI Hq .F//I p;q

i.e., E1 is p-forms on B with coefficients in the vector bundle of qth deRham cohomology along the fibers. The vector bundle of deRham cohomology along the fiber has a natural product structured locally (since deRham cohomology is identified with singular cohomology with real coefficients which is locally constant). That is to say the bundle of deRham cohomology along the fibers has a natural flat connection (called the Gauss–Manin connection). The condition that the fundamental group of the base acts trivially on the cohomology of the fiber is the condition that this connection is trivial, so that this bundle of deRham cohomology is identified with the constant coefficients of deRham cohomology of the fiber over the base point. One then shows that d1 becomes dB ˝ 1, i.e., differentiation in the base direction. Consequently, p;q

E2 D Hp .BI Hq .F//

.deRham cohomology/:

This is the C1 -version, or deRham version, of the Leray–Serre spectral sequence. The last description we give of this spectral sequence is the one originally given by Serre in [21]. One considers the singular cubical cochains Q .E/. The nth chain group is defined by taking the free abelian group generated by maps of the n-cube, In into E and setting equal to 0 the degenerate cubes, i.e., those that factor through In ! In1 defined by .t1 ; : : : ; tn / 7! .t1 ; : : : ; tn1 /: The cochains are dual to these chain groups. The boundary map is defined by restricting to the various codimension-1 faces with signs. These again calculate the usual singular cohomology. We say that ’ 2 Qn .E/ is in filtration level i; ’ 2 Fi , if ’ vanishes when evaluated on any cubical chain ®W In ! E which has the property that   ı ®W In ! B factors through the projection In ! Ii1 ; .t1 ; : : : ; tn / ! .t1 ; : : : ; ti1 /. Assuming that the fundamental group of B acts trivially on the cohomology of the p;q fiber E1 D Qp .B/ ˝ Hq .F/, d1 is identified with the usual coboundary map on Q . Since the non-degenerate cubical cochain complex calculates singular cohomology, p;q one finds that E2 D Hp .BI Hq .F// (with Z-coefficients). 

48

5 Spectral Sequence of a Fibration

Remarks. (i) There is also a homology spectral sequence for a fibration, fErp;q g, with E2p;q D Hp .BI Hq .F//. Over Q the two spectral sequences are dual in the obvious sense. (ii) If  1 .B; b0 / acts nontrivially on H .F/, then there is still a spectral sequence. p;q In this case E2 is Hp .BI Hq .F// where the coefficients are the locally trivial, but globally nontrivial coefficient system on the base which is given by the flat connection determined by the action of the fundamental group of the base on the cohomology of the fiber. (iii) There is a ring structure on the terms in the cohomology spectral sequence such that the differentials di are derivations. In particular, if we use, say Q-coefficients, and assume the action of the fundamental group of the base on the cohomology of the fiber is trivial, then p;q

E2 Š Hp .B/ ˝ Hq .F/ p;0

0;q

Š E2 ˝ E2 and .p;q/

d2

.p;0/

D d2

.0;q/

˝ Id C .1/p Id ˝ d2 .p;0/

since d2

.0;q/

D .1/p Id ˝ d2

D 0:

This is proved by using the usual cup product on cochains and being careful about the formulae; for more details, cf. [23] pp. 490–498. If one is willing to work over R, then this multiplicative property is easily seen using the differential form version of the spectral sequence discussed above. It should be remarked that the most sophisticated computations involving spectral sequences seem to use the ring structure and derivation property.

5.5 Examples 1. Complex Projective Space. We shall compute the cohomology ring of CPn D Pn using the Hopf fibration S1 ! S2nC1 ! Pn . Using Z-coefficients, the E2 term is H .S1 / ˝ H .CPn / and E1 ) p;q  2nC1 H .S /. Since Hq .S1 / D 0 for q > 1; E2 D 0 for q > 1 and so d3 D d4 D : : : D 0 and E3 Š E4 Š : : : Š E1 . Thus, all the “action” in p;q p;q the spectral sequence occurs at the E2 level, and in particular E1 D E3 D 0 unless p C q D 0; 2n C 1. The picture of E2 is

5.5 Examples

49

1 1 where ’ 2 E0;1 2 Š H .S / is a generator. 1;0 1;0 To begin with, E2 D 0 since E1;0 1 Š E2 . Next the map d2

2;0 E0;1 2 ! E2 p;q

is an isomorphism since E3 D 0 for 1  p C q  2. Set “ D d2 ’ so that E2;1 Š Z.’ ˝ “/. Then d2 .’ ˝ “/ D “2 2 E4;0 Š H4 .Pn / since d2 is a 2 2 derivation. Continuing in this way, we see that H2q .Pn / D ZŒ“q

.q  n/

H2qC1 .Pn / D 0 So that H .Pn / Š ZŒ“ =.“nC1 /; i.e., H .Pn / it is a truncated polynomial algebra. 2. Rational Cohomology of K.Z; n/ We shall show in Chap. 7 that there is a space, unique up to homotopy equivalence, which has only one nonzero homotopy group, that being  n , and that group is isomorphic to Z. Such a space is denoted K.Z; n/. We shall prove the fundamental result here that H .K.Z; 2k/I Q/ Š QŒ’ ; and H .K.Z; 2k C 1/I Q/ Š Q.“/: (The first is a polynomial algebra and the second is an exterior algebra.) The proof of these results is by induction on n. For n D 1, K.Z; 1/ Š S1 and the result is immediate. Consider the inductive step from .2k  1/ to 2k. We study the path fibration over K.Z; 2k/: K.Z; 2k  1/ ! P.K.Z; 2k// ? ? y K.Z; 2k/ p;q

Since P is contractible, the fiber is K.Z; 2k  1/ and E1 D 0 if .p; q/ ¤ .0; 0/. Invoking the inductive hypothesis, we see that the E2 -term looks like

50

5 Spectral Sequence of a Fibration

H .K.Z; 2k/I Q/

. ; 2k  1/

    H .K.Z; 2k/I Q/

. ; 0/

Thus, E2 D E3 D : : : D E2k1 and E2k D E1 . First note that ( H .K.Z; 2k/I Q/ D i

0;

i < 2k

Q;

i D 2k

:

(This is a consequence also of the Hurewicz theorem and the universal coefficient theorem). Let “ be a generator of H2k1 .K.Z; 2k  1/I Q/, and let ’ D d2k .“/. Then ’ is a generator of H2k .K.Z; 2k/I Q/. Suppose inductively that we have shown that QŒ’ ! H .K.Z; 2k/I Q/ is an isomorphism for all degrees  t.2k/. Then we have

E2k D

0

0

0

0

0

0:::

0

“

0

’˝“

0

’2 ˝ “

0:::

’t ˝ “

0

0

0

0

0

0:::

0

0

0

0

0

0

0:::

0

0

0

0

0

0

0:::

0

1

0

’

0

’2

0:::

’t

p;q

It results from the fact that E2kC1 D 0 for all .p; q/ ¤ .0; 0/, that Hi .K.Z; 2k/I Q/ D 0 for 2kt < i < 2k.t C 1/, and that H2k.tC1/ .K.Z; 2k/I Q/ is isomorphic to Q and generated by d2k .“ ˝ ’t /. By the derivation property, d2k .“ ˝ ’t / D ’ ˝ ’t D ’tC1 . This completes the induction step .2k  1/ ! 2k. For the inductive step 2k ! 2k C 1, we use the following path fibration: K.Z; 2k/ ! P.K.Z; 2k C 1// ? ? y K.Z; 2k C 1/:

5.5 Examples

51

Here E2 is

An argument similar to the one in the other case shows that 8 < E2 D : : : D E2kC1 E D : : : D E1 : 2kC2 p;q E1 D 0 for p C q > 0: d2kC1

2kC1;0 It follows that E0;2k 2kC1 ! E2kC1 is an isomorphism, and so we set “ D d2kC1 ’. By the derivation property, d2kC1 .’2 / D 2 ’ ˝ d2kC1 ’ D 2 ’ ˝ “, and so d2kC1

2kC1;2k E0;4k 2kC1 ! E2kC1

is an isomorphism (this is where we need Q-coefficients). This gives that H4kC2 .K.Z; 2k C 1/; Q/ D 0, and continuing on we obtain the desired result. 3. Grassmannians. Let G.k; N/ be the Grassmann manifold of complex k-planes in CN and G.k/ D limN!1 G.k; N/ the infinite Grassmanian. Using Z coefficients, we want to prove that Hp .G.k// Š Z Œc1 ; c2 ; : : : ; ck ; cj 2 H2j .G.k//: When k D 1, we have G.l/ D CP1 D K.Z; 2/ and we have proved the result in this case. Assuming the result for k  1, we consider the two fibrations S2k1

G.k  1/ o

 F o  G.k/

S1

52

5 Spectral Sequence of a Fibration

where F D f.v;  / j   is a k-plane in C1 and v 2   is a unit vector}. The map F ! G.k1/ is given by sending .v;   / to the .k1/ plane in   perpendicular to v. Clearly, F ! G.k  l/ is a fibration with fiber S1 which is contractible. Hence, F Š G.k  1/. The fibration F ! G.k/ is given by sending .v;   / to  . The fiber here is p;q S2k1 . Thus, we have a spectral sequence such that E2 D Hp .G.k/I Hq .S2k1 // and  E1 converges to H .G.k  1//. It must be the case that E2 D E3 D : : : D E2k1 and that E2k D E1 . Let ck be d2k .’/ where ’ is the generator of H2k1 .S2k1 /. From the spectral sequence, we have exact sequences S

ck

HiC2k1 .G.k  1// ! Hi .G.k// ! HiC2k .G.k// ! HiC2k .G.k  1//: By induction H .G.k  1// is a polynomial algebra generated by c1 ; : : : ; ck1 . Also the inclusion G.k  1/ ,! G.k/ induces an isomorphism on Hi for i < 2k  1. This implies that the classes c1 ; : : : ; ck1 lift to H .G.k//. It follows that H .G.k// ! H .G.k  1// is onto. Consequently, the above sequence becomes S

ck

0 ! Hi .G.K// ! HiC2k .G.k// ! HiC2k .G.k  1// ! 0: Using this one proves easily that H .G.k// is a polynomial algebra on c1 ; : : : ; ck .

Chapter 6

Obstruction Theory

6.1 Introduction Recall, in the proof of the Whitehead theorem, we showed that if X is a CW complex and Y is a space with  i .Y/ D 0 for all i  0, then any map fW X ! Y is homotopic to the base point map X ! y0 2 Y. The proof was by induction over the skeleta of X. Obstruction theory is a generalization of this technique to the case when the homotopy groups of Y are not necessarily zero. It does not give a complete understanding of when a map fW X ! Y is homotopic to a constant, or more generally when two maps f1 ; f2 W X ! Y are homotopic, but it gives some insight. Obstruction theory not only deals with the uniqueness question “when are two maps f1 ; f2 W X ! Y homotopic?”; it also deals with the existence question of constructing maps from X to Y. In both cases, we work inductively over the skeleta. In the first, we suppose that we have a homotopy h from f1 jX.n/ to f2 jX.n/ hW X.n/  I ! Y and we ask if it extends to X.nC1/  I ! Y. In the second, we suppose that we have a map fW X.n/ ! Y, and we ask if it extends to a map OfW X.nC1/ ! Y. We will deal with the second case first. Lemma 6.1. Suppose that Y is simply connected. 1. There is a natural bijection  n .Y/

! ŒSn ; Y ;

where ŒSn ; Y means the free homotopy classes of maps of Sn into Y (with no restrictions on the image of base point).

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__6, © Springer Science+Business Media New York 2013

53

54

6 Obstruction Theory

2. Let U be the complement in SnC1 of the interiors of a finite disjoint collection of nC1 disks [i DnC1 , and denote the boundary by Sni . If a finite collection of i P of Di n maps fi W Si ! Y extends over U, then i Œfi D 0 in  n .Y/. Proof. We consider the first statement. There is always a function  n .Y/ ! ŒSn ; Y obtained by ignoring the base points. If  0 .Y/ D 0, then it is onto. This follows by applying the homotopy extension property (h. e. p.) to Sn  f0g [  I  Sn  I . is the base point of Sn ). If in addition  1 .Y/ D 0, then the map is 1–1. This follows by applying h. e. p. to .Sn  f0 [ lg [ I/  I in .Sn  I/  I. For the second statement, let FW U ! Y be an extension of the fi . There is a finite disjoint union of arcs ’j in U, with end points in the boundary of U such that the complement in U of an open regular neighborhood of the union of these arcs is an .n C 1/-disk D0 . We deform F so that the closure P of the neighborhood of these arcs maps to the base point of Y. Then we see that i Œfi D ŒFj@D0 in  n .Y/. The latter is obviously the trivial element. 

6.2 Definition and Properties of the Obstruction Cocycle We assume for the rest of this chapter that Y is simply connected. Let .X; A/ be a CW-pair. Suppose given fn W X.n/ [ A ! Y. We define the Q n / 2 CnC1 .X; AI  n .Y// as follows: If enC1 obstruction cochain O.f is an oriented ’ (nCl)-cell of .X; A/, then its attaching map c’ W Sn ! X.n/ [A composed with f gives f ı c’ W Sn ! Y, which determines an element of  n .Y/. If we reverse the orientation on enC1 (and hence on @enC1 ’ ’ ), then the resulting element in  n .Y/ changes sign. Thus, there is a well-defined homomorphism CnC1 .X; A/ !  n .Y/. We denote it Q n / and call it the obstruction cochain. by O.f Since Y is simply connected, the obstruction cochain groups are trivial for n D 0; 1. From now on we fix orientations for each cell of .X; A/ and use the symbol for the cell to represent this oriented cell. Q n / satisfies the following properties: Lemma 6.2. The obstruction cocycle O.f It is an invariant of the homotopy class of fn . It is 0 if, and only if, fn extends to a map fnC1 W X.nC1/ [ A ! Y. Q n /W CnC2 .X; A/ !  n .Y / is 0. It is a cocycle; i.e., ı O.f .n/ Q n /  O.f Q n / is a If gn W X [ A ! Y agrees with fn on X.n1/ [ A, then O.g coboundary. Q n/ (5) By varying the homotopy class of fn relative to X.n1/ [A, we can change O.f by an arbitrary coboundary.

(1) (2) (3) (4)

6.2 Definition and Properties of the Obstruction Cocycle

55

Proof. (1) and (2) are immediate from the definitions. Before proving (3), (4), and (5), we review a little of the terminology associated with cohomology and then prove a necessary lemma.  Let fC ; @g be a chain complex. Its homology is defined by Hn .C/ D

ker @n cycles : D Im @nC1 boundaries

If G is an arbitrary abelian group, then we define a cochain complex with coefficients in G by :::

 Hom.Cn ; G/

@Dı

 Hom.Cn1 ; G/

 ::::

Clearly, ı ı ı D 0. We define Hn .CI G/ D

kerın cocycles : D Imın1 coboundaries

From this description, it is easy to see that if n is an n-cochain and £nC1 is an .n C 1/ chain, then < ın ; £nC1 >D< n ; @£nC1 >. Thus, n is a cocycle if, and only if, it annihilates all boundaries. Q n // D 0. Proof of (3). ı.O.f Q n /; @”nC2 >D 0 for all .n C 2/To prove this,Pwe need to show that < O.f cells ”nC2 . Let ’ a’ Œe’ be the boundary of ”nC2 in the CW chain complex. Then the attaching map for ”, @”nC2 W SnC1 ! X.nC1/ [ A is homotopic to a map SnC1 ! X.nC1/ [ A with the property that the preimage of the interior of each e’ is a disjoint union of disks mapping homeomorphically onto the interior of the cell and the P sum of the orientation signs of these maps is a’ . It follows from Lemma 6.1 that ’ a’ Œ@ e’ D 0 in  nC1 .Y/. Thus, Q n /; @”nC2 > D < O.f Q n /; †’ a’ Œe’ ; @e’ > < O.f Q n /; Œe’ ; @e’ > D †’ a’ < O.f D †’ a’ Œfn ı @e’ D .fn / †’ a’ Œ@e’ D .fn / .0/ D 0:  Proof of (4). For each n-cell in X n A we have form the difference element “fn  gn ”W Sn ! Y. en’

fn j@en’

D

gn j@en’ .

Thus, we can

56

6 Obstruction Theory

This defines a cochain “fn  gn ”W Cn .X; A/ !  n .Y/.



Q n /  O.g Q n /. claim. ı.“fn  gn ”/ D O.f be an (n+l)-cell of .X; A/ with @enC1 D †a’“ en“ in Cn .X; A/. As Proof. Let enC1 ’ ’ nC1 n .n/ before we can deform @e’ W S ! X [A by a homotopy to a map b’ with the property that the preimage of each n-cell, e“ under b’ is a finite number of n-disks each mapping homeomorphically to e“ and counted with signs coming from the orientations the number is a’“ . The composites f ı b“ and g ı b“ agree on the complement of these disks. The “difference” element on each disk above e“ is ˙“fn  gn ”.e“ /, where, as before the sign is determined by the orientations. It is now an easy extension of the argument in Lemma 6.1 to show that Œf ı @e’  Œg ı @e’ D

X

a’“ “fn  gn ”.e“ /:

“

Thus, we have Q n /; @enC1 Q n /; @enC1 < O.f >  < O.g > D fn ı @enC1  gn ı @enC1 ’ ’ ’ ’ D †a’“ < “fn  gn ”; en“ > > D < .“fn  gn ”/; @enC1 ’ D < ı.“fn  gn ”/; enC1 >: ’  Proof of (5). Let en0 be an n-cell of.X; A/ and let g be an element of  n .Y/. Define a cochain 2 Cn .X; AI  n .Y/ by < ; en0 >D g and < ; en >D 0 for all other nQ n /  O.f Q 0n / D ı. /. Since cells en . We shall show how to change fn to f0n so that O.f these cochains generate the entire cochain group, this will suffice to establish 5. Choose a small ball Bn  int en0 By deforming fn by a homotopy, we can suppose that fn .Bn / D y0 Y. Define f0n to agree with fn on X.n/ -int Bn , and define f0n W .Bn ; @Bn / ! .Y; y0 / to represent g 2  n .Y; y0 /. Clearly, “f0n  fn ” D . Hence, by the previous Q 0n /  O.f Q n / D ı . result O.f  Let us consolidate our gains to this point. Theorem 6.3. Given fn W X.n/ [ A ! Y with Y simply connected, there is a Q n /. cohomology class O.fn / 2 HnC1 .X; AI  n .Y// constructed from the cocycle O.f This class vanishes if and only if fn jX.n1/ [ A can be extended to a map fW X.nC1/ [ A ! Y.

6.3 Further Properties

57

Let f; gW X ! Y be given and let HW .X.n/ [ A/  I ! Y be a homotopy between the restrictions of f and g to X.n/ [ A. The obstruction to extending the homotopy over .X.nC1/ [ A/  I lies in HnC1 .X  I; .X  .f 0g [ f1g// [ A  I/I  n .Y//: By the suspension isomorphism, this group is Hn .X; AI  n .Y//: Thus, the obstructions to constructing a homotopy between two maps fW X ! Y and gW X ! Y, given a fixed homotopy on A, lie in Hn .X; AI  n .Y//.

6.3 Further Properties As long as Hi .X;  i .Y// is 0, we have no obstructions to finding a homotopy between f and g both restricted to X.i/ . Suppose that Hn .X;  n .Y// is the first nonzero group. It is an exercise (using the theorem one step at a time over the skeleta X.i/ ; i < n/ to show that, given f and g W X ! Y, the first obstruction to finding a homotopy between them, O 2 Hn .X;  n .Y//, is well defined; i.e., does not depend on the step-by-step homotopy constructed from fjX.n1/ to gjX.n1/ this exhibits a universal phenomenon: The first obstruction lying in a nonzero group is always well defined. The higher obstructions are not defined until we make choices and then depend on these choices. ® As an unproven example, take CP2 ! S2 where ®jCP1 D ( D base P point in S2 ) ˜

˜

and ® on the 4-cell is given by a nonzero element in  4 .S2 /; namely, S4 ! S3 ! S2 where †˜ is the suspension of the Hopf map ˜. (We will do exercises of this type at a later time.) Take the point homotopy of ®jCP1 to . Then the obstruction to extending this to a homotopy of ® to is in H4 .CP 2 ;  4 .S2 // Š  4 .S2 / and is the nonzero element . 4 .S2 / Š Z2 /. But ® is homotopic to . (Notice that H4 .CP2 ;  4 .S2 // is not the first nonzero obstruction group since H2 .CP2 ;  2 .S2 // is nonzero.) Thus, obstruction theory is not the be all and end all of homotopy theory. It was once described by Sullivan as much like being in a labyrinth with a weak miner’s light attached to your forehead and being forced always to move forward. The light enables you to see if you may take your next step but is not strong enough to tell you which fork to take when you must make a decision. Also there is no guarantee that if you choose one path that eventually is blocked, then all of them are blocked. On the other hand, if you were a miner would you care to be without your light? Likewise, no topologist would forsake obstruction theory.

58

6 Obstruction Theory

6.4 Obstruction to the Existence of a Section of a Fibration There is another type of obstruction theory which we shall need later on. This is for the problem of constructing a section of a fibration . Let pW E ! B be a fibration with B path connected and with F D p1 .b0 /. A section W B ! E is a map for which p ı  D IdB . We assume that B is a CW complex, hat  1 .B/ acts trivially on F, and that  1 .F/ D 0. The obstructions to constructing a section lie in Hi .BI  i1 .F//. Given two sections the obstructions to constructing a homotopy of sections connecting them lie in Hi .BI  i1 .F//. Let us give the definition of the obstruction cocycle. Suppose W B.n1/ ! E is a section over the .n  1/-skeleton. For each n-cell of B, we have a trivialization of p1 .en / ! en as en  F, as we saw in Chap. 5. To construct such a trivialization, one must connect en to b by a path in B. Since  1 .B/ acts trivially on the fiber, the identification is well defined, up to homotopy, independently of the path chosen. A section  on B.n1/ induces a section of p1 .en / ! en , denoted ¢jen . Using the above product structure, this gives Q W en ! en F . Projecting onto the second factor yields an element in  n1 .F/. The obstruction cocycle OQ n ./W Cn .B/ !  n1 .F/ assigns to Œen , the resulting homotopy element. Arguments similar to the ones above show that OQ n ./ is a cocycle and that one may vary OQ n ./ by an arbitrary coboundary by changing  on B.n1/ B.n2/ . Thus, the class On ./ 2 Hn .BI  n1 .F// is the obstruction to extending jB.n2/ over B.n/ . If .B; A/ is a CW-pair and we are given a section over B.n/ [ A, then the obstruction to extending its restriction to B.n1/ [ A over BnC1 [ A lies in HnC1 .B; AI  n .F//.

6.5 Examples Let us consider some examples of these obstruction classes. Example A. The Euler class: If En ! B is an n-dimensional vector bundle, then a nowhere zero section of En is the same as a section of the associated sphere bundle Sn1 (E). The first obstruction to finding a section is in Hn .B;  n1 .Sn1 // Š Hn .B; Z/. It is called the Euler class of En and is an unstable characteristic class of the vector bundle, unstable in the sense that taking the connected sum of the bundle with a trivial bundle kills the class. Some of the exercises deal with it in more detail. Notice that this shows that if dim B < n, then En ! B always has a nonzero section, and thus, as a vector bundle En Š En1 ˚ 1 , where 1 is the trivial line bundle. Example B. Eilenberg–MacLane spaces K. ; n/: Given n  2 and   an abelian group, we will show that up to homotopy equivalence, there is exactly one CW complex K. ; n/ such that

6.5 Examples

59

(  i .K .; n// D



if i D n ;

0

otherwise.

Suppose that   is finitely presented 0 ! F.r1 ; : : : ; rl / ! F.’1 ; : : : ’k / !   ! 0 where F.; : : : ; / is the free abelian group generated by the symbols inside the parentheses. Form the space ._kiD1 Sn’i /. Its nth homotopy group is F.’1 ; : : : ; ’k / (here we assume n > 1). Thus, the elements ri are represented by maps ®rj W Sn !     nC1/ where the attaching _ki D1 Sn’i . Form the .n C 1/-complex _kiD1 Sn’i [ [ljD1 erj map for enC1 is ®rj . Call this CW complex X.nC1/ . Clearly  n .X.nC1/ / D   and rj  i .X.nC1/ / D 0 if i < n. Choose a generating set f”s gs2S for  nC1 .X.nC1/ /, and use these elements to attach .n C 2/ -cells. Call the result X.nC2/ One sees that (  i .X

.nC2/

/D

0

if i < n C 2 and i 6D n,

 

if i D n.

Continuing in this manner, we construct X.n/  X.nC1/  X.nC2/  : : : so that  i .X.nCk/ / D 0 for i < n C k and i ¤ n, whereas  n .X.nCk/ / D  . Let X be the union of the X.nCk/ . It has the correct homotopy groups. There is a similar construction even if   is not finitely presented. Now let us show that if Y is a CW complex with the same homotopy groups as X, then Y is homotopy equivalent to X. To establish this, we construct fW X ! Y which induces an isomorphism on the homotopy groups. Begin with a map fn W _kiDl Sn’i ! Y which induces the composition:  n ._Sn’i / D F.’l ; : : : ; ’k / !   D  n .Y/: This map extends over X.nC1/ since ®rj W Sn ! _kiD1 Sn’i represents an element in  n ._Sn’i / which is trivial in   D  n .Y/. Any extension fnC1 W X.nC1/ ! Y induces the identity id

 n .X.nC1/ / D   !   D  n .Y/:

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6 Obstruction Theory

The obstructions to extending fnC1 over all of X lie in H .XI  1 .Y// for  nC2. Hence, all obstructions vanish and fnC1 extends to fW X ! Y. Since f W  n .X/ !  n .Y/ is an isomorphism, by the Whitehead theorem, f is a homotopy equivalence. Remark. K.Z; 1/ D S1 ; K.Z=pZ; 1/; 1/ is the infinite lens space, that is to say the quotient of the unit sphere in C1 by the action of the pth-roots of unity; and K.Z; 2/ D CP1 , but outside of these the K. ; n/ are usually not spaces encountered directly in geometry. In general, the price for having the  i so simple is that the homology Hi .K. ; n/; Z/ is now very complicated (this seems quite plausible from the construction, since we are adding cells in all dimensions according (roughly) to the pattern of  k .Sn /.k > n//. Using the K. ; n/, we can construct Qa space with preassigned homotopy groups f i .X/ D  i g. Simply take X D i K. i ; ni //. Later we will show that any simply connected CW complex is homotopy equivalent to an iterated fibration of the K. ; n/. For the moment we want to record one important fact about obstruction classes: the first possible nonzero class is well defined and natural. We establish this in the two contexts of extending a map and a section. Theorem 6.4. Let .X; A/ be a CW-pair, and let fW A ! Y be given. Suppose Hi .X; AI  i1 .Y// D 0 and Hi1 .X; AI  i1 .Y// D 0 for i  n. Also suppose  1 .Y/ D 0. The first obstruction O 2 HnC1 .X; AI  n .Y// to extending f over X is well defined. It is natural with respect to maps ®W .X0 ; A0 / ! X; A/. Proof. As we extend f over X.i/ [ A, all obstructions lie in the zero group until we get to X.n/ [ A. Let fn and f0n be two extensions over X.n/ [ A. Since Hi .X; AI  i .Y// D 0 for i  n  1; f0n j.X.n/ [ A/ is homotopic relative to A to fn j.X.n/ [ A/. Thus, by 5.2, O.fn / D O.f0n / in HnC1 .X/; AI  n .Y/. This proves that the primary obstruction is well defined.  Now suppose given ®W .X0 ; A0 / ! .X; A/ where H .X0 ; A0 I  1 .Y// and H .X0 ; A0    1.Y// are equal to zero for  n. Then the first obstruction to extending f ı ® W A0 ! Y lies in HnC1 .X0 ; A0 I  n .Y//. As above we see that it is well defined. Naturality means that ® .O.f// D O.f ı ®/ in HnC1 .X0 ; A0 I  n .Y//. To establish this, deform ® to a cellular map, ®0 , relative to A0 . Let fn W X.n/ [ A ! Y .n/ be an extension of fW A ! Y. Then fn ı ®0 W X0 [ A ! Y is an extension of f  ® on A0 . We claim that the cocycle obstruction for fn  ®0 is the pull back via ®0 of the cocycle obstruction for fn . If ®0 .1 / D †i ’1i £i in CnC1 .X; A/, then ®0 .@1 / D †i ’1i @£i in  n .X.n/ [ A/. Thus, 1

< O.fn  ®0 /; i > D fn ı ®0 .@i / D fn .†i ’1i @£i / D †i ’1i fn .@£i /

6.5 Examples

61

D †i ’1i < O.fn /; £i > D < O.fn /; †i ’1i £i > D < O.fn /; ®0 .1 / > D < .®0 / O.fn /; 1 > :



Other Examples f

C. If Sn ! Y, then the obstruction to f being homotopic to zero is in Hn .Sn I  n .Y// D £n .Y/. It is, of course, the homotopy class of f in  n .Y/. D. Let gW .CP2 /.3/ ! S2 be the identity map. The obstruction to extending g over CP2 is in H4 .CP2 I  3 .S2 // D  3 .S2 /. It is the Hopf map; i.e., the attaching map for the 4-cell of CP2 . This is true in general. If X0 D X[f en , then the obstruction to extending Id: X ! X over X0 is Œf 2  n1.X/ . E. Given fW Sk ! Y and gW S` ! Y, form f _ gW Sk _ S` ! Y. The only obstruction to extending f _ g to a map Sk  S` ! Y is an element in HkC` .Sk  S` ; Sk _ S` I  kC`1 .Y/ Š  kC`1 .Y/. Since this obstruction is primary, it is well defined. The obstruction, denoted Œf; g , is the Whitehead product of f and g. There is an analogous theorem for lifting maps in a fibration. Theorem 6.5. Let .X; A/ be a relative CW complex and let  W E ! X be a fibration with fiber F. Suppose that  1 .X/ acts trivially on F and that W A ! EjA is a section of   over A. If Hi .X; AI  i1 .F// D 0 and Hi .X; AI  i .F// D 0 for i  n, then the first obstruction to extending  to a section defined on all of X lies in HnC1 .X; AI  n .F//. It is well defined and natural. The proof is analogous to that of Theorem 6.4 and is left to the reader.

Chapter 7

Eilenberg–MacLane Spaces, Cohomology, and Principal Fibrations

7.1 Relation of Cohomology and Eilenberg–MacLane Spaces In the last section, we saw that there was a natural transformation Œ.X; A/; .K. ; n/; / ! Hn .X; AI  / which assigns to any map fW .X; A/ ! .K. ; n/; /, the primary obstruction to deforming f to a constant map relative to A. (Here, is the base point of K. ; n/:/ Actually, this should be viewed as an extension problem for the map f [ c [ c W X  f0g [ A  I [ X  f1g ! K. ; n/ (where c denotes the constant map to the base point). The primary obstruction is well defined and lies in HnC1 .X  I; X  f0g [ A  I [ X  f1g I  / Š Hn .X; AI  /: By the Hurewicz theorem, Hn .K. ; n/I Z/ D   and Hn1 .K. ; n/I Z/ D 0. Thus, by the universal coefficient theorem, Hn .K. ; n/I  / D Hom. ;  /. (Here, we are viewing K. ; n/ as a space together with an identification of  n .K. ; n// with  .) Let  2 Hn .K. ; n/I  / be the class corresponding to the identity homomorphism   !  . If fW .X; A/ ! .K. ; n/; /, then we have f ./ 2 Hn .X; AI  /. This defines a function Œ.X; A/; .K. ; n/; / ! Hn .X; AI  / f 7! f ./: Theorem 7.1. The class f  is the primary obstruction to deforming f to the constant map relative to A. The association Œf 7! f  is a bijection for all CW-pairs .X; A/.

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Proof. First we show that f  is the first obstruction to deforming f to a constant map relative to A. By naturality it suffices to show that the obstruction for the identity map K. ; n/ ! K. ; n/ is . This is immediate from the definitions. Since K. ; n/ has only one nonzero homotopy group, if f  D 0, then the one and only obstruction to deforming f to a point, relative to A, vanishes. It follows that in that case f is homotopic, rel A, to a constant map. This shows that 1 .0/ is exactly the trivial map. This does not yet suffice to prove that  is 1–1 because we have not defined a group structure on Œ.X; A/; .K. ; n/; / , cf. below. We show that  is onto. Let ’ 2 Hn .X; AI  / be a class, and let ’Q W Cn .X; A/ !   be a cocycle representative for ’. Define f’Q jX.n1/ [ A to be the constant map. For each n-call en define .f’Q jen /W en ! K. ; n/ to be a map representing ’.e Q n/ 2   D .n/  n .K. ; n//. This gives f’Q jX [ A. Since ’Q is a cocycle, if we compose f’Q jX.n/ [ A with the attaching map for an .n C 1/-cell, the result is homotopically trivial. Thus, f’Q jX.n/ [ A extends over X.nC1/ [ A. All higher obstructions to extending f’Q jX.nC1/ [ A vanish. Let f’Q W .X; A/ ! K. ; n/ be an extension of f’Q jX.n/ [ A. Clearly, the relative cocycle which measures the primary obstruction to a homotopy from f’Q to the constant map is exactly ’Q 2 Hom.Cn .X; A/;  /. Thus, f’Q ./ D ’. To see that  is 1–1, we assume that f and g map .X; A/ to .K. ; n/; / and f  D g . The primary (and only) obstruction to a homotopy, relative to A, from f to g is in Hn .X; AI  /. It is easily identified with f ./  g ./ D 0. Thus, f is homotopic to g relative to A.  The Additive Structure: There is a map

K. ; n/  K. ; n/j ! K. ; n/ so that   D  ˝ 1 C 1 ˝  2 Hn .K. ; n/  K. ; n/I  / Š Š ŒHn .K. ; n/I  / ˝ H0 .K. ; n/I Z/ ˚ ŒH0 .K. ; n/I Z/ ˝ Hn .K. ; n/I  / It is well defined up to homotopy. It acts as a commutative, associative group multiplication up to homotopy. Thus, K. ; n/ is a homotopy commutative, associative H-space. This map induces an abelian group structure on Œ.X; A/I .K. ; n/; / . with this group structure  becomes a group isomorphism. (Details of proof left as exercises.)

7.2 Principal K. ; n/-Fibrations A map pW E ! B is said to be a K. ; n/ fibration if it satisfies the homotopy lifting property and if all fibers, p1 .b/, are spaces of type K. ; n/. If B is path connected, then pW E ! B is a K. ; n/-fibration provided that p1 .b/ is a space of type K. ; n) for any b 2 B.

7.2 Principal K. ; n/-Fibrations

65

A K. ; n/ fibration is said to be principal if the action of the fundamental group of the base on the fiber is trivial up to homotopy. For each loop ” in the base B based at b there is a self-homotopy equivalence ” W p1 .b/ ! p1 .b/ well defined up to homotopy. The fibration is principal if all these self-equivalences are homotopic to the identity. Lemma 7.2. Let .X; A/ be a CW pair, let pW E ! X be a principal K. ; n/fibration, and let W A ! E be a section of E over A. There is a unique obstruction O.p; / 2 HnC1 .AI  / to extending  over all of X. Given any class O 2 HnC1 .X; AI  / is realized as the obstruction O.p; / for some principal fibration pW E ! X and for some section W A ! E. Proof. Since  i .fiber/ D 0 for i < n, according to Theorem 6.3, the first obstruction to extending the section lies in HnC1 .X; AI  /. It is well defined and natural. Since all the higher homotopy groups of the fiber vanish, O.p; / is the unique obstruction to extending the section over all of X. Given a class O 2 HnC1 .X; AI  /, there is a map fO W .X; A/ ! .K. ; n C 1/; / so that fO ./ D O. Over K. ; n C 1/ we have the principal fibration K. ; n/ D K. ; n C 1/ ! P.K. ; n C 1// ? ? y K. ; n C 1/ Since K. ; n C 1/ is simply connected, this fibration is principal. If we pull back the fibration by fO , then we get a principal K. ; n/ fibration over X. Any section over pulls back to a section over A. The obstruction to extending the section over 2 K. ; n C 1/ to one over all of K. ; n C 1/ is  2 HnC1 .K. ; n C 1/;  /. By naturality the obstruction to extending the induced section over A to one over all of X is fO ./ D O. This proves all classes arise as obstructions.  f

f0

Suppose that E ! B and E0 ! B are principal K. ; n/-fibrations. In particular, the fibers are identified with K. ; n/ uniquely up to homotopy. They are said to be equivalent if there is a map ˚W E ! E0 with f0 ı˚ D f and with ˚ inducing a map on the fibers which is compatible up to homotopy with the identification of the fibers with K. ; n/, meaning that after these identifications, the map given by the restriction of ˚ induces the identity on the nth homotopy group of the fibers. More generally given a pair .B; A/, two principal K. ; n/-fibrations E ! B and E0 ! B with sections  and  0 over A are equivalent if there is a homotopy equivalence ˚W E ! E0 computing with the projections to B with the property that ˚ ı  is homotopy to  0 as sections of E0 jA .

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There is also a relative version of these results summarized in the following commutative diagram. Œ.X; A/; .K. ; n/; /

Pull back the universal fibration

XXXXXX XXXXXX XXXXX XXXXXX XXXXXX ,

gg3 ggggg g g g g gggg ggggg g  g g g " # g principal K. ; n/-fibrations over X with a class of a section over A

;

HnC1 .X; AI  /

where diagonal arrows are the obstruction to a relative homotopy to the constant map and the obstruction to the extension of the section over A, respectively. All maps are bijections. To establish the relative version, use the fact that HnC1 .X; AI  / D HnC1 .X=fAgI  /, where X=fAg is the CW complex obtained from X by collapsing A to a point. Thus, elements in HnC1 .X; AI  / correspond to principal K. ; n/-fibrations over X=fAg. These fibrations have a section over the point fAg, and this section is unique up to homotopy. Pulling back via the natural map .X; A/ ! X=fAg; fAg/ gives the result. We say that a map fW E ! B is homotopy equivalent to a principal K. ; n/fibration if there is a principal K. ; n/-fibration EQ ! B and a homotopy equivalence ®W E ! EQ that commutes with the maps to B. It is an easy exercise using obstruction theory to show the following: Proposition 7.3. Let B and E be simply connected CW complexes. A map fW E ! B is homotopy equivalent to a principal K. ; n/-fibration if and only if the following two conditions hold: (i) H .B; E/ D 0 for 6D n C 1 and (ii) HnC1 .B; E/ D  . Suppose that B is a CW complex and that EQ ! B is a principal K. ; n/-fibration. Then EQ is homotopy equivalent to a CW complex. This is proved by induction over the cells of B; details are left to the reader. Proposition 7.4. Let pW EQ ! B be a principal K. ; n/-fibration over a simplicial complex. Then there is a simplicial complex E and a homotopy equivalence ®W E ! EQ so that the composition p ı ®W E ! B is a simplicial map. Proof. Since EQ is homotopy equivalent to a CW complex, there is a simplicial Q At the expense of subdividing complex E and a homotopy equivalence ®0 W E0 ! E. E, we can assume that with the map p ı ®0 W E ! B is homotopic to a simplicial map

7.2 Principal K. ; n/-Fibrations

67

§W E ! B. Applying the homotopy lifting property, we can deform ®0 to a map ®W E ! EQ with p ı ® D §.  Definition 7.5. We say that §W E ! B is a simplicial model for the principal K. ; n/-fibration over B.

Chapter 8

Postnikov Towers and Rational Homotopy Theory

Fibrations and CW structures should be viewed as dual, and a Postnikov tower for a space is a decomposition dual to a cell decomposition. In the Postnikov tower description of a space, the atoms of the space (which we think of as a molecule) are K. ; n/s. (These are atomic from the point of view of homotopy groups. The spheres are atomic from the point of view of homology groups.) The geometric configuration of the atoms in the molecule is exactly the information contained in the “k-invariants” of the space. These tell us how the various K. ; n/s are twisted together. We shall prove that all simply connected CW complexes have such towers. Given a simply connected space X, define X2 D K. 2 .X/; 2/, and define f2 W X ! X2 to be a map inducing the identity on  2 . Suppose inductively that we have a commutative diagram, up to homotopy, Xn E       ::   :          fn  : :  : X4   x; x  x  xx  xx  x  f4 xxx   xx  x X  xx kk5 3  xxx f3 kkkkkk  k x  xxx kkkkk  xxxkkkkkk f2 kk / X X2

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such that for every j  n, we have (1)  i .Xj / D 0 for all i > j. (2) The map Xj ! Xj1 is a principal fibration with fiber K. j .X/; j/ induced by some map kjC1 W Xj1 ! K. j .X/; j C 1/. (3) fj W X ! Xj an isomorphism on  i for all i  j. Invoking the mapping cylinder construction allows us to assume that fn W X ! Xn is an inclusion. The relative homology Hi .Xn ; X/ is zero for i  n C 1. Furthermore, 

HnC2 .Xn ; X/ Š  nC2 .Xn ; X/ !  nC1 .X/: By the universal coefficient theorem, HnC2 .Xn ; XI  nC1 .X// D Hom. nC1 .X/;  nC1 .X//: Let kQ nC1 2 HnC2 .Xn ; XI  nC1 .X// be the class corresponding to the identity homomorphism. This determines a principal fibration XnC1 ! Xn with a section over X. The section over X is equivalently and a lifting of fnC1 of fn as indicated in the diagram below. / K. nC1 .X/; n C 1/ h3 XnC1 hhhh h h fnC1 h h hhhh hhhh h h h h  hhhh fn hhhh / Xn X Lemma 8.1. The map fnC1 W X ! XnC1 meets conditions (1), (2), (3) above. Proof. Clearly (1) and (2) are satisfied. Also, .fnC1 / W  i .X/ !  i .XnC1 / is an isomorphism for i  n. It remains to show that .fnC1 / W  nC1 .X/ !  nC1 .XnC1 / is an isomorphism. We have a map of .Xn ; X/ ! .Xn ; XnC1 /. If this map  induces an isomorphism  nC2 .Xn ; X/ !  nC2 .Xn ; XnC1 /, then it follows that .fnC1 / is an isomorphism on  nC1 . Under the identification of  nC2 .Xn ; XnC1 / with  nC1 .K. nC1 .X/; n C 1// D  nC1 .X/, the map becomes evaluation of the cohomology class kQ nC2 . By definition this map is an isomorphism.  Remarks. (1) For each n let X0n be the CW complex obtained from X by inductively attaching cells of dimension  n C 2 so as to kill all homotopy groups in dimensions  n C 1, so that  i .X0n / D 0 for i > n. The inclusion X  Xn induces an isomorphism on  i for i  n. If fn W X ! Xn is the nth stage of a Postnikov tower for X, then a simple application of obstruction theory shows that fn extends to a map f0n W X0n ! Xn . This map induces an isomorphism on all homotopy groups, showing that there is

8 Postnikov Towers and Rational Homotopy Theory

71

a homotopy equivalence from X0n to the nth stage of the Postnikov tower of X, a homotopy equivalence that is compatible up to homotopy with the natural maps from X to these spaces. If gn W X ! Yn is the nth stage of another Postnikov tower for X, then we have a commutative diagram Xn ? O      f0n / X0 X n ?? ?? ?? ?? gn   Yn fn

with both vertical arrows being homotopy equivalences. Under the resulting identifications of HnC2 .Xn ; XI  nC1 .X// with HnC2 .Yn ; XI  nC1 .X// the k-invariants for the .n C 1/st-stages of the two towers correspond, as is easy to see. It is in this sense that the Postnikov tower is unique in the homotopy category. Q (2) If we form limfXi g, defined as the subspace of 1 iD2 Xi consisting of all  compatible sequences, then the maps ffn W X ! Xn g determine lim fn W X !  limfXi g. This map induces an isomorphism on all homotopy groups. To prove  this, one shows that  j .lim Xn / D lim  j .Xn / D  j .XN / for N  j. It is not   true in general for inverse systems that taking homotopy groups commutes with taking inverse limits. It is, however, true for this inverse system since  jC1 .XN / !  jC1 .XN1 / is onto for all N. (3) If Y is a CW complex and ®W Y ! lim Xn induces an isomorphism on all  homotopy groups, then the obstructions to lifting, up to homotopy, lim fn W X !  lim Xn to Y lie in H .XI   .lim Xn ; Y// D 0. Thus, there is a map W X ! Y   such that ® ı is homotopic to lim fn . In particular induces an isomorphism  on all homotopy groups, and hence, is a homotopy equivalence. This shows how to recover X, up to homotopy equivalence, from a Postnikov tower: It is the unique CW complex, up to homotopy equivalence, which maps to lim Xn inducing an isomorphism on the homotopy groups.  (4) Suppose X is a CW complex of dimension n. Once we have built the Postnikov tower for X through dimension n, the process of finishing the tower is formal (if complicated). Namely, fn W X ! Xn induces an isomorphism on Hi for i  n and is surjective on HnC1 . Thus, HnC1 .Xn / D 0, but HnC2 .Xn / may not be zero.

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Suppose it is  . Then the .n C 1/st homotopy group of X is   and the kinvariant knC2 2 HnC2 .Xn ;  / is the identity homomorphism. By the Serre spectral sequence for K. nC1 .X/; n C 1/ ! XnC1 ? ? y Xn one has HnC2 .XnC1 / D 0. We continue in this fashion: HnCkC1 .XnCk / D 0;

HnCkC2 .XnCk / D  nCkC1 .X/;

and the k-invariant is the identity map: HnCkC2 .XnCk / !  nCkC1 .X/: This ability to complete the Postnikov tower for X once we have Xn without making further reference to the space X is referred to. By saying that, after Xn , the process becomes formal. Thus, all simply connected spaces X of homological dimension n are formal after Xn . (5) Let us consider the case X D S2 . Then X2 D K.Z; 2/ D CP1 ; H3 .CP1 / D 0 and H4 .CP1 / D Z. Hence,  3 .S2 / D Z. If we form K.Z; 3/ ! X3 ? ? y CP1 with k-invariant the identity H4 .CP1 / ! Z, then H4 .X3 / D 0 and H5 .X3 / D Z=2. Hence,  4 .S2 / D Z=2. Continuing in this way gives a (theoretical) algorithm for calculating all the higher homotopy groups of S2 . This calculation has never been done and seemingly is impossibly complicated to do. It is an amazing theorem of E. Curtis that  i .S2 / ¤ 0 for all i  2. According to the last section given a Postnikov tower, we can replace each of the fibrations by a simplicial model.

8.1 Rational Homotopy Theory for Simply Connected Spaces

? ? y

? ? y ®n

Xn ? ? y

 Bn ? ?§ y n

Xn1 ? ? y

 Bn1 ? ? y

®n1

:

:

: ? ? y

: ? ?§ y 3

X2

73

®2

 B2 :

Here the Bi are simplicial complexes, the maps §i are simplicial maps, and the ®i are homotopy equivalences. We call the tower of simplicial complexes a simplicial model for the Postnikov tower.

8.1 Rational Homotopy Theory for Simply Connected Spaces We begin with a little of the theory of Q and Q-vector spaces. Let A be an abelian group (usually infinitely generated). Then A may be given the structure of a Q vector  space if and only if A Š A ˝Z Z ! A ˝Z Q. (This is equivalent to the equation ’x D “ having a unique solution x for all ’ 2 Z  f0g and “ 2 A:/ If 0 ! A1 ! A2 ! A3 ! 0 is a short exact sequence then so is 0 ! A1 ˝Z Q ! A2 ˝Z Q ! A3 ˝ QZ ! 0: Lemma 8.2. (a) If 0 ! A1 ! A2 ! A3 ! 0 is a short exact sequence, then if two of the three terms are Q-vector spaces, so is the third one. (b) If A is an abelian group and has a composition series A D A0 A1 : : : An 0 with successive quotients Q-vector spaces, then A is a Q-vector space.

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8 Postnikov Towers and Rational Homotopy Theory

(c) For any space X, we have H .XI Q/ Š H .X/ ˝Z Q and H .XI Q/ ' Hom Z .H .X/; Q/ Š ŒH .XI Q/  . Q  .X/ is a Q-vector space, then H Q  .XI G/ is a Q-vector space for any abelian (d) If H 1 group G. Proof. (a) There is a commutative ladder of short exact sequences: 0 !

A1 ? ? y

!

!

A2 ? ? y

A3 ? ? y

! 0

0 ! A1 ˝ Q ! A2 ˝ Q ! A3 ˝ Q ! 0 By hypothesis two of the three vertical maps are isomorphisms. Thus, by the 5 lemma, so is the third. (b) Follows immediately by induction from (a). 

(c) and (d) are left to the reader as exercises. Corollary 8.3. Hi .XI Q/ and Hi .XI Q/ are Q-vector spaces. Definition. A Q-space is a space, X, satisfying:

(1) X is homotopy equivalent to a CW complex (generally having 1-many cells in each dimension). (2)  1 .X/ D 0. (3)   .X/ is a Q-vector space for all  1. The following is an alternate definition. A space X satisfying (1) and (2) above Q  .X; Z/ is a Q-vector space. is a Q-space if in addition (3) H Theorem 8.4. The two definitions of a Q-space are equivalent. Proof. We will need the following lemma: Q  .K.Q; n/I Z/ is a Q-vector space. Lemma 8.5. (a) H  (b) H .K.Q; 2n/I Q/ is a Q-polynomial algebra on one generator, that generator being of degree 2n. (c) H .K.Q; 2n C 1/I Q/ is a Q-exterior algebra on one generator, that generator being of degree 2n C 1. Proof. We construct K.Q; 1/ D X as an iterated mapping cylinder: ×2

X =

×4 •••

S1

1Q

×3

H D reduced homology groups.

S1 × I

S1 × I

•

8.1 Rational Homotopy Theory for Simply Connected Spaces

75

Since homotopy commutes with direct limits,  1 .X/ D limfZ; ng D Q; !

and  i .X/ D limf0g D 0 for i > 1: !

Thus, X is a K.Q; 1/. Since homology also commutes with direct limits Q  .ZI Z/ D H



Q for D 1 0 for ¤ 1:

Q  .K.Q; n  1//I Z/ is a Q-vector Suppose inductively that we have shown that H space. Consider the Serre spectral sequence for   K.Q; n  1/ ! P K.Q; n/ ? ? y K.Q; n/ with Z-coefficients and with Q-coefficients. There is a comparison theorem for spectral sequences which says that, given two spectral sequences and a map between them inducing an isomorphism on E; 1 for 0;q p;q . ; / ¤ .0; 0/ and on E2 for all q > 0, then it is an isomorphism on E2 for all p;0 .p; q/ ¤ .0; 0/. In particular, it is an isomorphism on E2 for p > 0. This result is an algebraic one proved by induction, cf. Exercise (42). Applying it to the Serre spectral sequences for the above fibration with Z and Q  .K.Q; n  1/I Z/ is a Q-vector space, then so is Q-coefficients, we see that if H Q  .K.Q; n/I Z/. H Parts (b) and (c) are proved by the same inductive argument given in Chap. 5 to calculate H .K.Z; n/I Q/:  Corollary 8.6. K.Z; n/ ! K.Q; n/ induces an isomorphism on rational cohomology and thus on rational homology. Using Lemma 8.5 we will show that if  1 .X/ D 0 and  i .X/ is a Q-vector space Q i .X/ is a Q-vector space for all i. Since H .X/ Š H .Xn / for  n, for all i, then H where Xn is the nth stage in the Postnikov system for X, it suffices to prove by Q  .Xn / is a Q-vector space. Since X2 D K. 2 .X/; 2/ and  2 .X/ is induction that H Q  .X2 / is a Q-vector space. a rational vector space, from Lemma 8.5, we see that H Q  .Xn1 / is a Q-vector space and consider the Serre spectral sequence for Suppose H

76

8 Postnikov Towers and Rational Homotopy Theory

K. n .X; n// ! Xn ? ? y Xn1 p;q

p;q

Now E2 is a rational vector space for all .p; q/ ¤ (0,0), and consequently E1 is also a rational vector space for all .p; q/ 6D .0; 0/. Thus, there are composition Q i .Xn / with successive quotients Q-vector spaces. This proves H Q i .Xn / is series for H a Q -vector space for all i. Q  .X/ a Q-vector Conversely, suppose we have a space X with  1 .X/ D 0 and H space. We will show by induction that  i .X/ is a Q-vector space. Suppose we know that   .Xn1 ) are Q-vector spaces. From what we have just established, it follows Q  .Xn1 / is a rational vector space. Hence, so is H Q  .Xn1 ; X/. In particular, that H  n .X/ D HnC1 .Xn1 ; X/ is a rational vector space. Thus, all the homotopy groups of Xn are rational vector spaces. Continuing the induction, we establish that all the homotopy groups of X are rational vector spaces.  Theorem 8.7. Let X and X.0/ be simply connected CW complexes with X.0/ a Qspace and fW X ! X.0/ . The following three conditions are equivalent: Q  .X; Q/ ! H Q  .X.0/ I Q/ D H Q  .X.0/ / is an isomorphism (a) f W H (b) f# W   .X/ ˝ Q !   .X.0/ / ˝ Q D   .X.0/ / is an isomorphism (c) f is universal for maps of X into Q-space; i.e., given gW X ! Y.0/ with Y.0/ a Q-space, then g factors uniquely up to homotopy through X.0/

g

X f



{

{

{

/ Y.0/ {=

h

X.0/ Proof. We show that (a) implies (c). Given f satisfying (a) and gW X ! Y0 , then the obstructions to extending g over all of X.0/ (considering f now as an inclusion) lie in H .X.0/ ; XI   .Y.0/ //. Since f W H .X/ ! H .X.0/ / is an isomorphism with Q-coefficients and   .Y.0/ / is a Q-vector space HC1 .X.0/ ; XI   .Y0 // D 0. Thus, such a map h exists. The obstruction to any two maps being homotopic relative to g is in H .X.0/ ; XW   .Y.0/ // D 0. This proves that (a) implies (c). Now we show that (c) implies (a). Apply universality to Y.0/ D K.Q; n/ and we see that there is a 1–1 correspondence

8.1 Rational Homotopy Theory for Simply Connected Spaces

ŒX; K.Q; n/ x ? Š? Hn .X; Q/

77

Š

 ŒX.0/ ; K.Q; n/ f x ? ?Š Š

 f

Hn .X.0/ ; Q/

and thus, f is an isomorphism on Q-cohomology and thus on Q-homology. To prove (a) and (b) are equivalent, we need the following lemma: Lemma 8.8. If X is a simply connected CW complex, then   .X/ ˝ Q D 0 if and Q  .XI Q/ D 0. only if H (This is a variant of the Hurewicz theorem modulo the Serre class of abelian groups A such that A ˝ Q D 0. ) Proof. Q  .K. ; n/I Q/ D 0. To see this, notice Step 1: Suppose that   D Z=k Z. Then, H k

that we have a fibration K.Z=kZ; n/ ! K.Z; n C 1/ ! K.Z; n C 1/. From the fact that H .K.Z; n C 1/I Q/ is either a polynomial algebra or an exterior algebra, it follows that .k/W H .K.Z; n C 1/I Q/ ! H .K.Z; n C 1/ Q/ D 0. is an isomorphism. A simple application of the Serre spectral sequence shows that Q  .K.Z=k Z; n/I Q/ D 0. this implies H Q  .K. ; n/I Q/ D 0. Step 2:   ˝ Q D 0 ) H If   is a sum of cyclic groups, then the result follows from Step 1. In general   is a direct limit of sums of cyclic groups. Since homology commutes with direct limits, Step 2 is true for all groups  . Q i .XI Q/ D 0. Step 3: If  i .X/ ˝ Q D 0 for all i and  1 .X/ D 0, then H We build the Postnikov tower for X and prove the result inductively for the Xn . Q  .Xn1 I Q/ D 0. Consider the Serre spectral sequence Suppose we know that H for K. n .X/; n/ ! Xn ! Xn1 with Q-coefficients. E2p;q D 0 for all .p; q/ ¤ Q (0,0). Hence, E1 p;q D 0 for all .p; q/ ¤ (0,0). Thus, Hi .Xn I Q/ D 0 for all i. Since Hi .Xn / D Hi .X/ if n > i, this establishes Step 3. Q i .XI Q/ D 0 for all i, then  i .X/ ˝ Q D 0 for all i. Step 4: If  1 .X/ D 0 and H Consider the Serre spectral sequence for K. n .X/; n/

/ Xn  Xn1

for homology with Q-coefficients. At the E2 -term, we have

78

8 Postnikov Towers and Rational Homotopy Theory

Thus,  n .X/ ˝ Q D E1 0;n D Hn .XI Q/. But Hn .Xn / Š Hn .X/ and by assumption Hn .XI Q/ D 0. Thus,  n .X/ ˝ Q D 0. This completes the lemma.  From this we prove (a) and (b) are equivalent. Namely, let fW X ! Y and let Ff be the homotopy theoretic fiber. Then f W   .X/ ˝ Q !   .Y/ ˝ Q is an isomorphism ,   .Ff / ˝ Q D 0 if and only if H .Ff I Q/ D 0 if and only if f

H .XI Q/ ! H .YI Q/ is an isomorphism. (This last equivalence comes from the Serre spectral sequence.) This completes the proof of the theorem.  Definition. Given X and fW X ! X.0/ with X.0/ a Q-space and f satisfying (1), (2), or (3) above (and hence all of them), we call fW X ! X.0/ the localization at 0 of X. The terminology comes from the fact that if we localize Z at 0 we get Q. We will not, in this course, consider any other localization, although it is possible to localize at any prime ideal. In fact, there is a Hasse–Minkowski principle which allows one to recover the whole space from its various localizations. Theorem 8.9. If ®W X ! X.0/ and ®0 W X ! X0.0/ are localizations of X, then there is a homotopy equivalence hW X.0/ ! X0.0/ such that X.0/ > L } } }} }} } } '

X

1

AA h 0 AA ' AA AA

h

X0.0/

is a homotopy commutative diagram. Moreover, h is unique up to homotopy. Proof. The proof follows immediately from the universal property of localization. 

8.2 Construction of the Localization of a Space

79

8.2 Construction of the Localization of a Space The construction of the localization of a space goes by induction on the Postnikov tower of the space. We will assume that X is a CW complex and is simply connected. The idea of the proof is to tensor both the groups and the k-invariants with Q. ®n1 Suppose inductively that we have a localization Xn1 !.Xn1 /0 . Then

Xn

 Xn1

P qq8 q q qq qqq q q qqq

'n

 K. n .X/; n C 1/ q8 knC1 qqq q q qq qqq ' n1

/ .Xn /.0/

/ P n6 .0/ n n nnn nnn n n n nnn

 / K. n .X/ ˝ Q; n C 1/ 6 .knC1 /.0/ nnn nn n n nn nnn i

 / .Xn1 /.0/

Since K. n .X/ ˝ Q; n C 1/ is a Q-space, i ı knC1 factors uniquely through .Xn1 /.0/ . Let .Xn /.0/ be the fibration induced over .Xn1 /.0/ from .knC1 /.0/ . This defines ®.n/ W X.n/ ! .Xn /.0/ . By the commutativity of the diagram, we see that .®.n/ / W  n .X/ !  n ..Xn /.0/ / is an isomorphism when tensored with Q. This proves that ®.n/ W X.n/ ! .Xn /.0/ is the localization of X.n/ at 0. To complete the proof, we need the following lemma: Lemma 8.10. Let Y be a topological space. There is a CW complex X and a map W X ! Y which induces an isomorphism on all homotopy groups. If 0 W X0 ! Y is another such, then there is a homotopy equivalence hW X ! X0 so that 0 ı h is homotopic to . The proof of this lemma is left as an exercise for the reader. We apply the lemma with Y D lim.Xn /.0/ . Let X.0/ be a CW complex with  W X.0/ ! Y. The maps Xn ! .Xn /.0/ define a map lim Xn ! lim.Xn /.0/ . Thus, we   have X.0/

f'n g

X

/ lim Xn 

 / lim.Xn /.0/ 

80

8 Postnikov Towers and Rational Homotopy Theory .n/

The obstructions to lifting the map X ! lim X.0/ to a map X ! X.0/ lie in  H .XI  1 .F// where F is the homotopy theoretic fiber of . Since induces an isomorphism on homotopy groups,   .F/ D 0. The resulting map ®W X ! X.0/ is a localization at 0. Calculations: 1. We have ( H .K.Q; 2n  1/I Z/ D

Q if D 2n  1 0

otherwise:

Thus, the map S2n1 ! K.Q; 2n  1/ is a localization at 0, and consequently (  2n1  Q  S ˝QŠ 0

if D 2n  1 otherwise:

Since  i .S2n1 / is always finitely generated (  i .S2n1 / Š

Z

it i D 2n  1

finite group for i ¤ 2n  1.

2. S2n ! K.Q; 2n/ induces an isomorphism in rational cohomology through degree .4n  1/. The kernel of H4n .K.Q; 2n/I Q/ ! H4n .S2n I Q/ is generated by 2 . Form the principle fibration K.Q; 4n  1/ ! E ! K.Q; 2n/ with k-invariant 2 . A calculation using the Serre spectral sequence shows that H .EI Q/ Š H .S2n I Q/. Furthermore, we can lift S2n ! K.Q; 2n/ to a map S2n ! E. This proves that 8 ˆ for i D 2n ˆ D n ¨. This map is a map of cochain complexes by Stokes’ theorem (which is valid in our setting). Theorem 9.1 (p.l. deRham Theorem). ¡ induces an algebra isomorphism on cohomology. We will deduce the p.l. deRham theorem from the following proposition: Proposition 9.2. (i) Let ® 2 An .K/ satisfy d® D 0; ¡.®/ D 0 (i.e.,

R

® D 0 for all n ). Then, there

n

exists § 2 An1 .K/ such that d§ D ®; ¡.§/ D 0. p (ii) A ..K/ ! C .K/ is onto. Proof of p.l. deRham Theorem Assuming Proposition 9.2 (Additive Statement): Let B .K/ be the kernel of ¡. By (ii) we have a short exact sequence of cochain complexes ¡

0 ! B .K/ ! A .K/ ! C .K/ ! 0 leading to a long exact sequence of cohomology groups. The first part of the proposition says that H .B .K// D 0; and thus, by the five lemma, it follows that HdR .K/ Š H .KI Q/: The multiplicative statement will be considered later. We turn now to the proof of Proposition 9.2.

9.2 Lemmas About Piecewise Linear Forms

85

9.2 Lemmas About Piecewise Linear Forms Lemma 9.3 (Poincaré Lemma). Let c.K/ be the simplicial complex that is the cone over a finite complex K. Suppose ®` is a closed form in A .c.K//, ` > 0. Then, ®` D d§`1 for some §`1 2 A .c.K//. Proof. c.K/ is the join of the point c with K. Points in c.K/ are denoted by sk C .1  s/  c where 0  s  1 and k 2 K, with the proviso that 0  k C 1  c D c for all k 2 K. Define W c.K/  I ! c.K/ by .s  k C .1  s/  c; t/ D s.1  t/  k C .1  s C st/  c: If œ 2 A .c.K//, then  .œ/ is a form on c.K/  I. Restricted to any c.¢/  I, for ¢ a simplex of K, we have  .œ/ is a polynomial form with Q coefficients when expressed in terms of the barycentric coordinates, ti , of ¢ and the coordinate s on I. These forms are compatible. Claim. If ®` is a closed form of degree ` > 0 on c(K), then Z

tD1

d.

 .®` // D ®` :

tD0 i i Explanation: If we expand  .®` /jc.¢/  I as †N iD1 ’i .¢/t C “i .¢/t dt where  ’i .¢/ and “i .¢/ are in A .c.¢//, then the ’i .¢/ and “i .¢/ patch together to give forms ’i and “i in A .c.K//. If i i   ®` D †N iD0 ’i t C “i t dtwith’i ; “i 2 A .c.K//;

then we define Z

tD1 tD0

deg “i  ®` D †N iD0 .1/

We now check that  Z d 

tD1

® D ®: 

tD0

Since jtD0 is the identity ® D  ®j tD0 D ’0 : dtD0

“i : iC1

86

9 deRham’s Theorem for Simplicial Complexes

Since jtD1 is the constant map to c,  ®j tD1 D †N iD0 ’i D 0: dtD0

(Here is the only place we use ` > 0.) Lastly, since ® is closed, ¡ .d®/ D d  ® D 0; i.e., 

for i  0; and d’i D 0 .1/deg ’i i’i C d“i1 D 0 for i  1

Hence, Z d.

tD1

¡ ®/ D

tD0

N  X .1/deg “i C1 d“i iC1 i D0

D

N X

’i D ’0 D ®:

iD1

 `



n

Lemma 9.4 (Extension Lemma). Let ® be a form in A .@ /. There is a form §` 2 A .n / such that §` j@n D ®. Proof. Let ¢ be an .n  1/-dimensional face of n , say ¢ D f.t0 ; : : : ; tn /jtn D 0g. Let ’ 2 A .¢/. Let v be the vertex ftn D 1g and U  n be the complement of this vertex. There is a stereographic projection from the vertex pW U ! ¢

p.t0 ; : : : ; tn / D .

t0 tl tn1 ; ;:::; /: 1  tn 1  tn l  tn

The form p .’/, considered as a form on U, is a polynomial form with Qcoefficients in the variables t0 ; : : : ; tn1 ; 1=.1  tn /; dt0 ; : : : ; dtn1 ; d.1=1  tn /. Since d.1=1  tn / D 1=.1  tn /2 dtn the form p .’/ is a polynomial form with Qcoefficients in the variables. t0 ; : : : ; tn1 ; 1=1  tn , dt0 ; : : : ; dtn . Hence, for some N  0, the form .1  tn /N p .’/ D ’Q is a Q-polynomial form on n . It is the required extension of ’ to all of n . Note that if £ is a face of ¢ and if ’j£ D 0, then ’Q restricted to the join of £ and v is 0.

9.2 Lemmas About Piecewise Linear Forms

87

Now suppose that ® 2 A .@n /. Let ¢0 be the face ft0 D 0g, and let ®0 2 A .¢0 / be the restriction of ® to ¢0 . Extend this to a form §0 in n . The difference ®  §0 j@n vanishes on ¢0 . Call this difference ®1 . Let ¢1 be the face ft1 D 0g. Extend ®1 j¢1 to a form §1 on n . We know that §1 j¢0 D 0. Thus, ®.§0 C§1 j@n / vanishes on ¢0P [ ¢1 . Continue in this manner defining ®i on and §i on n In the end, we have ® D niD0 §i j@n .  

Lemma 9.5.  n .an / Let ®` be a closed vanishes on @n . If ` D n, then R form in A . /` which `1 assume also that ® D 0. Then, ® D d§ for some §`1 which vanishes n

on @n . `  n .bn / Let R ® be a closed form in A .@ /; ` > 0. Ifn ` D n  1, then assume that ® D 0. Then, ® D d§ for some § 2 A .@ /. @n

Proof. Clearly, .a0 / is true, and .b0 / and .b1 / are vacuous. Proof of .a1 /. Since k forms on a n-simplex vanish for k > n, to prove .a1 /, we need only consider functions and 1-forms. .a1 / is clearly true for functions. For 1-forms, it is just the fundamental theorem of calculus. If p.t/dt is a closed polynomial 1form on [0, 1], then there is a polynomial q.t/ so that q0 .t/ D p.t/ and q.0/ D 0. Its R1 value at 1 is 0 p.t/ dt. Thus, if the integral of p.t/dt over the interval is zero, then q.t/ vanishes on the boundary of the interval. Proof that .an1 / implies .bn /. Let ® be a closed form in A .@n /. Let ¢n D ftn D 0g be a codimension 1 face of n . By the Poincaré Lemma ®j.@n  int ¢n / D d§ for Q in some § 2 A .@n  int ¢n /. By the extension lemma, we can extend § to §  n n Q A .@ /. Then ®  d§ is a closed form on @ vanishing except on the face ¢ given by the equation ftn D 0g. On this face, it is a relative form; i.e., it vanishes when restricted to @¢ n1 . If ` D n  1, then Z Z Z n n Q ® D .®  d§/ D .®  d§/: 0D @n

@n

¢ n1

Q D d for some relative form on ftn D 0g. Let Q Thus, by .an1 /, we have ®  d§ Q C /. be the extension by 0 of to the rest of @n . Then, ® D d.§ Q Proof that .bn / implies .an /. Let ®` be a closed form on n with ®` j@n D 0. By the Poincaré lemma, ®` D d§`1 for some form §`1 which may not vanish on @n . Clearly §`1 j@n is closed. If ` D 1 and n  2, then §`1 j@n is a constant, and hence, by subtracting this constant, we can make §`1 j@n D 0. If ` > 1, then by .bn /; §`1 j@n D d `2 . (If ` D n, then Z

Z

Z

§`1 D @n

d§ D n

® D 0; n

which allows us to apply .bn /.) Extend to a form Q on all of n . Then, ® D n d.§  d / Q and .§  d /j@ Q D 0. 

88

9 deRham’s Theorem for Simplicial Complexes

This completes the proof of the lemmas. Let us return to the two statements in Proposition 9.2. R (i) Suppose ® 2 An .K/ satisfies d® D 0 and ® D 0 for all n-simplices ¢ n 2 K. ¢n R § D 0 for all Then, ® D d§ for some § 2 An1 .K/ with the property that £n 1

.n  1/-simplices £n1 2 K.

Proof. Given ® as above there is, according to Lemma 8.5, a form §¢ 2 An1 .¢ n / such that ®j¢ D d§¢ and such that §¢ j@¢ n D 0. The collection f§¢ n g defines an element in An1 .K.n/ / which vanishes on K.n1/ . Using the extension lemma repeatedly, we can extend this form to §n on all of K. Clearly ®  d§n vanishes on K.n/ . The difference .®  d§n /j¢ nC1 is closed, and for every .n C 1/-simplex ¢ nC1 , this form vanishes on @¢ nC1 . Hence, there is a form ¢ 2 An1 .¢ nC1 / such that d ¢ D .®  d§n /j¢ and ¢ j@¢ nC1 D 0. As before, the f ¢ g define a form on K.nC1/ vanishing on K.n/ . This can be extended to a form §nC1 on K which vanishes on K.n/ . The difference ®  d.§n C §nC1 / vanishes on K.nC1/ . Continuing in this manner, we define §n ; §nC1 ; : : : ; §i 2 An1 .K/, such that §i jK.i1/ D 0 and ®  d.§n C : : : C §nCk / vanishes on K.nCk/ . Because §i vanishes on K.i1/ , the  infinite sum †1  kD0 §nCk is an element of A .K/. It is the required element §. ¡

(ii) A .K/ ! C .K/ is onto. Proof. Given a simplex ¢ n 2 K, there is a form in ¢ n , (1/vol ¢ n ) .dt1 ^ : : : ^ dtn /, which has integral 1. This form can be extended by 0 to the rest of the n-skeleton of K. Using the extension lemma, we then extend it to all of K. The result is an n-form ® such that Z

Z ® D 1 and

¢n

® D 0 for £n ¤ ¢ n : £n

Since we can do this for any simplex, the map ¡ is onto.



9.3 Naturality Under Subdivision Let K be a linear cell complex. This means that K D UDi where Di is a convex linear cell. On the intersection of two cells, the linear structures agree. A subdivision of K is a new linear cell structure on K so that each cell in the new decomposition lies affine linearly in a cell of the old. Such subdivisions arise by choosing points which are to be the new vertices. If we choose exactly one point from each cell, the resulting subdivision is called barycentric subdivision. Barycentric subdivision always yields a simplicial complex.

9.4 Multiplicativity of the deRham Isomorphism

89

Lemma 9.6. Let K be a linear cell complex, and let K0 be a subdivision of K which is a simplicial complex. Then the integration map ¡W A .K0 / ! C .KI Q/ induces an isomorphism on cohomology. 0

Proof. The definition of ¡ is to integrate forms in A .K / on cells of K. Now apply the p.l. deRham theorem for each cell of K and argue by induction over the cells.  If K is a simplicial complex and K0 is a subdivision of K in which each new vertex has rational barycentric coordinates in K, then restriction induces a map of differential algebras defined over Q, A .K/ ! A .K0 /: By the p.l. deRham theorem, this map is an isomorphism on cohomology. If fW jKj ! jLj is a continuous map between the geometric realizations of simplicial complexes, then it is possible to subdivide K sufficiently to simplicial 0 complex K0 , with the property that there is a simplicial map ®W K ! L homotopic to f. This approximation induces j®jW jKj ! jLj which is homotopic to f. Since K’ could be any sufficiently fine subdivision of K, we can take it to be a rational 0 subdivision. Hence, we will have ® W A .L/ ! A .K / resulting from a continuous map fW jKj ! jLj. Of course ® will depend significantly on many choices (i.e., which subdivision and which approximation we take), but (as we shall see) in the homotopy category of differential algebras ® is well defined.

9.4 Multiplicativity of the deRham Isomorphism Let K be a simplicial complex, A .K/ the piecewise linear forms (p.l. forms) on K, and HdR (K) its cohomology. We have shown that ¡ W HdR .K/ ! H .jKjI Q/ is an additive isomorphism. Both HdR .K/ and H .jKjI Q/ are naturally graded rings: HdR from wedge product of forms and H .jKjI Q/ from the Alexander– Whitney formula for cup product of cochains. We want to prove that ¡ is an isomorphism of the graded cohomology algebras. The easiest path to this goal is to give a different description of the cup product in singular cohomology. Let K be a simplicial complex with underlying space jKj. The space jKj  jKj carries naturally the structure of a linear cell complex. The cells are all products ¢  £ with ¢ and £ simplices of K. This cell structure is not a simplicial complex structure. (If it were, then there would be a graded-commutative, associative cup product on simplicial cochains.) Let ’ and “ be simplicial cochains on K. We define ’ ˝ “ as a cellular

90

9 deRham’s Theorem for Simplicial Complexes

cochain on jKj  jKj. The value is given by < ’ ˝ “; ¢  £ > D < ’; ¢ >  < “; £ >. One sees easily that •.’ ˝ “/ D •’ ˝ “ C .1/deg ’ ’ ˝ •“. Consequently, if ’ and “ are cocycles, then so is ’ ˝ “. The class Œ’ ˝ “ 2 H .jKj  jKj//, and when restricted to the diagonal W jKj ! Kj  jKj, gives Œ’ [ Œ“ in H .jKj/. The Alexander–Whitney formula for multiplication arises from choosing a homotopy from W jKj ! jKj  jKj to a linear mapping (called a chain approximation to the diagonal). Let ®0 and ®1 be elements of A .K/. Let .jKj  jKj/0 be a rational subdivision 0 of jKj  jKj which is a simplicial complex. Define ®0 ˝ ®1 2 A ..jKj  jKj/ / as 0 follows. Each simplex £  .jKj  jKj/ lies in a product ¢0  ¢1 in such a way that its vertices are rational. The form .®0 j¢0 / ˝ .®1 j¢1 / is a polynomial form with rational coefficients in the product linear structure. Thus, .®0 j¢0 / ˝ .®1 j¢1 / restricts to give a form ®0 ˝ ®1 .£/ 2 A .£/. Clearly, these forms fit together to define ®0 ˝ ®1 2 A ..jK  jKj/0 /. Under the map ¡W A ..jKj  jKj/0 / ! C .jKj  RjKjI Q/, the R form ®0 ˝ ®1 goes to the cochain which evaluates on ¢0  ¢1 to give . ¢0 ®0 /  . ¢1 ®1 /. Thus, if ®0 and ®1 are closed forms, then ¡ .®0 ˝ ®1 / is the cocycle ¡.®0 / ˝ ¡.®1 /. Restricting to the diagonal, we see that the singular cohomology class of ®0 ^®1 is the cup product of the classes of ®0 and ®1 . This proves that ¡ W H .A .K// ! H .C .K/I Q/ is an algebra isomorphism.

9.5 Connection with the C1 deRham Theorem If M is a C1 manifold, then associated to it is the differential algebra of C1 forms. It is an algebra over R. The original theorem of deRham says that the cohomology of this differential algebra is naturally isomorphic (as a ring) to the singular cohomology with real coefficients. The connection between forms on singular cochains is once again achieved by integration. There are many proofs by now of deRham’s theorem. For example, one can use currents to give a proof (essentially deRham’s original proof); one can prove that the resulting homology groups satisfy the Eilenberg–Steen road axioms and hence must be singular homology. More in the spirit of the present discussion one can prove the Poincaré lemma and then establish the isomorphism by induction on a handlebody (instead of a triangulation). Similarly one could use a cover by convex subsets of Rn , but in this setup, the induction is more complicated and goes under the appellation of “sheaf theory.” Whatever method is chosen, the result is the following, cf. [27]: Theorem 9.7. Let AC1 .M/ denote the DGA of C1 forms on M. Then, a map of cochain complexes induced by integration ¡W AC1 .M/ ! C .M; R/ induces an isomorphism of cohomology rings.

9.5 Connection with the C1 deRham Theorem

91

Any C1 manifold has a C1 triangulation. This means that there is a simplicial complex KM and a homeomorphism ®W KM ! M which is C1 on each simplex. Consider the DGA of Q-polynomial forms A .KM /. This gives a second algebra of forms on M. We wish to relate these. Let 

piecewise polynomial forms on M with Q-coefficients  A .M/ D for the given triangulation AC1 .M/ D fC1  formsg 9 8 collections f®g of forms on the > ˆ > ˆ = < n simplices f g such that  Q .M/ D A 1 N n for each n-simplex > ˆ > ˆ .i/ ® is C in  ; : 0 0 0 n .ii/ ®j \  n D ® jn \  n : Q  .M/ is constructed in the same way as A .M/ only now using C1 forms. Thus, A Q  .M/ is graded algebra having a “d.” Also, This definition makes sense, and A Q  .M/ Stokes’ theorem for simplicial chains holds as before. We call forms in A 1 piecewise C -forms. We have the following inclusions: A .M/ ˝Q R LLL LLL LLL L%

Q  .M/ A

AC 1 .M/ u uu uuu u zu u

Q  .M/ by H 1 .M/, then the integration map If we denote the cohomology of A pC   induces a map Hp:C1 .M/ ! H .MI R/. Thus, we have the following commutative diagram of isomorphisms:

7 ooo o o ooo ooo

Hp:C 1 .M/

H .A .M// ˝Q R PPP PPP PPP PP'

 H .MI R/

fNNN NNN NNN N

H .AC 1 .M// p ppp p p pp w pp p

We shall show: Proposition 9.8. Integration induces an isomorphism Hp:C1 .M/ ! H .MI R/.

92

9 deRham’s Theorem for Simplicial Complexes

Q  .M/ and A1 .M/ ! A Q  .M/ Corollary 9.9. The inclusions A .M/ ˝Q R ! A C both induce isomorphisms on cohomology. Proof (of Proposition 9.8). The proof is essentially the same as in the p.l. case. The Poincaré lemma holds and is proved the same way. The extension lemma is again proved using stereographic projection, but instead of multiplying by a power of .1  tn /, we simply multiply by a C1 function of tn which is 1 for tn D 0 and is identically 0 near tn D 0. Once we have these lemmas, the argument given in the Q-polynomial case is valid, mutatis mutandis, in the piecewise C1 case.  Corollary 9.9 follows immediately from the commutative diagram and Proposition 9.8.

9.6 Generalizations of the Construction Suppose that X is a space made out of manifold “pieces.” This includes, for example, a simplicial complex. Then it is possible to define an algebra of piecewise C1 forms on X which will calculate the cohomology. We don’t prove this result in general, but rather confine ourselves to one important example. Let D1 ; : : : ; Dk be smooth submanifolds in an ambient manifold Y. Suppose that all intersections Di1 \ : : : \ Dit ; i1 < i2 < : : : < it are transverse. We define Ap:C1 .[kiD1 Di / to be compatible collections of forms f¨i 2 AC1 .Di /; i D 1; : : : ; t; satisfying ¨i jDi \ Dj D ¨j jDi \ Dj for all i; jg: We claim that the cohomology of Ap:C1

S k iD1

 Di is isomorphic to the singular

cohomology of [kiD1 Di with real coefficients. Triangulate the union so that all intersections are subcomplexes and each simplex in the triangulation is C1 . Let K be the resulting simplicial complex. There is a map Ap:C1 .[kiD1 Di / ! C .KI R/ given by integrating the form over the simplices of K. We claim that this map induces an isomorphism on cohomology. This is proved by induction on k. If k D 1, then this is exactly the deRham theorem. Suppose we have established the result k1 Di / [ Dk where the intersection is for all ` < k. We write [kiD1 Di as .[iD1 0 k1 k1 [iD1 .Di \Dk /. Let D D [iD1 Di and D D [kiD1 Di . We have a commutative diagram of short exact sequences

9.6 Generalizations of the Construction

93

0

/ A 1 .D/ p:C

/ A 1 .D0 ` Dk / p:C

/ A 1 .D0 \ Dk / p:C

/ 0

0

 / A1 .D/ C

 ` / A1 .D0 Dk / C

 / A1 .D0 \ Dk / C

/ 0

The result follows immediately by induction and the five lemma.

Chapter 10

Differential Graded Algebras

10.1 Introduction In this chapter, we shall study differential algebras in their own right. What we are doing, actually, is studying the homotopy theory of differential algebras. In fact, we shall construct an object (the minimal model) which should be considered the Postnikov tower of a differential algebra. Definition. A differential graded algebra (or DGA for short), A , is a graded vector space over Q; R, or C, A D ˚p0 Ap ; having (i) A differentiation d W A ! AC1 with d2 D 0. (ii) A multiplication Ap ˝ Aq ! ApCq satisfying ’“ D .1/pq “’: (iii) d.’“/ D d’“ C .1/p ’d“. Examples. (i) The C1 deRham complex AdR .M/ of a smooth manifold and the p.l. deRham complex Ap:l: .K/ of a simplicial complex are DGAs over R; Q, respectively. (ii) The cohomology H .X; Q/ of a space is a DGA (with d D 0), but the singular cochain complex C .X; Q/ is not (the signed commutativity fails). (iii) The commutative cochain problem. Perhaps the main genesis of the theory we are considering is the problem of commutative cochains. This was solved in an abstract manner by Quillen, and in an attempt to better understand this, Sullivan was led to the p.l. forms and the connection between differential forms and homotopy type. In retrospect one can already see much of the P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__10, © Springer Science+Business Media New York 2013

95

96

10 Differential Graded Algebras

theory in Whitney’s book [27]; however, one fundamental point was missing in that Whitney only constructs commutative cochains over R, and as already mentioned there is no way to build Postnikov towers over R and thus tie in the commutative cochains with homotopy type. Let X be a simplicial complex. The usual definition of the cup product ’’ [ “q between a p-cochain ’q and a q-cochain “q is < ’p [ “q ; pCq > D< ’p ; front p-face of pCq >  < “q ; back q  face of pCq > : This formula leads to the following properties: (i) •.’p [ “q / D •’p [ “q C .1/p ’p [ •“q . (ii) ’p [ .“q [ ”r / D .’p [ “q / [ ”r . Moreover, a somewhat grizzly computation shows that on the cohomology level, we have graded commutativity (iii) Œ’p [ Œ“q D .1/pq Œ“q [ Œ’p . However, it is obviously false that ’p [ “q D .1/pq “q [ ’p on the cochain level. Now one may attempt to modify the formula so as to have all three properties, but all such attempts are doomed to failure since, as realized by Steenrod 35 years ago, the failure to find commutative cochains over Z is reflected in the existence of cohomology operations, such as the Steenrod squares. These objections do not apply over Q (we have essentially proved this by calculating H .K.Z; n/; Q//; thus, it is reasonable to look for commutative cochains/Q. Before going on, it is time to precisely define what is meant by commutative cochains. Definition. Commutative cochains assign functorially to each simplicial complex X a DGA defined over Q, C .X/, satisfying (i)–(iii) above and such that: (iv) The cohomology of C .X/ is H .XI Q/. (v) Given a subcomplex Y  X, we have C .X/ ! C .Y/ ! 0: Example. The p.l. forms Ap:l: .X/ give an explicit solution to the commutative cochain problem. The cohomology H .X; Q/ does not give a solution because (v) is violated. The argument given in this chapter shows that Theorem. Let C .X/ be any solution to the commutative cochain problem over Q. Then the minimal model M for C .X/ gives the Q-homotopy type of X.

10.2 Hirsch Extensions

97

So now we have some full circle. The problem of commutative cochains is equivalent to finding not only the cohomology but also the Q-homotopy type of a space from a cochain complex. The p.l. forms explicitly solve this problem, and moreover, a simple comparison theorem shows that the C1 forms give the R-homotopy type of a smooth manifold. It is interesting to note that in [27], Whitney essentially showed that any solution to the commutative cochain problem over R satisfying a mild continuity condition is given by integration of suitable differential forms (the flat forms) over chains. Now, almost 25 years later, we have finally understood what he was driving at. Given a DGA, A , we denote by H .A / the cohomology algebra. It is again a DGA with d D 0. We assume throughout that H0 .A / is the ground field and that H1 .A / D 0. Thus, A is, so to speak, simply connected. Definition. A DGA A is said to be minimal if: (i) A is free as a graded-commutative algebra on generators of degrees  2. (ii) d.A /  AC ^ AC where A D ˚k>0 Ak . Condition (i) means that A is a tensor product of polynomial algebras on generators of even degrees and exterior algebras on generators of odd degrees. Condition (ii) says that d is decomposable. There is a notion of minimal DGAs which have generators in degree 1 (see Chap. 13). Given a DGA, A , we wish to construct a minimal model, M.A /, for A . By definition this means that M.A / is a minimal DGA and there is a map ¡W M.A / ! A of DGAs inducing an isomorphism on cohomology. One of the main results of this chapter is that every simply connected DGA has a minimal model. Remark. The construction of M.A / is motivated by the construction of the Postnikov tower of a space. In fact, the parallel is quite precise, as we shall see in Chap. 9.

10.2 Hirsch Extensions Actually the fundamental property of a minimal algebra is that it is an increasing sequence of subalgebras which are nicely related, one to the next. To illuminate this, we study these extensions separately. First Definition. Let A be a DGA. A Hirsch extension of A is an inclusion A ! A ˝d ƒ.Vk /: The notation means that (i) V is a (finite dimensional) vector space homogeneous of degree k; (ii) ƒ.Vk / is the free graded-commutative algebra with unit generated

98

10 Differential Graded Algebras

by V (the polynomial algebra on V if k is even and the exterior algebra on V if k is odd); and (iii) dW V ! AkC1 . The differential on the full algebra is determined by djA and djV. Two Hirsch extensions A ! A˝d ƒ.Vk / and A ! A˝d ƒ..V0 /k / are equivalent if there is a commutative diagram A ˝d .Vk / s9 ss ss s sss ss ® A K KKK KKK KKK K%  A ˝d0 ..V0 /k / with ® an isomorphism which is the identity on A. Lemma 10.1. A ! A ˝d ƒ.V/ and A ! A ˝d ƒ.V0 / are equivalent if and only if there is an isomorphism §W V ! V0 so that the following diagram commutes: d

V ! HkC1 .A/ ? ? ?D ? §y y d0

V0 ! HkC1 .A/ Q A ˝d0 ƒ.V0 / is an isomorphism extending the identity on Proof. If ®W A ˝d ƒ.V/! A, then ®.v/ D av C §.v/. For ® to be an isomorphism, § must be an isomorphism. Since ®.dv/ D d0 .®.v// D d0 av C d0 §.v/, Œdv D Œd0 § .v/ 2 HkC1 .A/: Q 0 so that Œdv D Œd0 §.v/ 2 HkC1 .A/, then Conversely, if we have §W V!V 0 dv  d §.v/ D dav for some av 2 A. We can choose av linearly in v. Define ®.v/ D av C §.v/. This defines a map ®W A ˝ ƒ.V/ ! A ˝ ƒ.V0 /, which is easily seen to be an isomorphism and to commute with the differentials.  To classify Hirsch extensions of A with a fixed vector space of new generators, V, we say that two are equivalent if the isomorphism ®W A ˝d ƒ.V/ ! A ˝d0 ƒ.V/ is the identity on A and sends v to av C v. Equivalence classes are then in natural one-to-one correspondence with maps

10.3 Relative Cohomology

99

dW V ! HkC1 .A/I or what is the same thing, the class of Œd 2 HkC1 .AI V /: Proposition 10.2. Let M be a DGA and let M.n/  M be the subalgebra generated in degrees  n. Then, M is a minimal DGA if and only if M.1/ D the ground field M.0/ D M.1/  M.2/  M.3/  : : : [n M.n/ D M; and for each n the inclusion M.n/  M.n C 1/ is a Hirsch extension with the new generators being in degree n C 1. Proof. Suppose that M is minimal. Then clearly, M.1/ is the ground field and M D [n M.n/. Since each M.i/ is free as a graded-commutative algebra, it is clear that as vector spaces M.n C 1/ Š M.n/ ˝d ƒ .VnC1 /: Since d.v/ is decomposable for v 2 VnC1 and M has no elements of degree 1, it must be the case that d.v/ is a sum of products of elements of degree  n; i.e., d.v/ 2 M.n/. This proves that M.n/  M.n C 1/ is a Hirsch extension. Conversely, if M D [n M.n/ where M.n/  M.n C 1/ is a Hirsch extension of degree n C 1 and M.1/ is the ground field, then by induction, one see that M.n/ is a minimal DGA. It follows that M is also a minimal DGA. 

10.3 Relative Cohomology Before beginning the actual construction of M.A /, we need a few basic facts about relative cohomology for a map between two cochain complexes. Let fW C ! D denote a degree-preserving map between two cochain complexes. Define Mnf D Cn ˚ Dn1 and let •W Mnf ! MnC1 be given by f 

•C 0 f •D

:

100

10 Differential Graded Algebras

One checks easily that •2 D 0. We define H .C; D/ to be H .Mf /. The maps .i2 /W D1 ! Mf and  1 W Mf ! C commute with the coboundaries. The resulting maps on cohomology give a long exact sequence f

i

•

f

! Hn .C/ ! Hn .D/ ! HnC1 .C; D/ ! HnC1 .C/ ! : : : : Clearly, this long exact sequence is functorial for commutative diagrams f1

C1 ! ? ? y

D1 ? ? y

f2

C2 ! D2 :

10.4 Construction of the Minimal Model Theorem 10.3. Every simply connected DGA has a minimal model. Proof. Given a DGA A, we shall construct an increasing sequence of Hirsch extensions ground field D M.0/ D M.1/  M.2/  : : : together with maps ¡n W M.n/ ! A so that ¡n jM.k/ D ¡k for k  n and the map ¡n W M.n/ ! A is an n-minimal model in the sense that: (i) M.n/ is minimal and generated by elements in degrees  n. (ii) ¡n is an isomorphism on cohomology in degrees  n. (iii) ¡n is an injection on cohomology in degree n C 1. To begin with, M.1/ is the ground field and ¡1 is the map sending 1 to 1. Obviously, this map is an isomorphism on cohomology in degrees  1 and an injection on cohomology in degree 2. Suppose for some n  1 that we have inductively constructed M.n/ and ¡n W M.n/ ! A as required. The relative cohomology Hi .M.n/; A// vanishes for i  n C 1. (This follows from the long exact sequence of the pair and the conditions on ¡n .) Let V D HnC2 .M.n/; A/. We will form M.nC1/ by taking the algebra M.n/ ˝ ƒ.VnC1 /:

10.4 Construction of the Minimal Model

101

To extend the differential on M.n/ to one defined on all of M.nC1/, by the Leibnitz rule, we need only define it on V subject to the condition that dv 2 AnC2 is closed for all v 2 V. Having done this, to define a map ¡nC1 W M.n C 1/ ! A extending ¡n , it is necessary only to define ¡nC1 jV subject to the condition ¡n .dv/ D d¡nC1 .v/ for all v 2 V: Both d and ¡nC1 are defined by a linear splitting s for the projection map from cocycles to cohomology: / HnC2 .M; A/

ZnC2 .M; A/ n s

This is equivalent to choosing, linearly in v, cocycle representatives .mv ; av / 2 M.n/nC2 ˚ AnC1 for the relative cohomology class v. For .mv ; av / to be a cocycle means dmv D 0 and ¡n .mv / D dav . Having chosen the splitting from relative cohomology back to the relative cocycles, we let d.v/ D mv and ¡nC1 .v/ D av : Then, d2 .v/ D d.mv / D 0 and ¡n .dv/ D ¡n .mv / D dav D d.¡nC1 .v//: This shows that M.n C 1/ is a differential algebra and that ¡nC1 is a map of differential algebras. Lastly, we must show that Hi .M.n C 1/; A/ D 0 for i  n C 2. For this we need a lemma. Lemma 10.4. (a) M.n/ has no elements of degree 1. (b) HnC2 .M.n/; M.n C 1// D V. (c) HnC3 .M.n/; M.n C 1// D 0. Proof. (a) But for any n  1, the DGAs M.n/ and M.1/ agree in degrees  1 so that M.n/ has no generators of degree  1. (b) Let us consider the relative cocycles of degree .n C 2/. These are all of the form .a; vCb/ where a; b 2 M.n/, and v 2 V. Furthermore, da D 0, and a D dvCdb.

102

10 Differential Graded Algebras

Varying such a cocycle by d.b; 0/ changes it to .a0 ; v/. No element of this form is exact unless v D 0. Conversely, given v 2 V, we have the cocycle .dv; v/. This proves that HnC2 .M.n/; M.n C 1// D V. (c) Since M.n/ has no generators of degree  1, M.n/ and M.n C 1/ are the same in degree .n C 2/. It follows easily from this and the fact that M.n/  M.nC1/ that HnC3 .M.n/; M.n C 1// D 0.  We have a map of pairs .id; ¡nC1 /W .Mn ; MnC1 / ! .Mn ; A/. The corresponding map of long exact sequences, the above fact, and the five lemma prove that ¡nC1 W H .M.n C 1// ! H .A/ is an isomorphism for  n C 1 and that ¡nC1 W HnC2 .M.n C 1// ! HnC2 .A/ is an injection. S This completes the inductive construction of the .M.n/ and ¡n . Define M D n M.n/ and define ¡W M ! A by ¡jM.n/ D ¡n . Since cohomology commutes with direct limits, it follows that ¡ W H .M/ ! H .A/ is an isomorphism. Thus, .M; ¡/ is a minimal model for A. 

Chapter 11

Homotopy Theory of DGAs

In this chapter we shall delve more deeply into the homotopy theory of DGAs. One consequence of this study will be to prove the uniqueness of the minimal model.

11.1 Homotopies Definition. Let f and g be maps of DGA from A to B. A homotopy from f to g is a map HW A ! B ˝ .t; dt/ satisfying HjtD0 D f and HjtD1 D g: Of course, when we restrict to t D 0; 1 we also set dt D 0. Explanation: .t; dt/ represents the tensor product of polynomials on t (degree of t D 0) with the exterior algebra on dt, where the degree of t is zero and the degree of dt is 1). It is the algebra of polynomial forms on the real line, R. The two restrictions correspond to evaluations of forms at the points {0} and {1}. The idea for this definition comes from dualizing the usual definition on the space level. To study homotopies, we introduce an additive operator Z 1 W B ˝ .t; dt/ ! B 0

by Z

1

b ˝ ti D 0 and 0

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__11, © Springer Science+Business Media New York 2013

103

104

11 Homotopy Theory of DGAs

Z

1

b ˝ ti dt D .1/deg b 0

b : iC1

Likewise, define Z

t

W B ˝ .t; dt/ ! B ˝ .t; dt/

0

by Z

t

b ˝ ti D 0 and

0

Z

t

b ˝ ti dt D .1/deg b b ˝

0

tiC1 : iC1

The following are proved directly from the definitions: (1) If “ 2 B ˝ .t; dt/, then Z

t

d

Z t “ C d“ D “  .“jtD0 / ˝ 1:

0

0

(2) If HW A ! B ˝ .t; dt/ is a homotopy from f to g, then Z

Z

1

1

H.a/ C

d 0

dH.a/ D g .a/  f.a/: 0

(Note that (2) follows from (1) by taking “ D H.a/ and restricting to t D 1.)

11.2 Obstruction Theory The basic result in the homotopy theory of Hirsch extensions is the following: Proposition 11.1. Given a diagram A ? ? y

g

! B ? ?® y f

A ˝d ƒ.Vk / ! C and a homotopy HW A ! C ˝ .t; dt/ from ® ı g to fjA, there is an obstruction class Q 2 HkC1 .B; CI V/ which vanishes if and only if there is an extension gQ W A ˝d Q of H to a homotopy from f to ® ı gQ . ƒ.Vk / ! B of g and an extension H

11.2 Obstruction Theory

105

Q Proof. For each v 2 V, we define Q.v/ 2 B kC1 ˚ C k by Q Q.v/ D .g.dv/; f.v/ C

Z

1

H .dv//: 0

Q This is the obstruction cocycle for extending g and H. First, we show that Q.v/ is Q indeed a cocycle, and then we show that if Q.v/ is exact for all v 2 V, the sought after extensions exist. Q dQ.v/ D .dg.dv/; ® ı g.dv/  df.v/  d

Z

1

H.dv// 0

Z D .g.d v/; ®Q ı g.dv/  f.dv/  d 2

1

Z

1

H .dv/ 

dH .dv//

0

0

D .0; 0/: Q Such a homomorLet ŒQ W V ! HkC1 .B; C/ be the homomorphism induced by Q. phism is the same as an element Q 2 HkC1 .B; CI V /. If ŒQ .v/ D 0 for all v 2 V, then there are relative cochains .bv ; cv / (depending Q linearly on v) so that d.bv ; cv / D Q.v/. Define gQ .v/ D bv and Q H.v/ D f.v/ C

Z

t

H.dv/ C d.cv ˝ t/:

0

Let us check that these equations define DGA maps gQ W A ˝d ƒ.V/k ! B Q A ˝d ƒ.V/k ! C ˝ .t; dt/ extending H. For this, it is necessary extending g and HW Q only that dQg.v/ D g.dv/ and dH.v/ D H.dv/. Clearly, dQg.v/ D dbv D g.dv/. Q Also, by (1), dH.v/ D df.v/ C H.dv/  H.dv/jtD0 . Since H is a homotopy from f to Q Q is a ® ı g, H.dv/jtD0 D f.dv/. Thus, dH.v/ D H.dv/. Lastly, we must show that H homotopy from f to ® ı gQ . But Q H.v/j tD0 D f.v/ and Z 1 Q D f.v/ C H.dv/ C dcv H.v/j tD1 Z

0

Z

1

D f.v/ C

1

H.dv/ C .®.bv /  f.v/  0

H.dv// 0

D ®.bv / D ® ı gQ .v/: Q of H such that H Q is Conversely, if we are given any extension gQ of g and H k k1 a homotopy from f to ® ı gQ , then define §.Qg;H/ by §.v/ D Q WV ! B ˚ C R1 Q .Qg.v/; H.v//. One checks directly that 0

106

11 Homotopy Theory of DGAs

Z d§.Qg;H/ Q D .g.dv/; f.v/ C

1

Q H.dv// D Q.v/:

0

 We shall also need a relative version of this lifting property. Lemma 11.2. Suppose given the following commutative diagram: ®

M

/ A

?? ?? ?? ??  f

 M ˝d ƒ.V /

§

n

?

 / B

C



where (1)  ı f D . (2) is onto. (3) ı ® D  ı §jM. H

˝1

(4) M ! B ˝ .t; dt/ ! C ˝ .t; dt/ is constant (i.e., . ˝ 1/ ı .H/ D . ı H0 / ˝ 1/. Then, the obstruction cohomology class ŒQ 2 HnC1 .A; B; V / vanishes if, and only Q M ! B˝.t; dt/ of H satisfying if, there is an extension ®Q of ® and an extension HW (1) ı ®Q D  ı § and Q is a constant homotopy. (2) .v ˝ 1/ ı H Q V ! CocyclesnC2 .A; B/ as before: Proof. Define QW Q Q.v/ D .®.dv/; Q ®.v/ C

Z

1

H.v//: 0

Q  d.av ; 0/ D .®.dv/ Q  Let av 2 A be such that .aV / D .®.v//. Consider Q.v/ R1 dav ; ®.v/  av C 0 H.dv//. This is a cocycle in .Ker /nC1 ˚ .Ker /n . If it is exact Q here, Q.v/  d.av ; 0/ D d.’v ; “v / with ’v 2 Ker and “v 2 Ker . Q using the cochain .aV C ’v ; “v /. Checking the formulas in Then, define ®Q and H Q is constant. Since is Proposition 11.1, one sees that ı®Q D ı§ and that .˝1/ıH Q  .Ker ; Ker/ ! H .A; B/ is an isomorphism. onto, the five lemma implies that H Q Thus, Q.v/  d.av ; 0/ is exact in .Ker ; Ker / if, and only if, ŒQ.v/ D 0 in HnC1 .A; B/. 

11.3 Applications of Obstruction Theory

107

Corollary 11.3. Given a commutative diagram ®

M ? ? y

! A ? ? yf §

M ˝d ƒ.Vn / ! B such that f is onto, the element ŒQ W V ! HnC1 .A; B/ is the obstruction to extending ® to a map ®W Q M ˝d ƒ.V/n ! A such that fı®Q D §. Proof. Apply Lemma 11.2 with C D B. The cohomology of Ker f is identified with the relative cohomology H .A; B/. 

11.3 Applications of Obstruction Theory Corollary 11.4. For M minimal, the relation on maps from M ! A of being homotopic is an equivalence relation. Proof. Let HW M ! A ˝ .t1 ; dt1 / be a homotopy from f0 to f1 and JW M ! A ˝ .t2 ; dt2 / be a homotopy from f1 to f2 . Let X be the differential algebra .t1 ; t2 ; dt1 ; dt2 /=ft2 .t1  1/ D 0; t1 dt2 D t2 dt1 D 0g: This algebra represents the piecewise polynomial forms on the variety t2 .t1 1/ D 0 in the .t1 ; t2 /-plane:

The homotopies H and J define a map “H C J00 W M ! A ˝ X . If H.m/ D †ai ˝ ti1 C bi ˝ ti1 dt1 and 0

j

0

j

J.m/ D †aj ˝ t2 C bj ˝ t2 dt2 ;

108

11 Homotopy Theory of DGAs 0

then since H.m/jt1 D1 D J.m/jt2 D0 , we have †i0 ai D a0 . The formula for “H C J00 .m/ is 0

0

j

j

†ai ˝ ti1 C hi ˝ ti1dt C †j1 aj ˝ t2 C †j0 bj ˝ t2 dt2 : One checks easily that “H C J” is a map of DGAs. Consider the diagram A ˝ .t1 ; t2 :dt1 ; dt2 / ? ?p y "HCJ"

M !

A˝X

The obstructions to lifting “H C J00 lie in H .A ˝ Œ.t1 ; t2 ; dt1 ; dt2 / ; X / D 0. Since p is onto, Lemma 11.2 says that there is a map ¡W M ! A˝.t1 ; t2 ; dt1 ; dt2 / such that p ı ¡ D “H C J00 . If we restrict ¡ to t1 D t2 , we find a homotopy from f0 D Hjt1 D0 to f2 D Jjt2 D1 . This proves that the relation of being homotopic is transitive. Reflexivity and symmetry are easily shown.  Definition. We denote the homotopy classes of maps of DGA from A to B by ŒA; B . Theorem 11.5. Let ®W B ! C induce an isomorphism on cohomology, and let M be a minimal differential algebra. Then ® ŒM; B ! ŒM; C is a bijection. Proof. If we have fW M ! C, then the obstructions to lifting f, up to homotopy, to B step by step over the natural increasing filtration of M lie in HnC1 .B; CI In .M/ /. (Here, In .M/ is the space of indecomposables of M in degree n. The “ ” means the dual vector space.) Since ®W B ! C induces an isomorphism on cohomology, H .B; CI V/ D 0 for all vector spaces V. Thus, there is a map g W M ! B and a homotopy from f to ® ı g. This proves that ® is onto. To show that ® is one-to-one, suppose given f0 and f1 W M ! B and a homotopy HW M ! C ˝ .t; dt/ between ® ı f0 and ® ı f1 . Let P be the kernel of

ŒC ˝ .t; dt/ ˚ B ˚ B ! C ˚ C ! 0; 0

where the map sends ” 2 C ˝ .t; dt/ to .”jtD0 ; ”jtD1 / and sends .b; b / 2 B ˚ B 0 to .f.B/; f.b //. There is a map ¡W B ˝ .t; dt/ ! P; which sends “ to .f ˝ Id.t;dt/ .“/; “jtD0 ; “jtD1 /. This map induces an isomorphism on cohomology. To see this, notice that one has a long exact sequence: 

! H .P/ ! H .C/ ˚ H .B/ ˚ H .B/ ! H .C/ ˚ H .C/: !

11.4 Uniqueness of the Minimal Model

109

The map  sends .c; b0 ; b1 / to .c  f b0 ; c  f b1 /. Hence,  is onto and H .P/  H .C/ ˚ H .B/ ˚ H .B/ is represented as all triples .c; .f /1 .c/; .f /1 .c//. From this, one sees easily that ¡W B ˝.t; dt/ ! P induces an isomorphism in cohomology. The maps f0 and f1 W M ! B together with HW M ! B ˝ .t; dt/ define a map ¡W M ! P: Apply Lemma 11.2 to the diagram B ˝ .t; dt/ KK .tD0;tD1/ v: KK v KK v KK v K%  v ¡ / / B˚B M P ¡Q

We see that there are no obstructions to lifting ¡ to ¡Q W M ! B ˝ .t; dt/ such that ¡Q jiD1 D  Bıi ¡ D fi . Thus, f0 and f1 W M ! B are homotopic. 

11.4 Uniqueness of the Minimal Model Theorem 11.6. If A is a differential algebra and M

BB BB ¡ BB BB }> }} } } }}

A

¡0

M0

are minimal models for A, then there is an isomorphism IW M ! M0 and a homotopy H from ¡W ¡0 ıI. The isomorphism I is itself determined by these conditions up to homotopy. Proof. Applying Theorem 11.5, we see that there is a map IW M ! M0 with ¡0 ı I is homotopic to ¡, and that such an I is well defined up to homotopy. It remains to show that any such IW M ! M0 is an isomorphism. Since, on the level of cohomology, 0 I ı .¡ / D ¡ and .¡0 / and ¡ are isomorphisms, it follows that any such I W M ! M0 induces an isomorphism on cohomology. To conclude the proof, we have the following lemma:

110

11 Homotopy Theory of DGAs

Lemma 11.7. Let M and M0 minimal DGAs and suppose that IW M ! M0 induces an isomorphism on cohomology. Then I is an isomorphism of DGAs. Proof. Clearly, I induces In W M.n/ ! M0 .n/. Assuming inductively that In is an isomorphism, we shall show that InC1 is an isomorphism. Consider the exact sequence of cohomology: HnC1 .M.n// ! HnC1 .M/ ! HnC2 .M.n/; M/ ! HnC2 .M.n// ? ? ? ? ? ? ? ? yIn yI y.In ;I/ yIn HnC1 .M0 .n// ! HnC1 .M0 / ! HnC2 .M0 .n/; M0 / ! HnC2 .M0 .n//

By the inductive hypothesis and the five lemma, now prove that I is an isomorphism. We claim that HnC2 .M.n/; M/ D HnC2 .M.n/; M.n C 1//. As we have seen before, HnC2 .M.n/; M.n C 1// Š VnC1 , the vector space of new generators added in going from M.n/ to M.n C 1/. To prove the claim, note that in degree  n C 2 the relative cochains for .M.n/; M.n C 1// and .M.n/; M/ are equal. In degrees > nC2, the former are a subgroup of the latter. The equality of the relative cohomology groups up through degree .n C 2/ follows immediately from this. Since these identifications are natural, we have a commutative diagram: .In ;I/

HnC2 .M.n/; M/ ! HnC2 .M0 .n/; M0 / x x ? ? ? ? VnC1 .M/

I

VnC1 .M0 /

!

Thus, I induces an isomorphism on the vector space of new generators, and hence InC1 induces an isomorphism: InC1

0

M.n C 1/ ! M .n C 1/:  This completes the proof of Theorem 11.6.



Corollary 11.8. Let ¡A W MA ! A and ¡B W MB ! B be minimal models, and let fW A ! B be a map of differential algebras. There is map OfW MA ! MB and a homotopy from ¡B ı Of ! f ı ¡A . The map Of is determined up to homotopy by these properties.

11.4 Uniqueness of the Minimal Model

111

Proof. This is immediate from Theorem 11.5 applied to the following diagram:

z

MA

z

z fı¡A

z
2, by induction we have constructed M.k/ ! A .Bk / for every  k < n and maps ¡k W M.k/ ! A .Bk / such that 1. ¡k W M.k/ ! A .Bk / is a minimal model. 2. M.k  1/  M.k/ and this inclusion is a Hirsch extension with new generators in degree k. 3. ¡k jM.k  1/ D pk ı ¡k1 .  Then according to Theorem 12.1 letting M.n 1/  M.n/ be a Hirsch extension dual to the rational form of the K. n .B/; n/-fibration Bn ! Bn1 , there is a map ¡n W M.n/ ! A .Bn / satisfying the three properties listed above. Since this Hirsch extension is dual to the principal fibration, the vector space of new generators is dual to  n .Bn / ˝ Q D  n .B/ ˝ Q. This completes the inductive construction. We set M D [1 nD2 M.n/. To complete the proof of the corollary, we need to show that M is the minimal model for A .B/. Since pn

   ! Bn ! Bn1 !   

116

12 DGAs and Rational Homotopy Theory

is a simplicial model for the Postnikov tower of B. Fix n. There is a map fn W B ! Bn which, after a rational subdivision of B can assume to be simplicial and which is homotopic to the map from B to the nth-stage of its Postnikov tower. Let Kn be this subdivision. We have the composition fn ı ¡n W M.n/ ! A .Bn / ! A .Kn /: This is an n-minimal model in the sense that the relative cohomology vanishes in degrees n C 1. Of course, there is the inclusionA .B/  A .Kn /, and the relative homology groups of this pair are trivial. Hence, by Proposition 11.1, there is a map n W M.n/ ! A .B/ which when composed with the inclusion into A .Kn / is homotopic to fn ı ¡n . The map n is also an n-minimal model. Now, we extend to M.n C 1/. There is a rational subdivision KnC1 of Kn and a simplicial map fnC1 W KnC1 ! BnC1 which is homotopic to the map from B to the .nC1/st-stage of its Postnikov tower. In particular, the following diagram commutes up to homotopy:

u uu uu u u u zu  A .Kn / O

A .B/

sn

A .Bn / O

/ A .BnC1 / O ¡nC1

¡n

M.n/

rnC1

KK s KK nC1 KK KK K% / A .KnC1 / O



/ M.n C 1/;

where sn , snC1 , and rnC1 are the maps induced by the various rational subdivisions. We have rnC1 ı sn D snC1 . Now, deform fnC1 ı ¡nC1 by a homotopy to an .n C 1/-minimal model nC1 W M.n C 1/ ! A .B/. It follows that nC1 jM.n/ is homotopic to n . Using Proposition 11.1, we can deform nC1 by a homotopy until its restriction to M.n/ agrees with n . In this way, we construct a tower of maps k W M.k/ ! A .B/ so that for over every k0 < k we have k jM.k0 / D k0 . Also, each k is a k-minimal model. Then, the collection of these k define a map W M ! A .B/ that is a k-minimal model for every k; that is to say, M is the minimal model for A .B/. Corollary 12.3. Let B be a simply connected simplicial complex whose rational homology groups are finite-dimensional rational vector spaces in each degree, and let M be a minimal model for the p.l. forms A .B/. Then: 1. I .M/ and   .B ˝ Q/ are dual rational vector spaces. 2. H .M.n// is identified with H .Bn I Q/.

12.4 The Minimal Model of the deRham Complex

117

3. The rational k-invariant .knC1 ˝ 1/ 2 HnC1 .Bn1 I  n .B/ ˝ Q/ is identified with the class of the Hirsch extension M.n  1/  M.n/; that is to say, the fibration Bn ! Bn1 is dual to the Hirsch extension M.n  1/  M.n/. It follows from this corollary that we can build the rational Postnikov tower of B from the minimal model of the p.l. forms A .B/ and vice-versa. This means that all the rational homotopy theoretic information about a space B is determined by the minimal model for the p.l. forms on B.

12.4 The Minimal Model of the deRham Complex The result connecting the minimal model for the p.l. forms on a simplicial complex with the Postnikov tower of the complex has an analogue of the minimal model of the smooth forms on a smooth manifold. Corollary 12.4. The minimal model of the C1 , forms on a simply connected C1 manifold M, M1 .M/, is isomorphic to MM ˝Q R where MM is dual in the sense of Theorem 12.1 to the rational Postnikov tower of M. Proof. Choose a C1 -triangulation of M. This gives a diagram: A .M/ ˝Q R ! Ap:C1 .M/

 AC1 .M/:

where both inclusions induce isomorphisms on cohomology. Thus, the minimal models of AC1 .M/ and Ap:C1 .M/ are isomorphic, and the minimal model of A .M/ tensored with R is isomorphic to that of Ap:C1 .M/. Hence, we have an isomorphism, well defined up to homotopy M.A .M// ˝Q R Š M.AC1 .M//: The corollary now follows immediately from Corollary 12.2.



Corollary 12.5. Let Y be a smooth manifold and D1 ; : : : ; Dk  Y smooth submanifolds which intersect transversally. Let D D [kiD1 Di , and let Ap:C1 .D/ be the DGA constructed in Sect. 9.6 of Chap. 9. Suppose that D is simply connected. The minimal model of Ap;C1 .D/ is the real form of a rational minimal model which is dual to the rational Postnikov tower of D. Proof. Let K be a C1 -triangulation of D. There is an inclusion map Ap:C1 .D/ ! Ap:C1 .K/ which gives rise to a commutative diagram of cochain complexes

118

12 DGAs and Rational Homotopy Theory

By the result in Sect. 9.6 of Chap. 9, the integration map induces an isomorphism in  cohomology H .Ap:C1 .D// ! H .KI R/. By the piecewise C1 deRham theorem, 

integration induces an isomorphism H .Ap:C1 .D// ! H .KI R/. Thus, the inclusion Ap:C1 .D/Ap:C1 .K/ is a homotopy equivalence of DGAs. The corollary now follows from Corollary 12.2. 

Chapter 13

The Fundamental Group

So far we have been considering simply connected spaces. In this chapter, we broaden the definition to include information about the fundamental group. We‘find that the theory developed in the previous chapter has a natural analogue which relates DGA of p.l. forms to the universal nilpotent quotient of the fundamental group.

13.1 1-Minimal Models Let A be a connected, but not necessarily simply connected, DGA. A 1-minimal model for A is DGA M.1/ and a map ¡ W M.1/ ! A inducing an isomorphism on H1 and an injection on H2 where M.1/ is an increasing union of Hirsch extensions of degree 1: Ground Field D M.1; 0/  M.1; 1/  M.1; 2/  : : : : [1 nD0 M.1; n/ D M.1/: Theorem 13.1. Let A be a connected DGA. Then, A has a 1-minimal model. Given two such: M.1/

M.1/0

DD ¡ DD DD DD " z< zz z zz zz ¡0

A

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__13, © Springer Science+Business Media New York 2013

119

120

13 The Fundamental Group

there is an isomorphism IW M.1/ ! M.1/0 and a homotopy H from ¡ to ¡0 ı I. Proof. Let M.1; 1/ D ƒ.H1 .A// and let ¡1 W M.1; 1/ ! A be defined by sending each cohomology class to a closed form representing it. Given ¡n W M.1; n/ ! A with ¡n inducing an isomorphism on H1 , define V1nC1 to be the kernel of ¡n W H2 .M.1; n/ ! H2 .A/. (This is also H2 .M.1; n/; A/:/ We choose a splitting sW H2 .M.1; n/; A/ ! Cocycles2 .M.1; n/; A/ for the natural quotient map Cocycles2 .M.1; n/; A/ ! H2 .M.1; n/; A/; We write s.v/ D .mv ; av /. Define dW V1nC1 ! M.1; n/ by d.v/ D mv ; define ¡W V1nC1 ! A1 by ¡.v/ D av As Before, one sees that these formulae lead to M.1; n C 1/ D M.1; n/ ˝d ƒ.V1nC1 / and ¡nC1 W M.1; nC1/ ! A extending ¡n . One sees that the kernel of ¡n W H2 .M.1; n/ ! H2 .A/ is contained in the kernel of n W H2 .M.1; n/ ! H2 .M.1; n C 1// where n is the inclusion M.1; n/  M.1; n C 1/. Thus, in creating M.1; n C 1/, we have killed the kernel of ¡n on H2 . But in doing so, we may have created a new nonzero kernel in H2 at the next stage. We keep repeating the process until we arrive at a stage N with ¡N W H2 .M.1; N/ ! H2 .A/ with trivial kernel. If this never happens, then we construct M.1; n/ for all 1  n < 1, and let M.1/ D [n M.1; n/ ¡W M.1/ ! A being defined by ¡jM.1; n/ D ¡n for every n. One sees easily, from the fact that, on H2 ; Ker¡n  Ker ?n , that ¡ W H2 .M.1// ! H2 .A/ D 0 is an injection. On the other hand, since ¡n W H1 .M.1; n// ! H1 .A/ is an isomorphism for all n, ¡ W H1 .M.1// ! H1 .A/ is also an isomorphism. The proof of the existence of I and a homotopy H is exactly the same as in the simply connected case; see Chap. 11. 

13.2  1 ˝ Q Let   be a finitely presented group. We wish to define “  ˝ Q”. The method is to replace   by a tower of nilpotent groups, fNi . /g, and tensor each of them with Q. Thus,   ˝ Q will be a tower of rational nilpotent groups. Let fi . /g be the terms of the lower central series, i.e., 2 . / D Œ ;   and inductively nC1 . / D Œn . /;   . Let Ni . / be  = i . /. Each Ni . / is a nilpotent group of index of nilpotence i; i.e., all i-fold commutators in Ni . / vanish. The Ni . / fit together in a tower: 

/ N4 . /

/ N3 . / / N2 . / D  =Œ ;   dII O oo7 II ooo II o o II ooo I ooo  

We have short exact sequences (where n D n . // 0 ! n1 = n ! Nn . / ! Nn1 . / ! f1g where n1 = n is an abelian group and is the center of Nn . /.

13.2  1 ˝ Q

121

Such extensions correspond to fibrations K .n1 = n ; 1/ ! K.Nn . /; 1/ # K.Nn1 . /; 1/: The fact that n1 = n is central in Nn . / means that in the fibration, the fundamental group of the base, Nn1 . /, acts trivially on the fundamental group of the fiber. (In general, the action is conjugation in Nn . /.) Hence, central extensions correspond exactly to principal fibrations with base, total space, and fiber having only fundamental groups. The method for tensoring principal fibrations with Q leads to a method for tensoring nilpotent groups with Q. If inductively, we have K.Nn1 . /; 1/ ! K.Nn1 . / ˝ Q; 1/ inducing an isomorphism on rational cohomology, and then we define K.n1 = n ; 1/ ! K..n1 = n / ˝ Q; 1/ ? ? ? ? y y K.Nn . /; 1/ ? ? y

!

K.Nn   ˝ Q; 1/ ? ? y

K.Nn1 . /; 1/ !

K.Nn1   ˝ Q; 1/

where the fibration on the right has k-invariant in H2 .K.Nn1 . / ˝ Q; 1/I n1 = n ˝ Q/ D H2 .K.Nnl . /; 1/I n1 = n // ˝ Q equal to k ˝ 1Q where k is the k-invariant on the left. Taking fundamental groups produces a ladder of maps between two central extensions: f0g

/ n1 = n

/ Nn . 

f0g

 / n1 = n ˝ Q

 / Nn . / ˝ Q

”n

/ Nn1 . /

/ f1g

”n1

 / Nn1 . / ˝ Q

/ f1g:

The extension class for the lower sequence one is just the extension class for the upper one taken with Q-coefficients. Induction on n and a simple comparison theorem of spectral sequences shows that ”n induces an isomorphism on rational cohomology.

122

13 The Fundamental Group

It is a simple argument to show that the elements of finite order in a nilpotent group form a subgroup Tor N. Mal’cev [14] proved that if N is a nilpotent group, then N=Tor N can be embedded in a uniquely divisible nilpotent group N.0/ . (Uniquely divisible means xn D a has exactly one solution x 2 N for all a 2 N and n 2 ZC .) If we take N.0/ to be minimal among all uniquely divisible nilpotent groups containing N , then N.0/ is determined up to isomorphism by N. It is called the Mal’cev completion of N. This is another way to construct the group denoted N ˝ Q above. Notice that if the abelianization of   tensored with Q is a finite-dimensional rational vector space, then for every n  2 the rational vector space n1 = n ˝ Q is finite dimensional. Mal’cev, however, found N.0/ by associating to N a rational Lie algebra LN and then used the Baker–Campbell–Hausdorff formula, cf, [1], to define a nilpotent group structure, N.0/ , on LN . Note that since LN is nilpotent, the B–C–H formula becomes a polynomial with rational coefficients and hence defines a group structure on the Q-vector space. This approach has the advantage of proving that the Qnilpotent groups have Q-nilpotent Lie algebras and hence are the rational points of algebraic groups defined over Q. We have shown that we can associate to any finitely presented group,  , a tower of rational nilpotent groups, fNn . / ˝ Qg, and tower of rational, nilpotent Lie algebras fLn . /g. These two towers determine each other via the B–C–H formula and its inverse. Theorem 13.2. Let X be a simplicial complex with H1 .XI Q/ finite dimensional, and let ¡1 W M.1/ ! A .X/ be a 1-minimal model. Then M.1/ is dual to the tower of Lie algebras fLn . 1 X/g. Proof. M1 is an increasing union of DGAs M.1; 1/  M.1; 2/  : : :. Let Wn be the generators of M.1; n/. The differential in M.1; n/ is determined by .djWn /W Wn ! Wn ^ Wn . By definition, the rational Lie algebra dual to M.1; n/ has underlying vector space .Wn / . The bracket Œ ; W .Wn / ^ .Wn / ! Wn is dual to d. The Jacobi identity for Œ;  is dual to the equation .d2 jWn / D 0. We call this Lie algebra structure Ln . The inclusion M.1; n/  M.1; n C 1/ gives an inclusion Wn  WnC1 . Dualizing gives a map of rational Lie algebras: 0 ! KnC1 ! WnC1 ! Wn ! 0: The fact that .djWnC1 /W WnC1 ! Wn ^ Wn dualizes to the fact that KnC1 is an abelian Lie algebra and is the center of the Lie algebra LnC1 . Since L0 is the trivial Lie algebra, it follows by induction that each Ln is a nilpotent Lie algebra. This gives the duality between 1-minimal models and towers of nilpotent Lie algebras. It remains to show that this duality carries the 1-minimal model for the p.l. forms on X to the tower of Lie algebras associated to its fundamental group. This

13.3 Functorality

123

is first proved inductively for towers of principal fibrations of K. ; 1/’s. The proof uses the Hirsch lemma and is the complete analogue of Corollary 11.3. Given X, we map it to the following tower: :: 9t : tt tt t t tt tt t  tt / K.N3 .X/ ˝ Q; 1/ X L LLL LLL LLL LL%  K.N2 .X/ ˝ Q; 1/ The minimal model for the tower is the one dual to the tower of Lie algebras associated with it. On the other hand, H1 .K.Ni . / ˝ Q; 1/I Q/ D H1 .XI Q/ for all i, and limfH2 .K.Ni . / ˝ Q; 1/I Q/g ! H2 .XI Q/ !

is an injection. Thus, the minimal model for this tower pulls back to the 1-minimal model of X.  Definition. By the real fundamental group of X, we mean the tower of real nilpotent Lie algebras: fNn . 1 .X/ ˝ Rg.

13.3 Functorality It is when we consider functorality that the base point makes its appearance. We define the base point of a differential algebra to be a map A0 ! k ( k D ground field). If X is a simplicial complex and p 2 X is a base point (with rational barycentric coordinates), then it defines A0 .X/ ! Q by evaluation. If A ! k and B ! k are differential algebras with base points, then a base point preserving map ®W A ! B is one that commutes with the maps to k. A homotopy HW A ! B ˝ .t; dt/

124

13 The Fundamental Group

is base point preserving if A ! B ˝ .t; dt/ ? ? ? ? y y k ! k ˝ .t; dt/ commutes. Notice that if f B 0 D k, and if H is base point preserving, then H.a/ D P1 i 1 i D0 bi ˝ t for every a 2 A . A closer look at the argument given in the proof of Theorem 13.1 shows the following: Theorem 13.3. (i) If A ! k is a differential algebra with base point, then there is a 1-minimal model M.1/ which automatically has a unique base point and a base point preserving map M.1/

CC CC CC CC !

/ A

k

inducing an isomorphism on H1 and an injection on H2 (ii) Given two such ¡W M1 ! A and ¡0 W M.1/0 ! A, then there is an isomorphism IW M.1/ ! M.1/0 (automatically base point preserving) and a base point preserving homotopy H from ¡01 ı I to ¡. The isomorphism I is well defined up to base point preserving homotopy. Completely analogously to 11.2, we have the following: Theorem 13.4. If fW X ! Y is a base point preserving map, then it induces OfW M.1/Y ! M.1/X well defined up to base point preserving homotopy. At this point, a miraculous thing happens. Lemma 13.5. If HW M.1/ ! N.1/ ˝ .t; dt/ is a base point preserving homotopy between 1-minimal DGAs, then H is constant in the sense that H D H0 ˝ 1. Proof. We prove by induction that HjM.1; k/ D .H0 jM.1; k// ˝ 1: Suppose we know P this for k < n, and let x be an element of degree one in M.1; n/. Then, H.x/ D i i ˝ ti C ci ˝ ti dt for one forms i and constants ci . Since H is base point preserving and the base pointP map N ! k is an isomorphism P in degree zero,P the ci D 0 for all i. Thus, H.x/ D i i ˝ ti , and hence, H.dx/ D i d˜i ˝ ti  i ˜i ˝ ti1 dt. But dx 2 M.1; n  1/ and hence by induction H.dx/ is of the

13.4 Examples

125

form ˝1. Consequently, i D 0 for i > 0, and hence, H.x/ D ˜0 ˝1. Since M.1;n/ is generated by elements in degree 1, it follows that HjM.1; n/ D H0 jM.1; n/ ˝ 1. Taking the union over all n gives the result.  Thus, we have shown that assigning to based simplicial complexes their 1minimal models is a functor from based homotopy category of simplicial complexes to 1-minimal models and base point preserving maps between them. If we dualize to Lie algebras and then exponentiate, the result is the functor that assigns  1 .X/ ˝ Q to X and f# W  1 .X; p/ ˝ Q !  1 .Y; q/ ˝ Q to fW .X; p/ ! .Y; q/. Thus, from the Q-polynomial forms on X, one has a purely algebraic way using p.l. forms on X to recover the homotopy functor “ 1 ˝ Q”.

13.4 Examples (1) S1 _ S1 . Consider ev

0 ! A ! AC1 .S1 / ˚ AC1 .S1 / ! R ! 0: where the map ev is evaluation at 0. The differential algebra A can be used to calculate the real fundamental group of S1 _ S1 . There is a map of differential algebras fH .S1 _ S1 I R/; d D 0g ,! A which induces an isomorphism on cohomology. Hence, to construct the 1-minimal model for A , it suffices to construct the 1-minimal model for H .S1 _ S1 I R/. We begin with ƒ.fx; yg/ ! H .S1 _ S1 / where x and y are a basis for H1 .S1 _ S1 /. This map induces an isomorphism on H1 , but it has a kernel in degree 2 generated by x ^ y. The next stage is ¡2 W ƒ.fx; y; g/ ! H .S1 _ S1 / where d˜ D x ^ y and ¡2 .˜/ D 0. This map has a kernel in degree 2 generated by x ^ ˜ and y ^ ˜. As we continue the construction, the kernels which we encounter in degree 2 keep growing larger and the construction must be repeated ad infinitum. Actually, one is constructing the increasing system of 1-minimal DGAs dual to the tower of free nilpotent Lie algebras on 2 generators. Clearly, this is the tower associated to the free group on 2 generators. In general, when H2 .X; Q/ D 0, the tower of Lie algebras constructed is a tower of free nilpotent Lie algebras.

126

13 The Fundamental Group

(2) Let N be the nil-manifold obtained by dividing the group of upper triangular real matrices 80 19 < 1xz = @0 1 yA : ; 001 by the lattice of such integral matrices. This gives a compact 3-manifold whose cohomology ring is the same as that of S1  S2 #S1  S2 . When we build the 1-minimal model for the forms on S1  S2 #S1  S2 , we get the infinite process described in Example 1. When we build the 1-minimal model for N, we find that the result is ƒ.fx; yg/ ˝d ƒ.˜/I

d˜ D x ^ y:

Thus, H2 .N/ is generated by the “Massey Products” hx; x; yi D Œx ^ ˜ and hx; y; yi D Œy ^ ˜ . In fact, the 1-minimal model for N is its minimal model.

Chapter 14

Examples and Computations

14.1 Spheres and Projective Spaces Let us consider an odd sphere S2nC1 . The first stage in building the minimal model for the forms on S2nC1 is to construct an exterior algebra ƒ.e/ on a generator of degree .2n C 1/. Clearly, the cohomology of this DGA maps isomorphically to H .S2nC1 / when we send e to a closed form on S2nC1 which is not exact. Thus, ƒ.e/ is the minimal model for the forms on S2nC1 . By Corollary 12.4, this implies that  i .S2nC1 / ˝ Q is zero for i ¤ 2n C 1 and equal to Q for i D 2n C 1. Let A .S2n / be the forms on S2n . The first stage in building the minimal model for  2n A .S / is the polynomial algebra, P.x2n /, on a generator of degree 2n. Clearly, x22n is cohomologous to zero in A .S2n /. Thus, we tensor in an exterior algebra ƒ.y4n1 / with dy D x2 The product P.x2n / ˝d ƒ.y4n1 / is the minimal model. It follows that  Q i D 2n; 4n  1  i .S2n / ˝ Q Š 0 i ¤ 2n; 4n  1: If fW Sn ! X is a simplicial map (with n > 1 and X simply connected), then there is induced OfW MX ! MSn well-defined up to homotopy. In degree n we have In .MX / ! In .MSn / Š Q. The isomorphism In .MSn / Š Q is via integration of the form over the fundamental cycle of Sn . The induced map In .MX / ! Q depends only on the homotopy class of Of, and hence only on the homotopy class of f. This defines a map  n .X/ ! HomQ .In .MX /; Q/; and hence a map  n .X/ ˝ Q ! ŒIn .MX /  : As we shall see in Chap. 15, this is exactly the duality between In .MX / and  n .X/ ˝ Q.

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__14, © Springer Science+Business Media New York 2013

127

128

14 Examples and Computations

Let us consider the minimal model for the forms on complex projective n-space, CPn . The first stage of the minimal model is the polynomial algebra on a 2-dimensional generator P.x2 /. The ideal generated by the class .x2 /nC1 is the kernel of the map P.x2 / ! H .CPn I R/. Thus, the minimal model for the forms on CPn is P.x2 / ˝d ƒ.y2nC1 /I dy D xnC1 . In particular,   i .CPn / ˝ Q D

0 Q

i ¤ 2; 2n C 1 i D 2; 2n C 1:

One can also deduce this from the calculation of the homotopy groups of S2nC1 and the fibration S1 ! S2nC1 ! CPn .

14.2 Graded Lie Algebras Suppose L D ˚n Ln is a graded vector space (over a field of characteristic 0). Let Œ ; W L ˝ L ! L be a map which is homogeneous of degree 0. We say that .L ; Œ ; / is a graded Lie algebra if: (1) Œx; y D .1/.pC1/.qC1/Œy; x for x 2 Lp and y 2 Lq (graded skew symmetry). (2) Œx; Œy; z D ŒŒx; y ; z C.1/pq Œy; Œx; z for x 2 Lp and y 2 Lq (Jacobi identity). An ordinary Lie algebra L is a graded Lie algebra in which L0 D L and Ln D 0 for all n ¤ 0. If X is a simply connected space, then the Whitehead product (see Exercise 34) is a bilinear map: Œ;

 p .X/ ˝  q .X/ !  pCq1 .X/ The formulae in Exercise 34 show that if we define Ln D  nC1 .X/ ˝ Q, then the Whitehead product makes L D ˚n1 Ln into a graded Lie algebra. Another source of examples of graded Lie algebras is DGAs. Let M be a DGA that is free as a graded commutative algebra on positive dimensional generators. Let I.M/ D MC =MC ^ MC where MC is the ideal of elements of positive degree. Denote by .MC /k the kth -power of this ideal; i.e., .MC /k D f¨j¨ D †i ’i1 ^ : : : ^ ’ik with ’ij 2 MC g: If dW M ! M is decomposable, then dW .MC /k ! .MC /kC1 . Hence, it induces a map MC =.M?C /2 ! .MC /?2 =.MC /3 yD yD d W I.M/ ! I.M/ ^ I.M/

14.3 The Borromean Rings

129

The fact that d2 D 0 implies that the composition d

d^idC.1/deg id^d

I.M/ ! I.M/ ! I.M/ ^ I.M/ ^ I.M/

(14.1)

is zero. Let Ln D ŒInC1 .M/  . Dual to d is a map Œ ; W L ˝ L ! L which is homogeneous of degree 0. The fact that d maps into I.M/ ^ I.M/ dualizes to the fact that Œ; satisfies the symmetry condition to be a graded Lie algebra bracket. The dual to Eq. (14.1) is the Jacobi identity. There is a connection between these two examples. If X is a simply connected space, and if MX is its minimal model, then the graded Lie algebras (  C1 .X/ ˝ Q; Whitehead product/ and .˚ InC1 .MX / ; d / are isomorphic under the map: 

 nC1 .X/ ˝ Q !ŒInC1 .MX /  : As an example of this, let us consider S2 _ S2 . Its minimal model begins with two generators 1 and 2 of degree 2. We then add 3-dimensional generators to kill all 4-dimensional cohomology, and so on. The dual graded Lie algebra to this DGA is the free graded Lie algebra on two 1-dimensional generators 1 and 2 . More generally, Sn1 _ : : : _ Snk has a minimal model whose indecomposables are dual to a free graded Lie algebra on generators of degrees .n1  1/; : : : ; .nk  1/ (assuming each ni  2).

14.3 The Borromean Rings There is a geometric form of Poincaré duality. It allows one to do the calculations involved in building the minimal model with submanifolds instead of forms. The basis for the duality is the following. Let Mk  Nn be an embedded submanifold. Suppose that Mk and Nn are both closed and oriented. By the tubular neighborhood theorem there is a neighborhood .M  N/ which is diffeomorphic to a disk bundle over M:

Let D0nk be the fiber over a point m0 2 M. Since M and N are oriented, D0nk receives an orientation. By the Thom isomorphism theorem (see theR exercises), there is a unique class UM 2 Hnk ..M  N/; @.M  N/I Z/ so that UM D 1. (Here D0

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14 Examples and Computations

we assume that M is connected.) There is a C1 differential form representing U which vanishes identically near @.M  N/. If we extend by 0 to N  .M  N/, Q M on all of N. Its cohomology class is the Poincaré then we have a closed C1 -form U dual of ŒM 2 Hk .N/. There is a form of this duality for manifolds with boundary. If .M; @M/  .N; @N/ with M meeting @N normally in @M, then the same construction yields a Q M 2 Hnk .N/ which is the Lefschetz dual of the class ŒM; @M 2 Hk .N; @N/. class U Q M , transverse intersection of manifolds Under the correspondence M ! U corresponds to wedge product of forms. Thus, if Mk0 and M`1 are transverse in kC`n Q 0 and U Q 1 are Thom forms for M0 and M1 Nn with intersection M0;1 , and if U Q Q 1 is a Thom form for M0;1 . This supported in sufficiently small tubes, then U0 ^ U Q Q means that U0 ^ U1 is a closed form supported in a tube about M0;1 and integrating to 1 over each fiber. Q M , given M and U Q M , corresponds The operation of finding a solution for d˜ D U k to finding a submanifold of N whose boundary is M. Thus, if M D @LkC1 , then Q L supported in a tube about L, closed outside the tube about M, there is a form U integrating to 1 over fibers of .L  N/ which are outside the tube about M, and so QL D U QM that dU

Z

Q L D 1; dU QL D U Q M: U D0

We consider now an explicit example of the operation of building the minimal model via submanifolds, namely, the ambient space in S3 -(Borromean rings).

We can think of this as a manifold with boundary by taking out open solid tori around each of the circles, or more simply we work with ordinary cohomology and proper submanifolds The first cohomology of S3  B has rank 3 and is generated by classes which are dual to 2-disks spanning the components. We choose these disks as pictured below:

14.4 Symmetric Spaces and Formality

131

Q 1; U Q 2 , and U Q 3 be the dual Thom classes in H1 .S3  B/. The first stage in Let U Q 1; U Q 2; U Q 3 /. the minimal model is ƒ.U The geometric fact that the linking number of any pair is zero in S3  B means Qi [U Q j D 0 in H2 .S3  B/ for all i ¤ j. Clearly, U Q2 ^U Q 3 D 0 as a form since that U Q1 ^U Q 2 is the Thom form for the interval I12 . To solve the D2 \ D3 D ;. The form U Q1 ^U Q 2 , we must find a proper 2-dimensional submanifold whose equation d D U boundary is I12 . We take this to be the part of D1 cut off by I12 which lies above D2 , denoted C12 in the above drawing. To form the Massey product (see Sect. 14.9 for a Q 1; U Q 2; U Q 3 >, we must take 12 ^ U Q3 CU Q 1 ^ 23 definition of Massey products) < U Qi ^U Q j . In this case, we take 23 D 0 and we take 12 to be supported where dij D U Q3 CU Q 1 ^ 23 is represented by the Thom form of C12 \ D3 . near C12 . Thus, 12 ^ U This intersection is I123 . Since I123 is an arc with end points on different components Q 1; U Q 2; U Q 3 >¤ 0. If we of B, the class ŒI123 2 H1 .S3  B/ is nonzero. Thus, < U 3 0 0 do a similar calculation for S  B where B is three unlinked circles, then all Q i1 ; U Q i2 ; U Q i3 > are trivial. Thus, the third stages of the minimal Massey products < U 3 models for the forms on S  B and S3  B0 are different. In particular, B and B0 are not isotopic. In fact, since the minimal models differ at the third stage, this implies that  1 .S3  B/= 5 and  1 .S3  B0 /= 5 are not isomorphic groups. The group  1 .S3  B0 / is free. The existence of nonzero Massey products in S3  B means that its fundamental group is not free, and indeed its 5th nilpotent quotient is different from that of the free group on 3 generators.

14.4 Symmetric Spaces and Formality A space is said to be formal if the homotopy type of the DGA of forms on the space is the same as the homotopy type of the cohomology ring of the space. Thus, if X is formal and MX is a minimal model for the forms on X, then there is a DGA map MX ! .H .X/; d D 0/ which induces the identity on cohomology. One can always define a map of cochain complexes .H .X/; d D 0/ ! A .X/

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14 Examples and Computations

which induces the identity on cohomology by choosing (linearly) closed form representatives for each cohomology class. Usually, this map will not be multiplicative. If it is possible to choose this map to be multiplicative, then .H .X/; d D 0/ and A .X/ have the same minimal model. Thus, this gives algebraic topological conditions on a space which must be satisfied if there is to be a multiplicative mapping from the cohomology to the forms inducing the identity on cohomology. If X is a closed Riemannian manifold, then there is a canonical map .H .X/; d D 0/ ! A .X/ which assigns to each cohomology class its unique harmonic representative. From the above discussion, we see that for X to admit a Riemannian metric in which the wedge product of harmonic forms is harmonic, it must be the case that X is formal (over R). There is one class of Riemannian manifolds in which wedge product of harmonic forms is harmonic. These are the Riemannian locally symmetric spaces.

14.5 The Third Homotopy Group of a Simply Connected Space Let A be a DGA with H1 .A / D 0. The first stage in constructing the minimal model of A is the polynomial algebra on H2 D H2 .A /. We denote this DGA by fPŒH2 ; d D 0g. The next stage is created by tensoring in an exterior algebra on the relative 4th -cohomology group H4 .PŒH2 ; A /. If X is a simply connected space, then  3 .X/ ˝ Q is dual to H4 .PŒH2 ; A .X//. The long exact sequence of the pair .PŒH2 ; A .X// leads to a long exact sequence: h

[

0 ! H3 .X/ ! Hom. 3 .X/; Q/ ! P2 ŒH2 .X/ ! H4 .X/ where h is dual to the Hurewicz homomorphism, P2 ŒH2 .X/ is the vector space of quadratic polynomials in H2 .X/, and the map to H4 .X/ is the cup product mapping. If fW S3 ! X is a continuous mapping, then we can deform it until it becomes simplicial. At this point, it induces a map f W A .X/ ! A .S3 /, and hence, OfW MX ! M There is the evaluation map M

S3

S3

D ƒ.e/:

! Q given by sending ¨3 2 M

S3

to

R S3

¨3 2

Q. As we said in part B of this chapter, the map  3 .x/ ! ŒI3 .M.X/ /  defined by R sending Œf to Œ ı Of 2 I3 .MX / gives the duality between  3 .X/ ˝ Q and I3 .MX /. Suppose ¨1 ; : : : ; ¨k 2 H2 .X/ is a basis and ¨ Q i is a closed 2-form in A .X/ representing ¨i . Suppose .†aij ¨i ^ ¨j ; ˜/ 2 P2 ŒH2 ˚ A3 .X/ is a relative cocycle (i.e., Q i ) and that fW S3 ! X. We give a formula for the value of P d˜ D † aij ¨Q i ^ ¨ . ij aij ¨i ^ ¨j ; ˜/ 2 H4 .P2 ŒH2 ; A .X// D I3 .MX / on Œf 2  3 .X/. To do this, we must deform

14.5 The Third Homotopy Group of a Simply Connected Space

133

f

X

MX ! A .X/ ! A .S3 / by homotopy to factor through M

S3

D ƒ.e/. Define HW PŒH2 ! A .S3 / ˝ .t; dt/

as follows: For each ¨i 2 H2 , f ¨ Q i D d i for some 1-form i 2 A .S3 /. Define H.¨i / D f ¨ Q i ˝ 1  d. i ˝ t/: Let OfW MX .2/ ! ƒ.e/ be trivial, except in degree zero. At this point, we have f

A .X/ ! A .S3 / x x ? ?¡ 3 ¡X ? ?S MX .2/ ! .e/ Of

a homotopy commutative diagram. We wish to extend this to a homotopy Pcommutative diagram on MX .3/. The map OfW MX .3/ ! ƒ.e/ must send the class a ¨ ƒ ¨j ; / to some multiple –  e. (We R ij ij i assume here that e is chosen so that S3 .e// D 1:) If the given homotopy is to S3

extend over MX .3/, then – must be equal to Z D Z D

f ˜ 

D

S3

f ˜  †aij

Z

1Z 0

Z H.†aij ¨i ^ ¨j / S3

    f ¨ Q i ˝ 1  d. i ˝ t/ ^ f ¨ Q j ˝ 1  d. j ˝ t/

S3

f ˜

S3



1 0

s3

Z

Z

X

Z

1

Z

aij 0

.f ¨ Q i 1  d i ˝ t C i ˝ dt/ ^ .f ¨ Q j 1  d j ˝ t C j ˝ dt/:

S3

The only terms in the second part which will contribute to the integral are f ¨Q i ^ j ˝ dt  d i ^ j ˝ tdt C i ^ f ¨ Q j ˝ dt  i ^ d j ˝tdt Q j dt: D f ¨Q i ^ j ˝ dt  d i ^ j ˝ tdt C i ^ f ¨

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14 Examples and Computations

Thus, Z



f ˜

D S3

X

X

D

4aij . 1 2

S3

f ˜ 

f ¨ Q i ^ j C i ^ f ¨ Qj

aij S3

2

Z

3

d i ^ j / C i ^ d j 5

S3

Z

Z

X

2 4aij .

Z

3

1 f ¨ Q i ^ j /  d i ^ j C i ^ f ¨ Qj 2

s3

1  i ^ d j 5 2 Z D S3

2 1 f ˜  † 4aij . 2

Z

3 f ¨ Q i ^ j / C i ^ f ¨ Q j5

S3

This formula is a generalization of Whitehead’s formula for the Hopf invariant. Whitehead showed that if fW S3 ! S2 is a C1 map, then the Hopf invariant H.f/ is given by Z

f ¨ ^

S3

where

R

¨ D 1 and d D f ¨. Here,  3 .S2 / Š Z and the Hopf invariant of an

S2

element is its class under this isomorphism.

14.6 Homotopy Theory of Certain 4-Dimensional Complexes Let X be a space which is homotopy equivalent to ._TiD1 S2 / [® e4 , where ®W S3 ! _TiD1 S2 . Any simply connected 4-manifold is so represented. The homotopy type of such a space is determined by Œ® 2  3 ._TiD1 S2 /. This group is a free abelian group with generators the Whitehead products Œxi ; xj ; i  j where xi is the identity map of S2 onto the ith factor in the wedge. There is another description of the element [®] in terms of the cohomology ring of X. This ring is determined by the symmetric pairing: Š

H2 .X/ ˝ H2 .X/ ! H4 .X/ ! Z;

14.7 Q-Homotopy Type of BUn and Un

135

where the last map is evaluation on the fundamental cycle. Since H2 .X/ has a natural basis dual to the spheres, the cup product mapping is given by a symmetric integral matrix (ij ) It turns out that Œ® D †ij ij Œxi ; xj . This shows that such homotopy types are classified by equivalence classes of symmetric bilinear pairings. The condition that X satisfy Poincaré duality is that the pairing be nonsingular over Z. It is unknown which pairings arise from closed simply connected smooth 4-manifolds but for closed, simply connected topological 4-manifolds all pairings satisfying Poincaré duality over the integers occur; see [5]. The diffeomorphism classification of simply connected 4-manifolds is much more complicated than the classification up to homotopy equivalence, see [4]. Classifying such complexes up to rational homotopy equivalence is the same as classifying the symmetric pairings up to rational equivalence (including a change of scale in the value group). Thus, if X and Y are two simplicial complexes of this form, then MX and MY are isomorphic if and only if the pairings are rationally equivalent. The algebras of piecewise C1 forms on X and Y have isomorphic minimal models if and only if the pairings for X and Y are equivalent over the reals (again including an automorphism of the value group). Two symmetric forms are equivalent over the reals if and only if the pairings are of the same rank, have the same dimensional null space, and have the same signature up to sign.

14.7 Q-Homotopy Type of BUn and Un The Grassmannian of n-dimensional complex linear subspaces in C1 is the classifying space for complex n-plane bundles and is denoted BUn . (Thus, BUn D limN!1 G.n; N/). Recall that H .BUn I Z/ Š ZŒc1 ; c2 ; : : : ; cn where c` 2 H2` .BU˜ I Z/ is the `th Chern class of the universal vector bundle. This gives a map f

BUn ! …n`D0 K.Z; 2`/ which induces an isomorphism on Q-cohomology. Since .K.Z; 2`//.0/ D K.Q; 2`/, it follows that f.0/

.BUn /.0/ !

n Y `D1

is a homotopy equivalence.

K.Q; 2`/

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14 Examples and Computations

Remark. This gives a proof of the following rational isomorphisms:  2i .BUn / ˝ Q Š Q for 2i  2n  2iC1 .BUn / ˝ Q D 0: for 2i C 1  2n This is the unstable version of the Bott periodicity theorem over Q. In fact, if BU D lim Gr.n; 2n/ n!1

then H .BU; Z/ Š ZŒc1 ; c2 ; c3 ; : : : ; . The same argument as above gives that BU.0/ …1 `D1 K.Q; 2`/: Thus,  2i .BU/ ˝ Q Q  2iC1 .BU/ ˝ Q D 0 which is the Bott periodicity over Q. Bott periodicity is the result  2i .BU/ Z  2iC1 .BU/ D 0 together with the fact that cn W  2n .BU/ ! Z .n  1/Š is an isomorphism. This divisibility property seems to be closely related to deducing Bott periodicity over the integers from Bott periodicity over the rationals. Let Un be the unitary group. Recall that H .Un ; Z/ Š Zfx1 ; x3 ; : : : ; x2n1 g is an exterior algebra with odd-dimensional generators. From this, we deduce that the Q-homotopy type: 2`1 .Un /.0/ …n`D1 S.0/ :

Thus, over Q, Un looks like a product of odd spheres. Moreover, we see that  2i1 .Un / ˝ Q Š Q;  2i .Un / ˝ Q D 0

for 1  i  n

14.8 Products

137

The stable Z version for U D lim Un!1 is  2i1 .U/ Š Z  2i .U/ D 0: This is equivalent to the Bott periodicity above using the exact homotopy sequence of the fibration Un

/ S.n; 2n/  Gr.n; 2n/

for n large compared to the dimension, 2i or 2i  1, in which we are computing the homotopy groups . Here, S.n; 2n/ is the Stiefel manifold of n-frames in 2n-space and Gr.n; 2n/ is the Grassmannian of n-planes in 2n-space — cf, the exercises

14.8 Products Let A and B  be DGAs. The tensor product A ˝ B  naturally receives the structure of a DGA. If MA and MB are minimal models for A and B  , then MA ˝ MB is a minimal model for A ˝ B  . If X and Y are C1 manifolds, then there is a map AC1 .X/ ˝ A .Y/ ! AC1 .X  Y/ which, by the Künneth theorem, induces an isomorphism on cohomology. If X and Y are simplicial complexes and .X  Y/0 is a triangulation of the product cell complex, then there is a map A .X/ ˝ A .Y/ ! A ..X  Y/0 / which (again by the Künneth theorem) induces an isomorphism on cohomology. It follows that MXY Š MX ˝ MY . This should be viewed as a generalization of the Künneth theorem. It includes the Künneth theorem (by taking cohomology). It also includes the rational or real form of the result that  i .XY/ Š  i .X/˚ i .Y/, since I.MX ˝ MY / Š I.MX / ˚ I.MY /.

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14 Examples and Computations

14.9 Massey Products Given a space X and classes ’ 2 Hp .X/; “ 2 Hq .X/; ” 2 Hr .X/ that satisfy ’ [ “ D 0 in HpCq .X/;

“ [ ” D 0 in HqCr .X/;

there is defined the Massey triple product < ’; “; ” >2 HpCqCr1 .X/=.’  HqCr1 .X/ C ”  HpCq1 .X//: Using differential forms a, b, c to represent the classes ’; “; ”, respectively, we write a ^ b D dg;

b ^ c D dh

where g, h are forms of degrees p C q  1; q C r  1. The p C q C r  1 form k D g ^ c C .1/p1 a ^ h is closed and its cohomology class is well-defined modulo ’  HqCr1 .X/ C ”  HpCq1 .X/; as may be directly verified by making different choices in the above recipe. The Massey triple product is then the class in the quotient HPCqCr1 .X/=.’HqCr1 .X/C ”  HpCq1 .X//. Now, we consider the special case of compact Kähler manifolds. We recall the following: Definition. A Hodge structure of weight m is given by Q-vector space HQ together with a Hodge decomposition HC D ˚pCqDm Hp; q Hp;q D H

q; p

of its complexification HC WD HQ ˝ C D HQ ˝Q C. Suppose now that M is a compact K˝ahler manifold (cf. Sect. 7 in Chap. 0 of [8] for the relevant definitions and notations). If we set Ap;q .M/ D vector space of complex-valued C1 .p; q/ forms on M, and Hp;q .M/ D f® 2 Ap;q .M/ W d® D 0g=dApCq1 .M/ \ Ap;q .M/;

14.9 Massey Products

139

then it is a basic fact that the subspaces Hp;q .M/  Hm .M/;

pCqDm

define a Hodge decomposition on Hm .M/ D Hm .M; Q/ ˝ C. Consequently, the cohomology Hm .M/ of a compact K˝ahler manifold has a functorial Hodge structure of weight m (functorial means the obvious thing, to be explained momentarily). Actually, this statement is a consequence of the following lemma ([8], page 149) about forms on compact Kähler manifolds: Lemma 14.1 (The principle of two types). On a compact Kähler manifold M, suppose that ® 2 Ap;q .M/ is a form such that ® D d˜;

˜ 2 ApCq1 .M/:

Then, there are forms ˜0 and ˜00 such that ® D d˜0 and ˜0 2 Ap1;q .M/ and ® D d˜00 and ˜00 2 Ap;q1 .M/: We shall show that the lemma implies that all Massey triple products are zero on a compact Kähler manifold. For this, we remark first that the cup product Hm .M/ ˝ Hn .M/ ! HmCn .M/ is compatible with the Hodge structures, meaning that Hp;q .M/ ˝ Hr;s .M/ ! HpCr;qCs .M/: Thus, in forming the Massey Product < ’; “; ” >, it suffices to assume that ’; “, and ” are themselves homogeneous with respect to the Hodge decomposition; say, ’ 2 Ht;s .X/; “ 2 Hi;j .X/, and ” 2 Hu;v .X/. We shall construct two closed differential forms ®1 and ®2 which are cohomologous and which represent < ’; “; ” > so that ®1 2 AtCiCu; sCjCv1 .X/ and ®2 2 AtCiCu1; sCjCv .X/. In light of the fact that the Hodge decomposition is a direct sum decomposition, this will prove that ®1 and ®2 are exact and hence that < ’; “; ” >D 0. Choose closed forms a 2 At;s .X/; b 2 Ai;j .X/, and c 2 Au;v .X/ representing ’; “, and ”. We know that a ^ b is contained in AtCi;jCs .X/ and that it is exact. Choose y 2 AtCi;jCs1 .X/ and y0 2 AtCi1;jCs .X/ so that dy D dy0 D a ^ b. The form y  y0 is closed. Using the Hodge decomposition in degree t C s C i C j  1, we can vary y and y0 by closed forms preserving the bi-graded type so that y  y0

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14 Examples and Computations

becomes exact. Similarly, we choose z 2 AiCu;jCv1 .X/ and z0 2 AiCu1;jCv .X/ so that dz D dz0 D b ^ c and z  z0 is exact. The forms y ^ c C .1/tCs1 a ^ z and y0 ^ c C .1/tCs1 a ^ z0 both represent the Massey product < ’; “; ” >. The first lies in AsCiCu; tCjCv1 .X/ and the second lies in AsCiCu1;tCjCv .X/. Furthermore, they are cohomologous. This completes the proof that the Massey products vanish on a compact Kähler manifold. Remark. The above is a special case of the theorem proved in [3] that the rational homotopy type of a compact Kähler manifold is a formal consequence of the cohomology ring (cf. Sect. 14.4 above for the definition of formal). This result, in turn, is generalized to noncompact algebraic varieties in [18].

Chapter 15

Functorality

In Chap. 12, we made explicit the duality between minimal DGAs over the rationals and rational Postnikov towers. We showed that the minimal model of the p.l. forms of a simplicial complex, A .X/, is dual to the rational Postnikov tower of X. In this chapter, we discuss the functorial properties of this duality.

15.1 The Functorial Correspondence Let B and C be simply connected simplicial complexes and let fW B ! C be a continuous mapping. For each vertex v of the triangulation of C, let sVt.v/ be the open star of v. This is the union of all open simplices of C which have v as a vertex. This gives an open cover of C by fVst.v/gv2vertices.C/ . If B0 is an subdivision of B so that each simplex of B0 lies in f1 .Vst.v// for some vertex v 2 C, then there is a simplicial map ®W B0 ! C which is homotopic to f. Since the points with rational barycentric coordinates are dense in a simplex, we can always choose B0 to be a rational subdivision. The inclusion IW B0 ! B induces a restriction map on p.l. forms I

A .B/ ,! A .B0 /. Suppose that ¡B W MB ! A .B/ and ¡C W MC ! A .C/ are given minimal models. The composition I ı ¡B W MB ! A .B0 / is also a minimal model by Theorem 11.5. There is a map Of0 W MC ! MB , unique up to homotopy, so that .f0 /

A .C/ ! A .B0 / x x ? ?¡C ?I ı¡B ? MC

b f0 !

MB

commutes up to homotopy.

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__15, © Springer Science+Business Media New York 2013

141

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15 Functorality

Theorem 15.1. The association f ! fO0 defines a function ŒB; C ! ŒMC ; MB that preserves compositions and identities. Proof. Let us first show that the homotopy class of OfW MC ! MB depends only on the homotopy class of fW B ! C. Suppose we have two simplicial approximations f0 W B0 ! C and f00 W B00 ! C where both B0 and B00 are rational subdivisions of B. Form the linear cell complex B  I. There is a rational subdivision of the product cell structure on B  I which agrees with B0 at one end and B00 at the other. If we choose this subdivision fine enough, then there is a simplicial map HW B  I ! C ` that extends f0 f00 on the ends. This leads to a commutative diagram:

There is ®W MC ! MB such that   ı ¡B ı ® is homotopic to H ı ¡C . Thus, ¡B ı ® is homotopic to .f0 / ı ¡C , and ¡B00 ı ® is homotopic to .f00 / ı ¡C . Thus, by 10.8, ® is homotopic to Of0 and Of00 . This proves that ŒB; C ! ŒMC ; MB is well defined. Clearly, ŒB; B ! ŒMB ; MB sends the class of the identity to the class of the identity. Lastly, suppose

is a diagram in which each square homotopy commutes. Since homotopy is an equivalence relation of the maps from a minimal DGA, ¡B ıOf ı gO Š f ı ¡C ı gO Š f ı gı ¡D : Thus, Of ı gO is the map associated to f ı g . This proves that compositions are preserved.  Lemma 15.2. If fW M ! N is a map between simply connected, minimal DGAs, then it induces a map on indecomposables I.f/W I.M/ ! I.N/. As we vary f by

15.1 The Functorial Correspondence

143

homotopy, the induced map on indecomposables remains unchanged. Thus, there is a well-defined map ŒM; N ! Hom.I.M/; I.N//. Proof. Let HW M ! N ˝ .t; dt/ be a homotopy. Since M and N are minimal and have no generators in degree 1, it follows that H induces a map: HW M ^ M ! .N ^ N/ ˝ .t; dt/: For any v 2 M, we have dv 2 M ^ M. We write H.v/ D v0 C

X

vi ˝ ti C wi ˝ ti1 dt:

i1

Then, dH.v/ D H.dv/ D dv0 C

X

dvi ˝ ti C .˙ivi C dwi / ˝ ti1 dt:

i1

This expression and dwi ˝ti1 dt are each contained in .N ^ N/˝.t; dt/. Thus, so is P i1 dt. It follows that for every i  1, we have vi 2 N ^ N. Consequently, i1 ivi ˝ t the maps induced by H at t D 0 and t D 1 from I.M/ to I.N/ are the same.  As an example of Theorem 15.1, let B D Sn . We have a map: ŒSn ; X D  n .X/ ! ŒMX ; MSn Š Hom.I.MX /; I.MSn //: There is an isomorphism In .MSn / ! Q given by sending Z ’2M

Sn

to Sn

’ 2 Q:

Thus, we have a map: œX

 n .X/ ˝ Q ! Hom.In .MX /; Q/ D ŒIn .MX /  : This map agrees with the identification given in Chap. 12. If fW X ! Y is given and OfW MY ! MX is an associated map on minimal models, then the following diagram commutes:  n .X/ ˝ Q ! In .MX / œX ? ? ?f ˝Id ? n O y Q yI .f /  n .y/ ˝ Q ! In .MY / œY

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This follows immediately from the fact that the correspondence in 15.1 is natural and preserves compositions. We shall show later in this chapter that X is an isomorphism for all simply connected X.

15.2 Bijectivity of Homotopy Classes of Maps We begin now the proof that if C is a local space, then the map ŒB; C ! ŒMC ; MB given in Theorem 15.1 is a bijection. This is proved by induction on the stages of the Postnikov tower of C. The inductive step requires a detailed analysis of a long exact sequence. It is this analysis that the next 5 propositions develop. Proposition 15.3. Let pW E ! C0 be a simplicial map with homotopy theoretic fiber K. ; n/. Suppose  1 .C0 / D feg. Suppose that V D   ˝ Q is a finite-dimensional rational vector space. Let ¡C0 W MC0 ! A .C0 / be a minimal model. Let M0 D MC0 ˝d ƒ.V /n and suppose there is a map ¡W M0 ! A .E/ such that: (1) ¡jMC0 D p ı ¡C0 . (2) ¡ induces an isomorphism on rational cohomology. Then, there is a commutative diagram:

ŒX; C0

obst. to

/ HnC1 .XI  /

lifting ˝Q

 nC1 H .XI V/  ŒMC0 ; MX

¡ X

obst. to

 / HnC1 .MX I V/:

extending

Proof. The obstruction to lifting fW X ! C0 is f .k/ where k 2 HnC1 .C0 I  / is the k-invariant of the fibration. The obstruction to extending ®W MC0 ! MX over M0 is ® .Œd / 2 HnC1 .MX I V/. As we saw in 11.5, Œd is the class of k ˝ 1 2 HnC1 .C0 I  / ˝ Q under the identification ¡C0 W H .MC0 / ! H .C0 /. If f 

and ® correspond, then f D ® under the identifications ¡C0 W H .MC0 / ! H .C0 /. 

and ¡C0 W H .MX / ! H .X/. The proposition is immediate from these facts.



Proposition 15.4. Let pW CnC1 ! Cn be a principal fibration with fiber K. ; nC1/. For any CW-complex X, there is an exact sequence: p



HnC1 .XI  / ! ŒX; CnC1 !ŒX; Cn ! HnC2 .XI  /:

15.2 Bijectivity of Homotopy Classes of Maps

145

This means that Im.p / D  1 .0/, and that there is an action of HnC1 .XI  / on ŒX; CnC1 so that two maps are in the same orbit if and only if they have the same image in ŒX; Cn . The isotropy group of this action at f 2 ŒX; CnC1 is the subgroup ® Hf  HnC1 .XI  / consisting of all elements ’ for which there is a map X  S1 ! Cn with ®jXp D pıf and the obstruction to lifting ® to CnC1 is ’˝š 2 HnC2 .XS1 I  / where š 2 H1 .S1 / is the generator. Proof. Since  is the obstruction to lifting, clearly,  1 .0/ D Im.p /. Let us define the action of HnC1 .XI  / on ŒX; CnC1 . Given ’ 2 HnC1 .XI  / D HnC2 .X  I; X  @II  / and fW X ! CnC1 , define a map ’  fW X ! CnC1 such that pı’f D pıf and the obstruction to finding a homotopy of liftings of pıf connecting f to ’  f is ’. One checks easily that this is an action of HnC1 .XI  / on ŒX; CnC1 . Clearly, p ı f D p ı g if and only if Œf D ’  Œg for some ’ 2 HnC1 .XI  /. Suppose ’  f D f in ŒX; CnC1 . Let HW X  I ! CnC1 be a homotopy form f to ’f. Since p ı f D p ı ’f, the composition p ı H defines a map ®W X  S1 ! Cn . Clearly, ®jX  f0g D p ı f. We claim that the obstruction to lifting ® to CnC1 is ’ ˝ š 2 HnC2 .X  S1 I  /. Since we have a lifting HW X  I ! CnC1 , the obstruction to lifting ® as a map of X  S1 lies in HnC1 .XI  / ˝ H1 .S1 I Z/  HnC2 .X  S1 I  /. It is easily identified with ’ ˝ .  There is an analogous sequence from Hirsch extensions of DGAs which we construct now. Let M0 D M ˝d ƒ.V /nC1 be a Hirsch extension, with V being a finite-dimensional rational vector space. Let A be a DGA. Proposition 15.5. There is an exact sequence: i



HnC1 .AI V/ ! ŒM0 ; A !ŒM; A ! HnC2 .AI V/: This means that  1 .0/ D Im i and that the group HnC1 .AI V/ acts on ŒM0 ; A so that two elements are in the same orbit if and only if they have the same image in ŒM; A . The isotropy group for this action at ® 2 ŒM0 ; A is the subgroup of all ’ 2 HnC1 .A; V/ for which there is a map §W M ! A ˝ ƒ.e/, with pA ı § D ®jM (where pA is the projection to A obtained by setting e D 0) with the obstruction to extending § over M0 being ’ ˝ e 2 HnC2 .A ˝ ƒ.e/I V/. Proof. Since  is the obstruction to extending over M0 ,  1 .0/ D Im i . Let us define the action of HnC1 .AI V/ D Hom.V ; HnC1 .A// on ŒM0 ; A . Given ’W V ! HnC1 .A/ and ®W M0 ! A, we define ’  ®. On M  M0 , ’  ® D ®. Choose a lifting ’W Q V ! fclosed forms of AnC1 g for ’. Define ’Q  ®W V ! A by ’  ®.v/ D ®.v/ C ’.v/. Q One sees easily that this defines a map of DGAs. The homotopy class of ’® Q depends only on the homotopy class of ® and the cohomology class of ’. Q Let us show that ®1 ; ®2 have the same image under i if and only if there is ’ 2 HnC1 .A; V/ so that ’®1 D ®2 . The “if” direction is immediate from the definition of the action. Suppose, conversely, that we have maps ®1 and ®2 so that ®1 jM is homotopic to ®2 jM. Consider the diagram:

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15 Functorality

M ? ? y

®1 jM

! A ? ?D y

M ˝d ƒ.V / ! A ®2

It is homotopy commutative. Since H .A; A/ D 0, there is no obstruction to finding ®02 W M0 ! A which is homotopic to ®2 and such that ®02 jM D ®1 jM. Thus, we may assume that ®1 jM D ®2 jM. For each v 2 V , consider ®1 .v/  ®2 .v/ 2 AnC1 . Since dv 2 M, we have d®1 .v/ D ®1 .dv/ D ®2 .dv/ D d®2 .v/. Hence, ®1 .v/  ®2 .v/ is closed. If ’W V ! HnC1 .A/ is the resulting homomorphism, then one sees easily that ’ ı ®2 D ®1 in ŒM0 ; A . Lastly, we need to understand the isotopy groups of this action. Let ’W Q V ! {closed forms of AnC1 g be a representative for ’. Let ®W M0 ! ’. If ’Q ı ® D ® in ŒM0 ; A , then there is a homotopy H W M0 ! A ˝ .t; dt/ from ® to ’Q ı ®. If we restrict H to M, it becomes a homotopy from ®jM to ®jM. Let K  .t; dt/ be all forms † ai ti C bi ti dt such that †i1 ai D 0. This is a subDGA. The algebra A ˝ K  A ˝ .t; dt/ is the kernel of r1  r0 W A ˝ .t; dt/ ! A where ri is restriction at t D i.  Lemma 15.6. ƒ.dt/  K induces an isomorphism on cohomology. This is a straightforward computation. With this lemma in place, let us return to the proof of the proposition. As a consequence of the lemma, we have that jW A ˝ ƒ.e/ ! A ˝ K, defined by e ! dt, induces an isomorphism on cohomology. Consider the diagram: A ˝ ƒ.e/ GG p v; GG A v GG j v GG v G#  v H r / A˝K / A M H0

where rW A ˝ K ! A is restriction at t D 0, and pA is defined by setting e D 0. The obstructions to lifting H up to homotopy vanish. Thus, there is H0 W M ! A ˝ ƒ.e/ so that pA ı H0 D ® and so that j ı H0 is homotopic to H. There is no obstruction to extending H0 to a homotopy J0 W M0 ! A ˝ .t; dt/ such that: (1) J0 jM D H0 and (2) J0 is a homotopy from ® to ’Q ı ®. We claim that the obstruction to extending H0 to a map M0 ! A ˝ ƒ.e/ is exactly the homomorphism v ! Œ’.v/ Q ˝ e 2 HnC2 .A ˝ ƒ.e//. To see this, note that the obstruction to extending H0 sends v 2 V to the class H0 .dv/ 2 HnC2 .A ˝ ƒ.e//.

15.2 Bijectivity of Homotopy Classes of Maps

147

Since J0 W M0 ! A ˝ .t; dt/ is an extension of j ı H0 jM, we have H0 .dv/ D P J0 .dv/ D 0 0 0 dJ .v/. Thus, j ı H .dv/ is of the form av ˝ 1 C bv ˝ dt. Suppose J .v/ D av;i ˝ ti C bv;i ˝ ti dt. We see that X

dav;i ˝ ti C .1/deg.av;i / iav;i ˝ ti1 dt C dbv;i ˝ ti dt D av ˝ 1 C bv ˝ dt:

This implies that av D dav;0 and hence that av is exact. Since Z

tD1

d tD0

Z H.v/ C

tD1 tD0

H.dv/ D H jtD1 .v/  HjtD0 .v/ D ’.v/ Q

we have Z Q d .1/deg.bv / bv D ’.v/

tD1

H.v/ :

tD0

Q ˝ e/. This proves that, Thus, in cohomology, the class of H0 .dv/ is .1/deg.bv / .’.v/ up to sign, the obstruction to extending H0 is the homomorphism v ! Œ’.v/ Q ˝ e 2 HnC2 .A ˝ ƒ.e//. This completes the proof of Proposition 15.5. Let pW CnC1 ! Cn be a simplicial model for a principal fibration with fiber K.V; n C 1/, where V is a finite-dimensional rational vector space. Let ®W M ! A .Cn / be a minimal model. Let M0 D M ˝d ƒ.V / and let ®0 W M0 ! A .CnC1 / be a map extending p ı ® and inducing an isomorphism on cohomology. Let X be a simplicial complex, and let ¡X W MX ! A .X/ be a minimal model. There is a commutative diagram of exact sequences: HnC1 .XI V/ ! ŒX; CnC1 ! ŒX; Cn ! HnC2 .XI V/ ? ? ? ? ?¡ ? ? ?¡ yX y y yX HnC1 .MX I V/ ! ŒM0 ; MX ! ŒM; MX ! HnC2 .MX I V/ We check now that the action of HnC1 .XI V/ on ŒX; CnC1 and action of HnC1 .MX I V/ on ŒM0 ; MX are compatible. Suppose ’ 2 HnC1 .X; V/ and fW X ! CnC1 are given. The maps ®0

f

®0

.’f/

M0 ! A .CnC1 / ! A .X/ and M0 ! A .CnC1 / ! A .X/

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15 Functorality

are homotopic on M. The obstruction to extending the homotopy over M0 is exactly ’ 2 HnC1 .XI V/. Thus, if ®f and ®’f are maps M0 ! MX representing f and ’f, then they are homotopic on M, and the obstruction to extending the homotopy over M0 is ’. Hence, ®’f D ’ ı ®f in ŒM0 ; MX . This completes the proof of Proposition 15.5 Theorem 15.7. Let C be a local space with homotopy groups that are finitedimensional rational vector spaces, and let X be a simplicial complex. The functor ŒX; C ! ŒMC ; MX is a bijection. Proof. The proof is by induction on the Postnikov tower of C. We show that ŒX; Cn ! ŒMC .n/; MX is a bijection for all n, where Cn is the nth stage in the Postnikov tower for C and MC .n/ is the subalgebra of MC generated in degrees  n. The inductive step follows by the 5 lemma from the commutative diagram ¡ X

above, together with the fact that the identification HnC1 .X; V/ ! HnC1 .MX I V/ induces group isomorphisms on the corresponding isotropy groups.  Corollary 15.8. The functorial mapping œX W  n .X/ ˝ Q ! In .MX / is an isomorphism for all simply connected spaces X with finite-dimensional rational cohomology in every degree. Proof. We show that œX is onto. Let ®: In .MX / ! Q be given. There is a map of DGA s ®W O MX ! MSn which realizes ®. O It is clear how to define ®O in degrees  n. In the higher degrees, there are no obstructions to extending ®O since H .MSn / D 0 for > n. Once we have ®, O Theorem 15.7 tells us there is ®W Sn ! X.0/ which realizes ®. O Thus, œX.0/ W  n .X.0/ / ! In .MX / is onto. But  n .X.0/ / D  n .X/ ˝ Q, and by naturality œX.0/ D œX ˝ IdQ . To see that œX is 1-1, suppose œX .f/ D 0. This implies that Of W MX ! MSn induces the zero map on indecomposables in degree n. Elementary obstruction theory implies that Of itself is homotopic to zero. Hence, by 15.7, the composition f

Sn ! X  X.0/ is homotopically trivial. This means that Œf 2  n .X/ is trivial in  n .X.0/ / D  n .X/ ˝ Q.  Corollary 15.9. Let X and Y be simply connected spaces, each with finitedimensional rational cohomology in every degree, and let fW X ! Y a continuous map. From MX and MY , we can construct spaces X.0/ and Y.0/ which are localizations for X and Y. From Of 2 ŒMY ; MX , we can construct a map f.0/ W X.0/ ! Y.0/ which is a representative, up to homotopy, of the localization of f.

15.3 Equivalence of Categories All of the results of this section can be formalized by saying that a functor between two categories is an equivalence of categories.

15.3 Equivalence of Categories

149

Definition. Let C1 and C2 be categories and let FW C1 ! C2 be a functor. F is said to be an equivalence of categories if the following hold: 1. Every object B of C2 is isomorphic to an object F.A/ for some object A of C1 . 2. For all objects A1 and A2 of C1 the map induced by F HomC1 .A1 ; A2 / ! HomC2 .F .A1 /; F.A2 // is a bijection. Given a category C and a set of morphisms S of C that include all isomorphisms and are closed under compositions, we define the localization of C at S by formally inverting all morphisms in S. Thus, the objects of the new category are the same as the objects of C, but the morphisms from A to B in the new category consist of a string of morphisms: A D A0 ! A1

 A2    ! Ak D B;

where all the arrows to the ‘left’ are elements of S. If successive arrows in the same direction can be replaced by the composite arrow, then by definition the morphism is unchanged. The first category with which we shall deal is the rational homotopy category of simply connected spaces of finite type. We begin with the category whose objects are simply connected, simplicial complexes whose rational homology groups are finite dimensional in each degree and whose morphisms are homotopy classes of maps. Then, we localize this category by localizing at those maps that induce isomorphisms on rational homology or equivalently on rational homotopy. We denote this category by HOM.0/ . The other category is the category of simply connected DGAs over Q with finite-dimensional cohomology in each degree, localized at the collection of DGA maps that induce isomorphisms on cohomology. This category is denoted DGA.0/ We have a functor “p.l. forms” from HOM.0/ to DGA.0/ . The existence of a minimal model shows that every object of DGA.0/ is isomorphic to a minimal DGA. Associated to a minimal DGA, there is a rational Postnikov tower or equivalently a simplicial complex whose p.l. forms have a minimal model isomorphic to the given one. This shows that every object of DGA.0/ is isomorphic to the DGA of p.l. forms on some simplicial complex. The fact that the p.l. forms functor induces a bijection on the sets of morphisms follows easily from Theorem 15.7. This shows that the functor of p.l. forms induces an isomorphism from HOM.0/ to DGA.0/ . That is to say, the rational homotopy category of simply connected spaces of finite type is isomorphic to the homotopy category of simply connected DGAs over Q with finite-dimensional cohomology in each degree, and the isomorphism between these rational homotopy categories is given by associating to a simplicial complex is p.l. forms.

Chapter 16

The Hirsch Lemma

The purpose of this chapter is to prove Theorem 12.1. Namely, we shall show the following. Let E ! B be a principal K. ; n/-fibration, and let f0 W E0 ! B0 be a simplicial model for it. Given a minimal model M for the p.l. forms on B0 , the Hirsch extension of M dual to the fibration is a minimal for the p.l. forms on E0 .

16.1 The Cubical Complex and Cubical Forms Definition. Let X beQ a topological space. For each q  0, let Iq be the unit cube q q q in R , namely, I D i D1 Œ0; 1 . For any 0  i  q and 2 f0; 1g, we define the q .i; /-face of I denoted f i .Iq / the subset where it h coordinate is equal to . This face is identified with Iq1 by the map f i W .x1 ; : : : ; xq1 / 7! .x1 ; : : : ; xi1 ; ; xi ; : : : ; xq1 /: We define the boundary of Iq by @Iq D

q 1 X X

.1/i .1/ f i ;

D0 iD1

considered as a formal linear combination of maps of the Iq1 -cube into Iq . A singular q-cube in X is a continuous map Iq ! X. A singular cube in X is a singular q-cube in X for some 0  q < 1. For any q  0, we denote the set of singular q-cubes in X by Cq .X/. Let ¢ be a singular q-cube in X. For any 0  i  q and 2 f0; 1g, we define the .i; /-face of ¢, denoted f i .¢/ to be the composition f i

¢

Iq1 ! Iq  ! X: Lastly, we say that a q-cube ¢ in X is degenerate if ¢ factors through the map pq

Iq ! Iq1 defined by pq .x1 ; : : : ; xq / D .x1 ; : : : ; xq1 /. (Notice that degeneracies P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__16, © Springer Science+Business Media New York 2013

151

152

16 The Hirsch Lemma

O are defined only with respect to the last coordinate.) We denote by Q.X/ the chain complex whose qt h chain group is the free abelian group generated by the set Cq .X/ and with the boundary map given by @¢ D

q 1 X X

.1/i .1/ f i .¢/:

D0 iD1

It is easy to see that the boundary of a degenerate cube is a linear combination of degenerate cubes of one lower dimension, so that the degenerate cubes generate a O subcomplex. We denote by Q.X/ the quotient chain complex Q.X/ by the subchain complex generated by the degenerate cubes. Lemma 16.1. The homology of the chain complex Q.X/ is naturally identified with the singular homology of X. Proof. Just as in the case of singular homology, one shows that all the axioms for homology are verified. [Excision requires subdivision of a cube into smaller cubes. The computation of the homology of a point is straightforward, but it is worth noting O is isomorphic to Z in every degree that the homology of a point computed using Q since the boundary of a cube has an even number of faces, half counted with a positive sign and half counted with a negative sign. Thus, it is crucial that we set degenerate cubes equal to zero in order to compute singular homology.]  O q .X/j the disjoint union over ¢ 2 Given a topological space X, denote by jQ O q .X/j is denoted .x1 ; : : : ; xq I ¢/, with the xi being in the Cq .X/ of Iq . A point of jQ interval Œ0; 1 and ¢ being an element of Cq .X/. We define O jQ.X/j D

1 a

O q .X/j: jQ

qD0

O We introduce an equivalence relation on jQ.X/j as follows. It is generated by two elementary equivalences: 1. For ¢ 2 Cq .X/ and any .x1 : : : ; xq1 / 2 Iq1 , we identify .x1 ; : : : ; xq1 I f i .¢// with .x1 : : : ; xi1 ; ; xi ; : : : ; xq1 I ¢/. 2. For any ¢ 2 Cq .X/ if ¢ D  ı pq , then for any .x1 ; : : : ; xq / 2 Iq , we identify .x1 ; : : : ; xq I ¢/ with .x1 ; : : : ; xq1 I /. The quotient space is denoted jQ.X/j. Notice that this space is a union of compact O and is a CW subsets which are the images of the various closed cubes in jQj complex with one cell for each nondegenerate singular cube in X. The closures of the open cubes are not necessarily embeddings of the closed cube because some of the faces of a nondegenerate cube can be degenerate and hence collapsed to lower dimensional cubes in jQ.X/j. Still, the topology of jQ.X/j is generated by the images of the closed cubes, meaning that a subset of jQ.X/j is open if and only if its intersection with the image of each closed cube is an open subset of O that image. There is the obvious map from jQ.X/j ! X which is compatible with

16.1 The Cubical Complex and Cubical Forms

153

the identifications and hence factors to give a continuous map jQ.X/j ! X. The fact that the chain complex Q.X/ computes the homology of X implies that this map is a weak homotopy equivalence; i.e., this map induces an isomorphism on the homology and homotopy groups. If X is the homotopy type of a CW complex, then it is a homotopy equivalence. There is a DGA, denoted  .X/, defined over Q, of piecewise linear forms on jQ.X/j. One way to think about this is that associated to the cube Iq , we have the DGA, denoted  .Iq /, of polynomial forms on the cube with rational coefficients ƒ .x1 ; : : : ; xq ; dx1 ; : : : ;` dxq /. Then an element of n .X/ is a collection of forms f¨¢ g¢ indexed by ¢ 2 q Cq .X/ where ¨¢ 2 n .Iq / when ¢ is a singular q-cube in X. This collection is required to satisfy two compatibility conditions: 1. ¨¢ jf i .¢/ is identified with ¨f i .¢/ under the natural identification of Iq1 with f i .Iq /. 2. If ¢ D  ı pq , then ¨¢ D pq .¨ /. In an obvious way,  .X/ can be viewed as piecewise polynomial forms on the cubical complex jQ.X/j. Since jQ.X/j is a cell complex, it has cellular chains: There is a generator in degree q for each nondegenerate singular q-cube in X with the boundary map induced by @Iq D

q 1 X X

.1/i .1/ f i .Iq /:

D0 iD0

This chain complex computes the homology of jQ.X/j and hence of X. The following result is proved along the same lines as the corresponding result for p.l. forms on a simplicial complex: Proposition 16.2. Integration defines a map from  .X/ to the cellular cochains on jQ.X/j with rational coefficients. This map induces a ring isomorphism from the cohomology of  .X/ to the singular cohomology of jQ.X/j. Hence, the cohomology of  .X/ is identified with the singular rational cohomology X. We can associate a simplicial complex S.X/ to jQ.X/j by taking the barycentric subdivision. The natural map S.X/ ! jQ.X/j is a homeomorphism so that the composition S.X/ ! jQ.X/j ! X is a weak homotopy equivalence and is a homotopy equivalence if X is homotopy equivalent to a CW complex. Then there is a natural restriction map of DGAs  .X/ ! Ap:l: .S.X//. This map induces an isomorphism on cohomology (which can be proved by induction on the cells of jQ.X/j). It follows that the minimal model for  .X/ is identified with that of A .S.X//, uniquely up to homotopy. Hence, if X is simply connected and its rational homology is finite dimensional in each degree, then the minimal model for  .X/ is dual to the rational Postnikov tower of X. To formulate the rational differential form version of the Serre spectral sequence for a Serre fibration, it will be convenient to have a restricted type of base. Recall that if Iq is a cube, then a face of Iq is the subspace given by fixing a subset i1 <    < ik  q and setting xij D j , where j is either 0 or 1. Any face of dimension q  k is identified with Iqk via the complementary set of coordinates (in the same order).

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16 The Hirsch Lemma

Definition. A cubical complex X is a topological space together with set of maps of closed cubes ¢W Iq ! X with the following properties: 1. Each ¢ is an embedding of a closed cube into X. 2. For each closed cube ¢W Iq ! X in the set, the restriction of ¢ to any face of Iq is an element of the set of closed cubes in X. 3. The union of the images of the closed cubes is all of X. 4. The intersection of two closed cubes is either empty or a face of each. 5. When the intersection of two closed cubes ¢1 and ¢2 is nonempty, the induced identifications of ¢1 \ ¢2 with a lower dimensional cube coming from ¢1 and ¢2 agree. Lemma 16.3. If X is a CW complex, then X is homotopy equivalent to a cubical complex. Proof. First, suppose that n isPa standard n-simplex with coordinates .t0 ; : : : ; tn / subject to ti  0 for every i and i ti D 1. For i D 0; : : : ; n, we set C.i/ equal to the Š

subset of n where ti  tj for every j 6D i. We define an isomorphism In ! C.i/ by  si1 1 si sn s1 ; ;:::; ; ; ;:::; .s1 ; : : : ; sn / 7! 1 C jsj 1 C jsj 1 C jsj 1 C jsj 1 C jsj P where jsj D j sj and, for each j, the variable sj ranges from 0 to 1. It is easy to see that these cubes define a cubical complex on n . Furthermore, the induced cubical complex on any face of n agrees with this construction applied to that face. Notice that the identification In ! Ci depends on the ordering of the vertices of n . Now suppose that K is a simplicial complex and suppose that we have a total ordering of the vertices of K. Then performing the above construction in each simplex produces a cubical complex structure on K. From this, it follows that any simplicial complex is homotopy equivalent to a cubical complex; in fact, it is piecewise linearly isomorphic to a cubical complex. To complete the proof, we appeal to the result Lemma 2.2 that says that any CW complex is homotopy equivalent to a simplicial complex. 

16.2 Hirsch Extensions and Spectral Sequences We begin by recalling the notion of a Hirsch extension. Let A be a connected DGA over Q with differential d, let n be a natural number, and let V be a finite-dimensional rational vector space. Given a map “W V ! AnC1 whose image lies in the kernel of d, we have a Hirsch extension in degree n of DGA’s by forming O A D d; and d.v/ O A ! A ˝“ ƒ .V /; with dj D “.v/: Here ƒ .V / is the free graded-commutative algebra generated by V in degree n. O We denote the Hirsch extension by A

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155

There is a spectral sequence associated to this construction. We define a O  by defining Fp .A O  / D Ap ˝ ƒ .V /. Clearly, this decreasing filtration on A O is a multiplicative filtration preserved by d. It is easy to identify the first few terms of the spectral sequence: E0 D Ap ˝ .ƒ .V //q I d0 D 0I p;q

E1 D Ap ˝ .ƒ .V //q I d1 D dI p;q

E2 D Hp .A I .ƒ .V //q / D Hp .A/ ˝ .ƒ .V //q : p;q

(The identification of E2 requires that A be simply connected.) This is a multiplicative spectral sequence and the induced multiplication on E; is the (graded) 2 tensor product of the multiplication in H .A / induced by the multiplication in A and the usual multiplication in ƒ .V /. In particular, when A is simply connected, multiplication induces an isomorphism p;0

0;q

p;q

E2 ˝ E2 ! E2 : p;kn

pCnC1;.k1/n

The differentials d2 ; : : : ; dn vanish and dnC1 W EnC1 ! EnC1

is given by

Hp .A / ˝ .ƒ .V //q ! HpCnC1 .A / ˝ .ƒ .V //qn where the map is a ˝ .v1 ^    ^ vk / 7!

X

.1/n.i1/ a [ Œ“.vi / ˝ .v1 ^    ^ vi1 ^ viC1    ^ vk /;

i

and Œ“.vi / 2 HnC1 .A / is the class represented by the closed form “.vi /. In [26], Sullivan showed that if fW E ! B is a Serre fibration with fiber K. ; n/ with  1 .B/ acting trivially on the homology of the fiber, with V D  ˝Q being finite dimensional, and if W A !  .B/ is a map of DGA’s inducing an isomorphism O  D .A ˝ ƒ .V /; d/ O and a on cohomology, then there is a Hirsch extension A    O map W O A ! .E/ with j O A D f ı  inducing an isomorphism on cohomology. The degree of the extension is n. The proof was an inductive one over the cells of the base. Here, we propose a different argument (which also has induction over the cells in the internal parts of the argument) to prove that the Hirsch extension of the forms on the base models the total space. Our approach is to construct an DGA, denoted

 .E=B/, which is a sub-DGA of  .E/ of cubical polynomial forms on jQ.E/j. This sub-DGA has on it a decreasing filtration whose E1 -term is the differential forms on the base with coefficients in the cohomology of the fiber and whose E2 -term is the cohomology of the base with coefficients in the cohomology of the fiber. If the fiber is a K. ; n/, then a DGA model for the fiber is ƒ .V / where V D Hom. ; Q/ and V is in degree n. Thus, the E2 -term of the spectral

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16 The Hirsch Lemma

sequence for  .E=B/ is isomorphic to the E2 -term of the spectral sequence of the associated Hirsch extension of  .B/. It is then simply a matter of a comparison theorem for spectral sequences to prove that there is a map from the Hirsch extension

 .B/ ˝d ƒ.V / to  .E/ extending the natural map of  .B/ !  .E/ and inducing an isomorphism on cohomology. To motivate the construction that we shall do in the case of Serre fibrations, let us recall briefly what one does in the case of locally trivial smooth fibrations. Suppose that fW M ! B is a proper smooth submersion of smooth manifolds. Then fW M ! B is a smoothly locally trivial fibration. In this case, there is a filtration on the (smooth) differential forms  .M/ defined by letting Fk . n .M// be the subspace of forms ¨ such that for any point p in M and any collection of tangent vectors £1 ; : : : ; £n with at least n  k C 1 of the £i being vertical (i.e., in the kernel of df), we have ¨p .£1 ; : : : ; £n / D 0. This is a multiplicative filtration preserved by exterior differentiation, and hence, there is an induced spectral sequence whose p;q E1 -term is identified with p .BI Hq .FI R// where the coefficients are the local system given by the cohomology of the fibers of the map. This is a locally trivial coefficient system over B and d1 is identified with dB , the usual differential for p;q this locally trivial system, so that E2 is identified with the cohomology of B with coefficients in this local system, Hp .BI Hq .FI R//. In the case when  1 .B/ acts trivially on the cohomology of the fiber, the coefficients become a trivial local system identified with Hq .F0 I R/, for the fiber F0 over the base point. Then p;q E2 D Hp .BI Hq .F0 I R// D Hp .BI R/˝ R Hq .F0 I R/. In this case, the multiplication of forms induces a product structure on the spectral sequence which agrees with the obvious product on E; 2 .

16.3 Polynomial Forms for a Serre Fibration Now suppose that fW E ! B is a Serre fibration with the base B being a cubical complex. (Recall that this means that each of the closed cubes is embedded in B.) Our goal is to give a Q-DGA model f W  .B/ !  .E=B/ for fW E ! B with a filtration on  .E=B/ analogous to the one above giving the Serre spectral sequence (with rational coefficients) for this Serre fibration. To do this, we define sets of cubes associated to this fibration, C .E=B/. An element of Ck .E=B/ is a singular cube ¢W Ik ! E for which there is a closed cube ¢N of B, with identification '¢N W Is ! ¢N  B such that setting p equal to the projection pW Ik ! Is given by p.x1 ; : : : ; xk / D .x1 ; : : : ; xs /, we have f ı ¢ D '¢N ı p. In this case, we say that ¢ covers ¢N . Notice that ¢ covers only one cube of B. We denote by Dk .E=B/ the set of degenerate cubes in Ck .E=B/. Suppose that ¢ 2 Ck .E=B/ covers ¢N of dimension s. Then for D 0; 1 for any i > s, the face f i ¢ also covers ¢N and for i  s the face f i ¢ covers f i ¢N . It follows that if f is a face of ¢, then f covers a face Nf of ¢N and the dimension of the fiber of f ! Nf is less than or equal to the dimension of the fiber of ¢ ! ¢N . We let Q .E=B/ denote the chain complex whose chain group in degree k is the quotient of the free abelian

16.3 Polynomial Forms for a Serre Fibration

157

group generated by Ck .E=B/ by the free abelian group generated by Dk .E=B/ and whose boundary map is the one induced by taking sums in the standard way of codimension-1 faces. The subcomplex D .E=B/ is closed under taking boundaries; also if singular cube covering ¢N is degenerate, then its nondegenerate quotient also covers ¢N . Clearly, C.E=B/ is a subset of C.E/, and the resulting maps of chain groups embed Q .E=B/ as a subcomplex of Q .E/. Ideas from [21] can be used to show the following: Proposition 16.4. The inclusion of Q .E=B/ ! Q .E/ induces an isomorphism on homology so that Q .E=B/ computes the singular homology of E. Proof. We first prove the result for bases, B that are finite cubical complexes. The proof is by induction on the number of cubes in the cubical complex structure of B. If B has a single cell, then it is a point and Q .E=B/ is just the usual singular cubical complex of E which by [21] or Lemma 16.1 computes the singular homology of E. Now suppose by induction that we know the result for all bases with fewer than N cells and B has exactly N cells. Let B0  B be the sub-cubical complex of B obtained by removing the interior one of the top-dimensional cubes, say ¢N of dimension s. Let E0 D f1 .B0 /. Then E0 ! B0 is a Serre fibration and by induction Q .E0 =B0 / computes the singular homology of E0 . Clearly, Q .E0 =B0 / is a subcomplex of Q .E=B/ so that we have a commutative diagram of short exact sequences of chain complexes whose chain groups are free abelian groups: 0 ! Q .E0 =B0 / ! Q .E=B/ ! Q .E=B/=Q .E0 =B0 / ! 0 # # # Q .E; E0 / !0 0 ! Q .E0 / ! Q .E/ ! where the vertical maps are the inclusions. By induction and the 5 lemma to establish that the middle vertical arrow induces an isomorphism on homology, it suffices to show that the last vertical arrow does. Let F0 be the fiber over the initial vertex v of ¢N . By the fact that singular cubical theory computes the usual singular homology, we have H .Q .E; E0 // D H .N¢ ; @N¢ / ˝ H .F0 /. Now we define a map K¢N W C .F0 / ! CsC .f1 .N¢ /=¢N /;

(16.1)

denoted K¢N .£/ D £O , by induction satisfying the following properties: 1. For £W Ik ! F0 , the map £O W Is  Ik ! f1 .N¢ / satisfies £O jfvgIk D £. 2. If £ is degenerate with £0 as nondegenerate quotient, then £O is degenerate with nondegenerate quotient £O 0 . 3. For each face f i D f i .£/ of £, we have £O jIs f i D f i .£/.

b

Suppose that for each singular cube £W It ! F0 of dimension t < k, we have defined £O W Is  Ik ! E0 covering ¢N with these properties. We consider a singular cube £ of dimension k. If £ is degenerate, with nondegenerate quotient £0 , the second

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condition defines £O from £O 0 . Suppose that £ is nondegenerate. By induction, we have b Is  @Ik ! E0 covering @N¢ . We also have £W fvg  Ik ! F0  E0 . We extend this @£W b to I1 @Ik . to a map £O 1 W I1 Ik covering ¢N jI 1 and compatible with the restriction of @£ Such an extension is possible by the Serre homotopy extension property and the fact that I1 Ik deformation retracts onto fvgIk [I1 @Ik . We continue inductively in this way extending £ to £O i over Ii  Ik covering the restriction of ¢N to Ii and compatible with the given extension on the boundary. This uses the Serre homotopy extension property and the fact that for each i  s the cube Ii  Ik deformation retracts onto Ii1  Ik [ Ii  @Ik . In the end, we construct £O as required. We do this independently for each nondegenerate k cube, completing the induction. This map determines a chain map K¢N W Q .F0 / ! QsC .f1 .N¢ /=¢N /=Q .f1 .@N¢ /=@N¢ // D Q .E=B/=Q .E0 =B0 / commuting with the boundary maps. Claim. K¢N induces an isomorphism H .F0 / ! HsC .Q .E=B/; Q .E0 =B0 //: Proof. Given an element £W Is  Ik covering ¢N , then the restriction of £ to fvg  Ik is an element £v of Ck .F0 / and hence K¢N .£v / is an element of CkCs .E=B/ covering ¢N . Notice that this map is compatible with taking faces in the last k-directions. This defines a chain map L¢N W QsCk .E=B/=QsCk .E0 =B0 / to Cs .F0 /. Clearly, L¢N ı K¢N D IdQ .F0 / . We complete the proof by showing that K¢N ı L¢N is chain homotopic to the identity of Q .E=B/=Q .E0 =B0 /. To do this, we define a map H¢N W Q .E; E0 / ! QC1 .E; E0 / compatible with taking faces in directions greater than s and also compatible with degeneracies and satisfies @H¢N  H¢N ı @ D Id  K¢N ı L¢N : This chain homotopy is defined inductively in the same way as the map K¢N , above. This completes the proof of the claim.  It follows immediately from the claim that the third vertical arrow in the above diagram induces an isomorphism on homology. The inductive step follows immediately from this and the 5 lemma. This shows that provided that B is a finite cubical cell complex, the result holds. Now any cubic complex is a direct limit of its finite subcomplexes and homology commutes with direct limits, so the result follows in general from the result for bases that are finite cell complexes. This completes the proof of the proposition.  Corollary 16.5. Integration defines a map of cochain complexes  .E=B/ ! Q .EI Q/ that induces an isomorphism on cohomology. Proof. Since the inclusion Q .E=B/ ! Q .E/ induces an isomorphism on homology, the result is immediate from Proposition 16.2. 

16.4 Serre Spectral Sequence for Polynomial Forms

159

16.4 Serre Spectral Sequence for Polynomial Forms Now we introduce the decreasing filtration on  .E=B/ and compute the E1 - and E2 -terms in the resulting spectral sequence. A filtration analogous to the one in the smooth case makes sense on  .E=B/. We say that a form f¨¢ g¢ in n .E=B/ is contained in Fk . n .E=B// if for any singular cube ¢ in E covering a cube ¢N of dimension s of the cubical structure on B and tangent vectors .v1 ; : : : ; vn / to In at a point a 2 In the evaluation ¨¢ .v1 ; : : : ; vn / is zero any time, at least n  k C 1 of the vectors vi are tangent to the fiber of the projection pW In ! Is , given by p.x1 ; : : : ; xn / D .x1 ; : : : ; xs /. Notice that since the cubes in E are required to cover embedded cubes in B, the filtration is preserved as we pass to faces: If ¨¢ is in filtration level p, then the same is true for the restriction of ¨¢ to any face. This is a multiplicative filtration preserved by d. Hence, it induces a spectral sequence with a multiplication. For each ¢ 2 C .E=B/ covering a cube ¢N of the cubical structure of B, we denote by p¢ W In ! ¢N the natural projection and by F¢ the fiber p1 ¢ .v¢N / where v¢N is the initial vertex of ¢N . Here is the main theorem of this appendix. Theorem 16.6. Suppose that fW E ! B is a Serre fibration with B being a path connected cubical complex. Let b 2 B be a base point and let F0  E be the fiber over b. Suppose that  1 .B; b/ acts trivially on the homology of F0 . Then the filtration on  .E=B/ defined above induces a multiplicative spectral sequence whose E2 -term is given by p;q

E2 D Hp .BI Q/ ˝ Hq .F0 I Q/; and with the multiplication on E2 —being the (graded) tensor product of the cup p;0 product structures on the cohomology of the base and the fiber. In particular, E2 ˝ 0;q p;q E2 ! E2 is an isomorphism. Proof. We begin with a series of claims. k



kC1





Claim. F . .E=B//=F . .E=B// is naturally identified with collections of elements f¨¢ g¢ indexed by ¢ 2 C .E=B/ with ¨¢ 2 p¢ k .N¢ / ˝  .F¢N /   .¢/ and compatible under the face relations and degeneracies (the degeneracies being only along the fiber direction). The map induced by exterior differentiation on Fk =FkC1 is given by .1/k ˝ dF¢N on p¢N k .N¢ / ˝  .F¢N /. Proof. Let ¢N be the cube of B covered by ¢. Then Fk .  .¢=¢N // is identified with p¢N k .N¢ / ˝  .F¢N /, and we have an identification Fk .  .¢=¢N //=FkC1 .  .¢=¢N // D p¢N k .N¢ / ˝  .F¢N /: The face relations preserve the bi-degrees as do the degeneracies. The latter are only in the fiber direction since ¢N is nondegenerate. This proves that any element in the associated graded determines a collection of elements as stated in the claim.

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Conversely, given a compatible collection of elements, they define an element in

 .E=B/ which lies in Fk and the linear subspace of such elements is a complement for FkC1  Fk . This proves the first statement in the claim. The second is clear: On

 .¢/ D p¢N  .N¢ / ˝  .F¢N /, exterior differentiation is given by d D d ˝ 1 C .1/k 1 ˝ d. The first term raises the filtration degree by 1 the second preserves it; thus, on the associated graded, the first term vanishes.  Now consider  .B/; it is the DGA of homogeneous collections f¨¢N g¢N with ¨¢N 2  .N¢ / indexed by the cells ¢N of B and compatible under restriction to faces. (Since we use only the cells of B and not all singular cubes in B, there are no degeneracies to consider.) This DGA has an integration mapping to Q .BI Q/ which induces an isomorphism on cohomology. Of course,  .B/ is a quotient DGA of the

 .jQ.B/j/ dual to the cellular inclusion B ,! jQ.B/j, and hence, this projection of DGA’s induces an isomorphism on cohomology. At this point, we choose a base point b 2 B (say the initial 0-cube of B under some ordering) and we let F0 be the fiber over b. We suppose that  1 .B; b/ acts trivially on the homology of F0 . Given any b0 2 B, choosing a path from b to b0 identifies the homology or cohomology of F0 with that of the fiber F0 over b0 . Because of the assumption that  1 .B; b/ acts trivially on the homology of F0 , this identification is independent of the path chosen to connect b to b0 . The next step is to define a map of Ek; to k .B/ ˝ H .F0 I Q/ sending the 1 multiplication in E1 to the wedge product of forms tensored with the cup product in cohomology. Let f¨¢ g¢ 2 p¢N k .N¢ / ˝  .F¢N / be a compatible collection of forms determining an element in Ek; 0 . Suppose further that this element is in the kernel of d0 , meaning that for each ¢ the form ¨¢ lies in p¢N k .N¢ / ˝ Z .F¢N / where Z   k; is the kernel of d. Such an element determines a class in Ek; 1 and every class in E1 is determined by such elements. Lastly, two such elements determine the same class in Ek; 1 if and only if they differ by a collection of forms fd˜¢ C “¢ g where f˜¢ g¢ is a compatible collection of forms in k .N¢ / ˝ 1 .F¢N / and f“¢ g¢ is a compatible collection of forms in kC1 .N¢ / ˝  .F¢N /. Given a form f¨¢ g¢ with the property that ¨¢ 2 p¢N k .N¢ / ˝ Z .F¢N / for every ¢, then we restrict attention to the ¨¢ indexed by the ¢ covering a given cell ¢N of B. Denote the set of such ¢ by C .N¢ /. In this way, we are restricting to  .f1 .N¢ /=¢N /. We pull f¨¢ gC .N¢ / back via K¢N to give a class in k .N¢ / ˝ Z .F¢N /. Projecting modulo

k .N¢ / ˝ d 1 .F¢N / determines an element in k .N¢ / ˝ H .F¢N /. Claim. For any choice of K¢N as in Equation 16.1, the resulting map K¢N W  .f1 .N¢ /=¢N / !  .N¢ / ˝  .F¢N / has the following properties: 1. The push forward of the filtration F via K¢N is the filtration which is the product of the filtration by degrees on  .N¢ / and the trivial filtration on  .F¢N /. p;q 2. The E1 -term of the spectral sequence for F .  .N¢ / ˝  .F¢N // is naturally identified with p .N¢ / ˝ Hq .F¢N I Q/.

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161

3. The pullback by K¢N induces an isomorphism on the E; terms on the spectral 1 sequences associated to F .  .f1 .N¢ /=¢N // and F .  .N¢ / ˝  .F¢N //, so that the p;q E1 -term of the spectral sequence of F .  .f1 .N¢ // is also naturally identified with p .N¢ / ˝ Hq .F¢N I Q/. Proof. Clearly, the induced filtration on ImK¢N agrees with the filtration on  .N¢ / ˝

 .F¢N / which is given by the degree in the first factor. Using this filtration, both the map K¢N W  .f1 .N¢ // !  .N¢ / ˝  .F¢N / and the map r¢N W  .N¢ / ˝  .F¢N / !

 .f1 .N¢ // are filtration preserving as is the homotopy map H¢N W  .f1 .N¢ /=¢N / ! R sD1

 .f1 .N¢ /=¢N / ˝ ƒ.s; ds/. Thus, sD0 H¢N forms a filtration-preserving cochain homotopy from Id to r¢N ıK¢N . It then follows that K¢N and r¢N are inverse isomorphisms on the E; 1 -terms of the spectral sequences associated to these filtrations. Clearly for  .N¢ / ˝  .F¢N / with the filtration coming from the degree of the first factor, p;q the E1 -term is naturally identified with p .N¢ / ˝ Hq .  .F¢N //.  Now we wish to replace H .F¢N /I Q/ in the above lemma by H .F0 I Q/,where F0 is the fiber over the base point. To do this, we need only remark that choosing any path from v0 to v¢N identifies the cohomology of F¢N with that of F0 and this identification is independent of the choice of path. Thus, for each cell ¢N of B, we p;q have an identification of E1 .F .  .f1 .N¢ /=¢N /// with p .N¢ / ˝ Hq .F0 I Q/. Next, we need to see that these identifications are compatible under passing from ¢N to a face £N . First, notice that the maps of the E; 1 -terms determined by two product structures K¢N and K0¢N are equal since they are both inverses to the map induced by r¢N on the E; 1 -terms. Next, notice that the map induced by a product structure over ¢N from H .F¢N I Q/ to the rational cohomology of the fiber over any other vertex v0 of ¢N agrees with the isomorphism induced by taking a path from v¢N to v0 and hence commutes with the identifications of the cohomologies of these fibers with that of F0 . Using these two facts, the result is immediate. Hence, we have constructed p;q a map from E1 .  .E=B/ to p .B/ ˝ Hq .F0 I Q/. The last thing to see is that this map is an isomorphism. This requires a claim: Claim. The restriction mapping

 .f1 .N¢ /=¢N / !  .f1 .@N¢ /=@N¢ / induces a surjective map on filtration level p for every p  0. Proof. Suppose by induction, we have extended a form ¨¢ 2 Fp .  .f1 .@N¢ /=@N¢ // to a form in filtration level p defined on all cubes covering ¢N of dimension less than k, and let ¢ be a cube covering ¢N of dimension k. (By the inductive hypothesis, the form is defined on @¢ and is in filtration level p.) The extension result requires inductively taking linear combinations of forms defined on opposite boundaries. This does not change the filtration level and hence shows that we can extend the form over all of ¢ keeping the result in filtration level p. 

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Now we are ready to show that E1 .  .E=B// D p .B/ ˝ Hq .F0 I Q/. The proof is first for the case when B is a finite cell complex and is by induction on the number of cells of B, the case of one cell being obvious. Suppose we know the result when the base has fewer than N cells and B has N cells. Let B0 be the p;q result of removing a top-dimensional cell of B. By induction, E1 .  .E0 =B0 // D p q p q

.B0 / ˝ H .F0 I Q/. Given any element ’ 2 .B/ ˝ H .F0 I Q/, we fix an element q0 2 Fp .  .E0 =B0 // which is in the kernel of d0 and represents the p;q restriction ’0 of ’ in E1 .  .E0 =B0 //. Choose closed forms —1 ; : : : ; —n ; : : : ; in q

.F¢N / whose cohomology classes form a basis for Hq .F¢N I Q/. P Then the restriction p;q i of ˛ to E1 .  .f1 .@N¢ /=@N¢ // is represented as a finite sum K i D1 ¨ ˝ i with i p i p i ¨ 2P .@N¢ /. Let ¨ O 2 .N¢ / be an extension of ¨ . Then the element  D i¨ O i ˝ —i is an element in Fp .  .f1 .N¢ /=¢N // which represents an element p;q in E1 .  .f1 .N¢ /=¢N // whose restriction to the boundary agrees in E1 with the restriction of ’0 to f1 .@N¢ /. Thus, there difference of these restriction is contained in FpC1 C dFp of  .f1 .@N¢ /=@N¢ /. Using the surjective property established in the previous claim, we can vary  by a form in .FpC1 C d Fp /.  .f1 .N¢ /=¢N // to arrange that its restriction agrees with the restriction of ’0 in  .f1 .@N¢ /=@N¢ /. This then gives an extension of the form over all of B and proves that the map p;q E1 ! p .B/ ˝ Hq .F0 I Q/ is surjective. Now we must show that it is injective. Again, this is done inductively. Suppose that ’ 2 Fp .  .E=B// is in the kernel of d0 and maps to the trivial element in p .B/ ˝ Hq .F0 I Q/. By induction, the restriction, ’0 , of ˛ in Fp .  .E0 =B0 // is contained in FpC1 C dFp . Using the surjectivity result above, we can assume that ’0 D 0. This means that ’ is the extension by zero over E0 of a relative form ’1 in  .f1 .N¢ /=¢N ; f1 .@N¢ /=@N¢ / in filtration level p. The image of ’1 in E1 is contained in p .N¢ ; @N¢ / ˝ Hq .F0 I Q/ and ’ is contained in the image of .FpC1 C dFp /.  .f1 .N¢ /=¢N //. We write ’ D “ C d” where “ 2 FpC1 and ” 2 Fp . Since ’Š restricts to zero over @N¢ , we have p;q

.“ C d”/jf1 .@N¢ / D 0: p;q1

This means that ”jf1 .@N¢ / represents in element in E1 .@N¢ /. By the surjectivity result, there is ”O 2 Fp .  .f1 .N¢ // with the property that O jf1 .@N¢ / D  j@N¢ . Varying ’ by d ”O allows us to assume that ”jf1 .@N¢ / D 0 and hence that also that“jf1 .@N¢ / D 0. Thus, ’1 D “ C d” in ˝  .f1 .N¢ /; f1 .@N¢ //. Thus, ’ is equal to the extension of “ C d” by 0 outside of f1 .N¢ /. This shows that ’ is an element in FpC1 C d Fp and p;q hence represents 0 in E1 . This completes the proof when the base is a finite complex. When the base is an infinite complex,  .E=B/ and Fp .  .E=B/ are the projective limits over finite subcomplexes Bn  B of the  .f1 .Bn /=Bn / and Fp .  .f1 .Bn /=Bn //. Since the restriction mappings are surjective on all filtration levels, it follows that the E0 -term p;q commutes with the projective limits. Since E1 .f1 .Bn /=Bn / is identified with

p .Bn /˝Hq .F0 I Q/, it follows that the restriction mappings in the projective system p;q are surjective on the cohomology of .E0 ; d0 /. It follows that the E1 commutes with p;q the projective limits and hence E1 .  .E=B// D p .B/ ˝ Hq .F0 I Q/. Clearly, the p;q map d1 is given by dB ˝ 1, so that the E2 .  .E=B// D Hp .BI Hq .F0 I Q//.

16.5 Proof of Theorem 12.1

163

16.5 Proof of Theorem 12.1 Suppose that B is a cubical complex and that fW E ! B is a Serre fibration with fiber K. ; n/ with V D   ˝ Q a finite-dimensional rational vector space and with the action of  1 .B; b/ on the homology of the fiber being trivial. Then V is the relative .n C 1/st cohomology associated with the map f W  .B/ !  .E=B/. That is to say, there is a map V ! nC1 .B/ ˚ n .E=B/ whose image lies in the relative cocycles and which induces the identity on the relative cohomology in degree n C 1. This defines a map ¡W  .B/ ˝d ƒ .V / !  .E=B/: We have the decreasing filtration on  .B/ ˝d ƒ .V / coming from the degrees of forms in  .B/. With this filtration and the filtration described above on  .E=B/, p;0 the map ¡ preserves the filtration. Given the isomorphisms of E2 of each spectral p;0 sequence with Hp .B/, the induced map on E2 is the identity. Also, the E0;n 2 -term of each spectral sequence is identified with the relative .n C 1/st cohomology, and again the map between them is the identity. It then follows from the multiplicative structure that the induced map at E2 is an isomorphism, and hence, the map on cohomology is an isomorphism. Now suppose that M !  .B/ is a minimal model. Let M0 D M ˝d ƒ.V / be the Hirsch extension corresponding to fW E ! B. Then M0 is the minimal model for

 .B/ ˝d ƒ .V / and hence for  .E=B/ and hence also for  .E/. Lastly, suppose that f0 W E0 ! B0 is a simplicial model for fW E ! B. Then M is also a minimal model for the p.l. forms A .B/. Then the minimal model for the p.l. forms A .E0 / is identified with the minimal model for  .E/, and hence, the map M0 !  .E/ determines a map M0 ! A .E0 / up to homotopy, and this map induces an isomorphism on cohomology. The restriction of this map to M factors as ¡

.f0 /

M ! A .B0 / ! A .E0 /; where ¡ induces an isomorphism on cohomology. Also, the relative .n C 1/st cohomologies of both .M; M0 / and of .A .B0 /; A .E0 // are identified with V . Under these identifications, ¡ induces the identity on these relative cohomology groups. This proves that given a principal K. ; n/-fibration E ! B and a simplicial model E0 ! B0 with minimal model ¡W M ! A .B0 /, the Hirsch extension M0 D M ˝d ƒ .V / dual to the principal fibration maps to A .E0 / extending the composition .f0 / ı ¡ and inducing an isomorphism on cohomology. This is exactly the statement of Theorem 12.1.

Chapter 17

Quillen’s Work on Rational Homotopy Theory

Before Sullivan’s work on differential forms and rational homotopy theory, in [19] Quillen had established algebraic models for rational homotopy theory. Quillen worked dually from the way Sullivan does: Instead of using differential forms as the basic model, Quillen uses differential graded Lie algebras. One can, as Quillen does, associate a differential graded, co-commutative co-algebra to a differential graded Lie algebra. Dualizing this produces a differential graded algebra computing the rational homotopy type and thus in some sense solves the rational commutative cochain problem. Sullivan’s construction was much more directly tied to differential forms on the space, whereas Quillen’s went through more homotopy theoretic constructions, i.e., the simplicial set (of singular simplices) associated with the loop space of the given space. In this chapter we will briefly describe Quillen’s results and compare them with Sullivan’s. We begin by describing the homotopy theory of various categories of algebraic objects that enter into Quillen’s construction.

17.1 Differential Graded Lie Algebras Definition. A graded Lie algebra (GLA) is a graded vector space L D ˚k Lk together with a homomorphism Œ;  W L ˝ L ! L satisfying: 1. Œ;  is homogeneous of degree zero in the sense that Œ;  W Lk ˝ Ln ! LkCn . 2. (Graded skew symmetry): For homogeneous elements x; y, we have Œx; y D .1/jxjjyjC1 Œy; x . 3. (Graded Bianchi identity): For homogeneous elements x; y; z, we have

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__17, © Springer Science+Business Media New York 2013

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ŒŒx; y ; z C .1/jzj.jxjCjyj/ ŒŒz; x ; y C .1/jxj.jzjCjyj/ ŒŒy; z ; x D 0: (Here, for a homogeneous element x 2 Lk , the symbol jxj is defined to be k.) A differential graded Lie algebra (DGLA) is a graded Lie algebra .L; Œ;  / together with a differential @W L ! L, which is a linear map of degree 1 satisfying @2 D 0 and which satisfies @.Œx; y / D Œ@x; y C .1/jxj Œx; @y for homogeneous elements x; y. The homology of a DGLA is the homology of .L; @/. Clearly, the homology of a DGLA is a GLA. The DGLAs that come up in Quillen’s construction are reduced, meaning that Lk D 0 for k  0. A map (morphism) of DGLAs is a degree-preserving linear map that preserves the bracket and commutes with the differentials. A quasi-isomorphism (or homotopy equivalence) is a map that induces an isomorphism on homology. The homotopy category of DGLA is the category whose objects are DGLAs and whose morphisms are compositions of DGLA maps and formal inverses of quasi-isomorphisms. In passing from spaces to DGLAs, there is a shift in degree by 1 from the usual notion of degree in a space to the degree in the DGLA, so that reduced DGLAs are associated to simply connected spaces. Suppose that X is a 1-connected space with base point x and that .L.X/; @/ is the DGLA associated to it by Quillen’s construction (as will be described below). Let X denote the based loop space of X. Then, the homology Hk .L.X/; @/ is identified with the rational homotopy group  kC1 .X/˝Q D  k . X/˝Q. Furthermore, the induced graded Lie algebra structure on H .L.X/; @/ is identified with the Whitehead product on rational homotopy groups, which recall is a pairing  k .X/ ˝  n .X/ !  kCn1 .X/ and is graded skew symmetric after shifting down one degree. This product is identified with the product  k1 . X/ ˝  n1 . X/ !  kCn2 . X/ induced by the loop composition product

X  X ! X.

17.2 Differential Graded Co-algebras Co-algebras are not as familiar as algebras, but they are the dual objects to algebras, meaning that all the properties that are familiar for algebras have dual analogues for co-algebras. The dual properties are formulated by simply reversing the directions of all arrows in the defining diagrams for the property for algebras. Definition. We fix a ground field K and all vector spaces are K-vector spaces. A coalgebra is a vector space C with a co-multiplication W C ! C ˝ C:

17.3 The Bar Construction

167

The co-algebra is said to be co-associative if . ˝ IdC / ı  D .IdC ˝ / ı  as maps of C to C˝C˝C. A graded co-algebra is a graded vector space C D ˚k Ck with a co-multiplication that is homogeneous of degree zero, meaning that for each k W Ck ! ˚iCjDk Ci ˝ Cj : A co-algebra is (graded) co-commutative if .c/ D T ı  where TW C ˝ C ! C ˝ C is the map that sends c1 ˝ c2 7! .1/jc1 jjc2 j c2 ˝ c1 for all homogeneous elements c1 ; c2 . We always assume that our co-algebras are connected, meaning that Ci D 0 for i < 0 and C0 is the ground field and co-unital, meaning that part of the structure is a co-unit W C ! K so that ˝ IdC ı  is the natural map C ! K ˝K C and analogously for Id ˝ ı . In the case of connected co-algebras identifies C0 with K and the co-unital condition can be rewritten as .c/ D c ˝ 1 C 1 ˝ c C .c/ with .c/ 2

P i;;j>0

Ci ˝ Cj .

A differential graded co-algebra is a graded co-algebra together with a boundary map @W C ! C of degree satisfying @2 D 0 and satisfying the co-Leibnitz rule: P 1 0 Suppose that .c/ D i ci ˝ c00ki where each c0i is homogeneous. Then, .@c/ D

X

0

@c0i ˝ c00i C .1/jci j c0i ˝ @c00i :

i

A DGCC means a differential graded co-commutative co-associative co-algebra. A map (morphism) of DGCCs is a degree-preserving co-algebra map. A morphism is a quasi-isomorphism (homotopy equivalence) if it induces an isomorphism on the homology. The homotopy category of DGCCs has as objects DGCC and as morphism compositions of morphisms of DGCCs and formal inverses of quasi-isomorphisms.

17.3 The Bar Construction There are many variants of the bar construction. The one that is relevant for us here goes from a differential graded Lie algebra to a differential graded co-commutative, co-associative co-algebra. Definition. Let V be a graded vector space. A co-commutative, co-associative graded co-algebra C is co-free with V as co-generating vector space if it satisfies the following universal property. There is a degree zero linear map  W C ! V

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(degree zero meaning that it preserves degrees), and given any co-commutative, co-associative graded co-algebra C0 with a degree zero linear map fW C0 ! V , there is a unique degree zero co-algebra map OfW C0 ! C with   ı Of D f. Let us give a construction of a co-free algebra with V as co-generating vector space. We form the subspace of graded symmetric elements S .V / inside the tensor algebra by taking the sub-vector space of elements in the tensor algebra T .V / that are invariant under the signed action of the symmetric group: ¢W V ˝    ˝ V ! V ˝    ˝ V acts by ¢.v1 ˝ ˝vn / D –v¢.1/ ˝  ˝v¢.n/ where the sign – D –.v1 ˝  ˝vn / is the Kozul sign which is the sign homomorphism generated by assigning .1/jvi jjviC1 j to the interchange permutation of i and i C 1. This determines S .V /. (Notice that it is not a subalgebra of the tensor algebra.) Next we introduce a co-multiplication which will make S .V / co-free in the category of co-commutative, co-associative graded co-algebras. To do this, we introduce the shuffle coproduct: .¢.v1 / ˝    ˝ ¢.vk // D

X

–.v; ; /v ˝ v :

. ;/

Let us explain the notation. Here . ; / ranges over all shuffles of f1; : : : ; kg, meaning that D 1 < 2 <    < i and  D 1 <    < j , and f 1 ; : : : ; i ; 1 ; : : : ; j g is a permutation of f1; : : : ; kg. Furthermore, v D v 1 ˝   ˝v i and similarly for v . The sign .v; ; / 2 f˙1g is the Kozul sign associated with the permutation and the degrees of the vi . To see that S .V / is co-free with V as co-generating space, suppose that C0 is a graded co-commutative co-algebra and fW C0 ! V is a degree zero linear map. Then, we define fk W C ! Sk .V / by fk D .f ˝    ˝ f/.k2 ı k3 ı    ı 0 / „ ƒ‚ … ktimes

where r D 1 ˝    ˝ 1 ˝: „ ƒ‚ … rtimes

One sees directly that, provided that C0 is co-commutative and co-associative, the fk together define the required co-algebra map from C0 ! S .V /. Now we apply this construction beginning with a DGLA .L D ˚k Lk ; @/. We shift up a degree by forming V D ΣL, where by definition .ΣL/k D Lk1 . Then, we form the co-algebra S .ΣL/ with the shuffle coproduct. For any a 2 Lk , we denote by ¢.a/ the corresponding element in .ΣL/kC1 . This produces a co-commutative, co-free co-algebra with ΣL as co-free co-generating space.

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169

So far we have not made use of two of the data from the DGLA: the Lie algebra product and the differential @W L ! L. It turns out that each of these can be used to define a differential on S .ΣL/ and that these differentials commute. The differential @W L ! L defines a differential, denoted @int W S .ΣL/ ! S .ΣL/, by requiring that @int .¢.a// D ¢@.a/ for all a 2 L and by requiring that @ satisfy the co-Leibnitz rule. It follows immediately that .@int /2 D 0. The Lie bracket defines a differential @ext by requiring @ext .¢.a/ ˝ ¢.b/ C .1/jajjbj ¢.b/ ˝ ¢.a// D .1/jaj ¢Œa; b . The Jacobi identity implies that .@ext /2 D 0, and the fact that @ commutes with the Lie bracket implies that @int @ext C @ext @int D 0. We then define the differential @O D @int C @ext . The resulting co-associative, co-commutative, (co-free) differential graded O is the bar construction applied to .L; @/. This is a functor co-algebra .S .ΣL/; @/ from the category of DGLAs to the category of DGCCs and it sends quasiisomorphisms to quasi-isomorphisms.

17.4 Relationship Between Quillen’s Construction and Sullivan’s Before giving Quillen’s construction of a DGLA associated to a simply connected space, let us state the relationship of his construction to Sullivan’s. One thing to point out is that Sullivan’s construction assumes that the rational homotopy groups are finite dimensional in all dimensions so that passing from the rational homotopy groups to their duals and dualizing again reproduces the rational homotopy groups. Since Quillen works with co-algebras, he does not have to dualize and hence does not need to make this assumption. Apart from that proviso, the constructions are dual in the following sense: Theorem 17.1. Quillen assigns a reduced DGLA, .L.X/; @/, to a simply connected topological space X in a functorial way. Applying the bar construction to .L.X/; @/ produces a co-commutative differential graded co-algebra Q .X/. Dualizing this produces a DGA Q .X/ functorially associated to X. Suppose that the rational homotopy groups of X in each degree are finite dimensional. If K is a simplicial complex and fW K ! X is a weak homotopy equivalence, then the minimal model MK for the p.l. forms A .K/ and the minimal model MX for the DGA Q .X/ are identified by a map determined up to homotopy by the homotopy class of f.

17.5 Quillen’s Construction We have not yet described how Quillen associates a DGLA to a simply connected topological space X. To explain this requires more algebro-topological machinery.

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Simplicial categories. Quillen’s construction uses simplicial objects, so we begin with a brief introduction to them. There is a category  with objects the ordered sets n D f0; : : : ; ng; n D 0; 1; : : :. The set n is called the n-simplex. The morphism set Hom .k ; n / is the set of all set functions f from f0; : : : ; kg to f0; : : : ; ng that are weakly order-preserving, in the sense that if a  b then f.a/  f.b/. Composition of morphisms in the category is given by the usual composition of set functions. These morphisms can also be viewed as simplicial maps from the geometric k-simplex to the geometric n-simplex that weakly preserve the order of the vertices. Given a category C, the simplicial C category has as its objects’ covariant functors from the opposite category to , denoted opp , to C and as its morphisms natural transformations between functors. In particular, an object O in simplicial C consists of an indexed family fOn gn0 of objects of C. For each order-preserving simplicial map ’W k ! n , there is a dual morphism ’ W On ! Ok of C and compositions (with order reversed) are preserved. There is an especially nice generating set of morphisms: For each n, there are nC1 face maps from n1 ! n denoted f0 ; : : : ; fn . In terms of simplicial maps, fi is the inclusion of n1 as the codimension-1 face of n opposite the it h vertex of n . In terms of set functions fi .a/ D a if a < i and fi .a/ D a C 1 if a  i. Dually, there are n degeneracy maps n ! n1 , denoted s0 ; : : : ; sn1 , where si is the projection of n to its codimension-1 face opposite the it h vertex. In terms of set maps, si .a/ D a for a  i and si .a/ D a  1 for a i. For each object O in C there morphisms which are the images of these. The image of fi is a “boundary map” @i W On ! On1 and the image of si is a degeneracy map si W On1 ! On . The relations in  lead to the following relations: @i ı @j D @j1 ı @i if j  i C 1: si ı sj D sjC1 ı si if j  i: @i ı sj D sj1 ı @i if j  i C 1: @i ı sj D sj ı @i1 if j C 1 < i: @i ı si D @iC1 ı si D Id: All morphisms of the category are compositions of these generating morphisms, and two compositions of these generators give the same morphism of the category if there is a sequence of compositions connecting them with each step in the sequence being one of these relations (pre- and post composed by the same pair of compositions).

17.5 Quillen’s Construction

171

Some examples are in order: 1. A simplicial set K has as objects indexed families of sets fKn gn0 and for each n set functions @i W Kn ! Kn1 for 0  i  n and set functions si W Kn1 ! Kn for 0  i < n satisfying the relations given above. A morphism from a simplicial set K to a simplicial set K0 consists of set functions ®n W Kn ! K0n commuting with the boundary and degeneracy maps. A simplicial set K with a base point x0 2 K0 has homotopy groups, which are easiest to define when the simplicial set is a Kan complex, meaning that it satisfies the following extension property: Given 0  k  n and elements a0 ; : : : ; ak1 ; akC1 ; : : : ; an in Kn1 with @i aj D @j1 ai for all i < j with i 6D k and j 6D k, then there is an element a 2 Kn with @i a D ai for all i 6D k. (See below for the definition of the homotopy groups of a simplicial set.) 2. The category of simplicial spaces has as objects indexed families of topological spaces fXn gn0 and continuous maps @i W Xn ! Xn1 and si W Xn1 ! Xn satisfying the relations given above. Morphisms are indexed families of continuous maps commuting with the boundary and degeneracy maps. 3. There are simplicial categories associated with various algebraic categories, e.g., simplicial groups, simplicial (augmented and completed) Hopf algebras, and simplicial DGLAs. 4. The first and motivating example is that of the simplicial set of a topological space X. The simplicial set associated to X has Sn .X/ equal to the set of singular n-simplices of X, i.e., the set of continuous maps n ! X. The boundaries, @i , are given by restricting to the various codimension-1 faces. More generally, given any morphism ®W k ! n of , we can view ® as a simplicial map from k-simplex to the n-simplex. Precomposing with this map induces a map from Sn .X/ ! Sk .X/. Notice that S.X/ is a Kan complex. The notions of a simplicial set and of the simplicial set associated with a topological space were introduced by Kan in [11]. As we remarked above, the homotopy groups of a simplicial set K are easiest to define when the simplicial set satisfies is a Kan complex. For a Kan complex K with a base point x0 2 X0 , we define the homotopy groups as follows. Consider all x 2 Kn with the property that @i x D x0n1 WD sn2 sn3    s0 x0 for all 0  i  n. We say that two such, x; y 2 Kn , are equivalent if there is z 2 KnC1 with @n z D x, @nC1 z D y and with @i z D xn0 for all 0  i < n. Because of the Kan condition, this is an equivalence relation. We denote the set of equivalence classes by  n .K; x0 /. To define the multiplication of two elements x; y 2  n .K/, consider a0 D    D an2 D xn0 ; an1 D x; anC1 D y. This is a compatible family of elements of Kn , so that by the Kan condition, there is z 2 KnC1 with @i z D ai for all i 6D n. We define xy to be the equivalence class of @n z. This is a well-defined group operation on  n .K; x0 /, which is commutative if n > 1. In general, to define the homotopy groups, one has to replace a simplicial set by an equivalent Kan complex, which can always be done. Given a simplicial set, one can construct its geometric realization. This is a simplicial complex whose n-simplicies are the nondegenerate n-simplicies in the

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simplicial set and whose gluing maps are determined by the face maps in the simplicial set. If X is a topological space and jS.X/j is the geometric realization of the singular simplicial set of X, then there is a natural continuous map jS.X/j ! X which induces an isomorphism on the homotopy groups and the homology groups. We can then take the rational differential forms on this simplicial complex and hence form Sullivan’s minimal model construction for any topological space, by replacing the space by the geometric realization of its singular simplicial set. If X is a simplicial complex, then ordering the vertices of X identifies it with a subcomplex of S.X/ and hence there is a natural restriction map A .jS.X/j/ ! A .X/, a map that induces an isomorphism on cohomology, and hence these DGAs are homotopy equivalent and in particular the minimal models for these two DGAs of p.l. forms are identified, uniquely up to homotopy. From a simply connected based space to its 1-trivial simplicial set. The first step in Quillen’s construction uses a variant of the simplicial set of a space, a variant that can be defined for simply connected spaces. Let X be a path connected, simply connected space with a base point x0 and form a simplicial set fEi2;Sing .X/gi0 where Ei2;Sing .X/ is the set of all continuous maps of the i-simplex into X with the property that the one-skeleton of i maps to the base point x0 . This is an object in the category of 1-trivial simplicial sets K, i.e., those with the property that the set Ki is a single point for i D 0; 1. From a 1-trivial simplicial set to its reduced simplicial loop group. One then replaces a 1-trivial simplicial set K, for example, K D E2;Sing .X/, by a simplicial group G.K/ that is the simplicial analogue of the based loop space .X; x0 / when K D E2;Sing .X/. The group G.K/n is the free group generated by the set of all elements of KnC1 that are not contained in sn .Kn /. For 0  i < n, the boundary map @N i W G.K/n ! G.K/n1 is induced by @i W KnC1 ! Kn and @N n x D @n x@nC1 .x/1 . For 0  i  n, the degeneracy map sNi W G.K/n ! G.K/nC1 is induced by the degeneracy si W KnC1 ! KnC2 . Then, G.K/ is an object in the category of reduced simplicial groups, i.e., those with G0 D feg. Any simplicial group is a Kan complex, and therefore, we can directly define the homotopy groups. But there is another more direct description of the homotopy groups of a simplicial group. We define the normalized chains N.G/ of a simplicial group G to be the complex Nq .G/ D \0. These ideas generalize to allow one to construct a C1 -isomorphism from a C1 -structure on H .A / to A , an isomorphism inducing the identity on cohomology. There is an analogous theorem in the noncommutative case that says a (not necessarily commutative) DGA is equivalent in an appropriate homotopy category to an A1 -algebra structure on its cohomology.

Chapter 19

Exercises

1. If X, Y are finite CW complexes, then prove that X  Y is also. Show that the k-skeleton .X  Y/.k/ D

[ X.i/  Y.j/

iCjDk

From this, deduce that the cellular chain groups Q  .X/ ˝ CQ  .Y/ CQ  .X  Y/ Š C with boundary @XY D @X ˝ 1 ˙ 1 ˝ @Y : Using this, prove the Künneth theorem. Why is the Künneth theorem hard for singular or simplicial homology? 2. Let (X, A) be a CW pair where A is a finite subcomplex. Show that X=A D fX with A collapsed to the base pointg is again a CW complex. Is the same thing true if A is infinite? Show that Q  .X=A/ .use excision/ H .X; A/ Š H  n .X; A/ ¤  n .X; A/ in general Hint: Take X D 2-disk and A D S1 its boundary. Then, X=A Š S2 and so  3 .X=A/ ¤ 0. What about  3 .X; A/‹/ 3. (a) Show that  0 .X/ D {set of path components of X}. (b) Let X be the loop space of X. Show that composition of loops induces a group structure on  0 . X/. Show that P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4__19, © Springer Science+Business Media New York 2013

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 0 . X/ Š  1 .X/: (c) More generally, prove that  n1 . X/ Š  n .X/; and deduce that if by induction we define n X D . nC1 X/, then  0 . n X/ Š  n .X/: 4. Let X be a CW complex and f W Sn ! X.n/ a map. Let Y D X [f enC1 be the CW complex obtained by attaching .n C 1/-cell by f. Show that the homotopy type of Y depends only on the homotopy class Œf 2  n .X.n/ /: Note: We are not considering [f] in  n .Y/. Why not? 5. Given spaces X,Y, define the wedge X _ Y D .X  y0 / [ .x0  Y/  X  Y: Compute the cohomology ring of CPn1 _ S2n . Is this the same as the cohomology ring of CPn ? 6. Using the preceding exercise, show that  2n1 .CPn1 / ¤ 0: 7. Using the exact homotopy sequence of a fibration, show that  n .X  Y/ Š  n .X/ ˚  n .Y/: f Q! 8. Let X be a space, X X its universal covering. Using the homotopy lifting property, prove that f

Q !  n .X/  n .X/ is an isomorphism for n  2. Using this, let

19 Exercises

189

Show that  2 .X/ Š Z ˚ Z ˚ : : : infinite number of times. Thus, the homotopy groups of a finite complex may be infinitely generated. [Is  1 of a finite complex finitely generated?] 9. Consider the space X D fx D 0g [ fy D sin 1x g in the .x; y/-plane. Show that  0 .X/ has two elements, but X is connected as a topological space. Let Y D fag [ fbg be a space with two distinct points and f W Y ! X the map

Show that f is an isomorphism on homotopy groups, but f1 does not exist in the homotopy category. Thus, the Whitehead theorem is false for non-CW complexes. 10. (a) Assuming that n > 1, show that the inclusion Sn _ Sn ,! Sn  Sn induces an isomorphism  n .Sn _ Sn / Š  n .Sn  Sn /: (Hint: Using Exercise 7, show that the map is onto. If we have Sn J JJ JJ h JJ f JJ J%   / Sn  Sn Sn _ Sn  such that h is homotopic to a constant, then since dim.Sn  Sn / D 2n > n C 1, we may assume that the homotopy misses a point p 2 Sn  Sn (why? cf. the simplicial approximation theorem). Thus, you must prove that Lemma. Sn  Sn n fpg retracts onto Sn _ Sn . For this, write Sn D In =@In and use a picture like

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19 Exercises

(b) What is  1 .S1 _ S1 / (picture D )? 11. (a) Fill in the details of the proof of Theorem 3.2. (b) Why does the proof fail for n D 1? 12. Given A D B [f enC1 , show that  i .B/ Š  i .A/

i> i/

 2i1 .G.n; 2n// D 0 ¡

The map  2i .G.n; 2n// ! Z is given as follows: A homotopy element in  2i .G.n; 2n// is given by S2i ! G.n; 2n/. Pulling back the universal bundle  on G.n; 2n/ gives a vector bundle Cn ! E ! S2i . Then, ¡.f/ D ci .E/=.i  1/Š is an element of integral homology where ci .E / 2 H2i .S2i / is the ith Chern class of E, and we have identified H2i .S2i / Š Z. This divisibility property of the Chern classes of a vector bundle is extremely subtle and is the sort of thing not covered by rational homotopy theory. 19. Complete the proof of Corollary 4.6 as follows: Given a CW complex X with  1 .X/ D 0 and Hi .X/ D 0 for i > n and Hn .X/ free abelian, look at the .n 1/skeleton X.n1/ . Using that, Hi .X.n1/ / Š Hi .X/

in2

Hn1 .X.n1/ / ! Hn1 .X/ ! 0

19 Exercises

193

deduce the exact sequence i

0 ! Hn .X/ ! Hn .X; X.n1/ / ! Hn1 .X.n1/ / ! Hn1 .X/ ! 0: Now Hn1 .X.n1/ / is free abelian. (why?) and so is Hn .X/ by assumption. Thus, Hn .X; X.n1/ / is free abelian (why?). Show finally that  i .X; X.n1/ / D 0 for i  n-1 and  n .X; X.n1/ / Š Hn .X; X.n1/ /, so that we may attach n-cells fen’ g to have Y D X.n1/ [f’ fen’ g together with a map Y ! X inducing an isomorphism on homology. 20. Here is an example of CW complexes X, Y and a map f

X ! Y Š

such that f is isomorphism on H .X/ ! H .Y/ but f is not a homotopy equivalence. Consider

Q 1 has a picture The universal covering Y

where R is the real line (= universal covering of S1 ) and 2-spheres are attached Q1 ! Y Q 1 be translation by 1 on R viewed as at each integer point. Let T W Y Q Q 1 =fTg where T generates  1 .Y1 / as a an automorphism of Y1 . Thus, Y1 D Y Q group of covering transformations on Y1 . Q 1 by a map Attach a 3-cell to Y Qf

Q1 S2 ! Y and let the attaching map to Y1 be   ı Qf D f S2

Qf

/ Y Q1 @@ @@ f @@ @@     Y1

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19 Exercises

To give Qf, we map S2 to S2fTD0g with degree C2 and to s2fTD1g with degree 1

Let Y D Y1 [f e3 . Then, Hi .Y/ D 0

i¤1

H1 .Y/ Š Z .why‹/; and so the inclusion map i

X D S1 ,! Y induces an isomorphism on H and on  1 , but  2 .X/ D 0, and we will check that  2 .Y/ ¤ 0 so that i is not a homotopy equivalence. Observe that the Q 1 / are  1 .Y1 /-modules (why?) (the homology H .Y f1 / homotopy groups  i .Y Q 1 / corresponding is also a  1 .Y1 / module). Letting g be the generator of  2 .Y to the S2 at 0, we have as sets Q 1 / D fTn gm gn;m2Z   2 .Y

.why‹/:

Thus, as groups Q 1 / Š ZŒT   2 .Y

.why‹/:

Now Q Š H2 .Y/ Q  2 .Y/ Š  2 .Y/

.why‹/:

Finally, Q Š ZŒT =2T  1 ¤ 0 H2 .Y/ (since @e3 D 2S20  S21 / (why?). Remark. (i) There is a non-simply connected version of the Hurewicz theorem. It says that a map between two CW complexes, f W X ! Y, is a homotopy equivalence if and only if: (1) f W  1 .X/ !  1 .Y/ is an isomorphism.

19 Exercises

195

(2) f W H .XI ZŒ 1 .X/ / ! H .YI ZŒ 1 .Y/ / is an isomorphism. The point is that H .XI ZŒ 1 .X/ / is identified with the ordinary integral Q of X. Thus, condition (2) implies that homology of the universal cover X Qf W X Q !Y Q induces an isomorphism in homology and hence on homotopy Q D  1 .Y/ Q D 0/. groups (since  1 .X/ (ii) The dodecahedron group G is a perfect group (i. e., G D ŒG; G / acting freely on S3 , and so X D S3 =G is a 3-manifold with Hi .X/ D 0 .i ¤ 0; 3/; H0 .X/ Š H3 .X/ Š Z. Thus, X is a homology sphere, and the map f

X ! S3 obtained by collapsing the 2-skeleton of X to a point induces an isomorphism on homology. This example, due to Poincaré, gives another example showing that the homology does not suffice to determine the homotopy type of even a 3-manifold. 21. Problem on algebraic Euler characteristics. Recall that a finitely generated abelian group G has a rank ¡.G/ D dimR .G ˝Z R/. Suppose that .Cn ; @/ is a chain complex with Cn finitely generated and Cn D 0 for n < 0; n  n0 . Set cn D ¡.Cn / D rank of Cn Hn D nth homology of

fC ; @g

bn D ¡.Hn / D nth Betti number: Prove the relation (Eule–Poincaré-Hopf ) †n .1/n cn D †n .1/n bn : Hint: Use induction on n0 and the fact that if 0

00

0!G ! G!G ! 0 is an exact sequence of finitely generated abelian groups, then ¡.G/ D .G0 / C .G00 /:/ If X is a finite CW complex and cn D ¡.Cn .X// D # of n-cells in X bn D ¡.Hn .X// D nth Betti number of X ¦.X/ D †n .1/n bn D Euler characteristic of X; then you have proved

196

19 Exercises

†n .1/n cn D ¦.X/ .formula of Euler–Poincaré/: Exercises on Euler characteristic and spectral sequences. 22. Let F ! E ! B be a fibration where  1 .B/ acts trivially on H .F/. Suppose that H .B/ and H .F/ are finitely generated. Prove that (i) H .E/ is finitely generated. (ii) ¦.E/ D ¦.B/¦.F/ where “¦” means Euler characteristic. (Hint: Use the Serre spectral sequence to see that H .E/ can be “no larger” than H .B/ ˝ H .F/, and so H .E/ is finitely generated. To prove (ii) use Exercise (21) together with the relation ErC1 D homology of fEr ; dr g: The formula ¦.E/ D ¦.B/¦.F/ means that the Euler characteristic is multiplicative for fiber spaces. 23. Let F ! E ! Sn be a fibration over an n-sphere .n  2/. Deduce the exact sequence (Wang sequence) i

D

i

: : : ! Hp .E/ ! Hp .F/ ! HpnC1 .F/ ! HpC1 .E/ ! : : : where i W F!E is the inclusion and where D.u  v/ D D.u/  v ˙ uD .v/: (Hint. Since Hq .Sn / D 0 for q ¤ 0; n the E2 term of the spectral sequence looks like

Deduce that the only nonzero differential in the spectral sequence is dn . From this, conclude that

19 Exercises

197

(a) E2 D : : : D En1 ; EnC1 D EnC2 D : : : D E1 (b)

0

0

i

/ Hp .E/ O

/ Hp .F/ O

[

D

/ Ep;0 1

/ Ep;0 n

/ HpnC1 .F/ O

/ HpC1 .E/ O

D dn

/ 0

[

/ EpnC1;n n

/ EpnC1;n 1

/ 0

The point of this exercise is to illustrate the general principle: When the E2 -term of the spectral sequence has most terms zero, then it simplifies considerably.)  

24. Another problem on spectral sequences. Let Sn ! E ! B be a fibration with fiber a sphere and where  0 .B/ D 0 and  1 .B/ acts trivially on H .Sn /. Deduce the exact cohomology sequence (Gysin sequence) §

 

 

: : : ! Hp .E/ ! Hpn .B/ ! HpC1 .B/ ! HpC1 .E/ ! : : : where § is cup product with a class e 2 HnC1 .B/ (e is the Euler class, defined by obstruction theory in Exercise (27) below). (Hint. The E2 term in the spectral sequence is 0

n

0 0

0 dn+1

0 0

0

and so E2 D : : : D En ; EnC2 D : : : D E1 and dnC1 is the only nonzero differential. The Euler class e D dnC1 .1/ where 1 2 Hn .F/ Š E0;n 2 is the generator of Hn .Sn /, and where now the argument proceeds as in the previous exercise using dn .’ ˝ “/ D ’ ˝ dn .“/ for ’ 2 H .B/; “ 2 H .F//. 25. Thom isomorphism. Let E ! X be an orientable vector bundle of dimension n. Choose a metric on E. Let D.E/  E be the unit disk sub-bundle with respect to this metric, and let S.E/ be the unit sphere sub-bundle. We have a relative fibration .D.E/; S.E// ! X; the fibers are pairs .Dn ; Sn1 /. p; q There is a relative version of the Serre spectral sequence where E2 D p q n n1  H .XI H .D ; S // and E1 ) H .D.E/; S.E// Using this, establish the Thom isomorphism:

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19 Exercises

There is a class U 2 Hn .D.E/; S.E// so that [U

H .X/ Š H .D.E// ! H .D.E/; S.E// is an isomorphism. 26. Still another problem on spectral sequences. Let Pn ! E ! B be a fibration with complex projective space as fiber. Recall that H .Pn / Š ZŒx =.xnC1 /;

x 2 H2 .Pn /:

Suppose that there exists a class  2 H2 .E/ such that jPn D x. (Here jPn means restriction of  to fiber). Show that we have the ring isomorphism nC1q H .E/ Š H .B/Œ =. nC1  †nC1 cq  nC1q / qD1 .1/

where cq 2 H2q .B/. (Hint: There is a natural map H .B/Œ ! H .E/  

coming from H .B/ ! H .E/ and the ring structure on H .E/. By assumption 0;2 2 n there exists 0 2 E0;2 1 corresponding to x 2 E2 Š H .P /. Thus, d2 X D d3 X D : : : D 0: Since E2 Š H .B/Œx =.xnC1 /, deduce that E2 D E3 D E4 D : : : D E1 : Thus H .B/Œ ! H .E/ is surjective. Additively there is an isomorphism H .E/ Š ˚nqD0 H .B/ q ; and so the kernel of the restriction mapping H .E/ ! H .Pn / is ˚nqD1 H .B/ q . But  nC1 jPn D 0, and so we obtain a relation in the ring H .E/ of the form nq cq  nC1q ;  nC1 D †nC1 qD1 .1/

Deduce finally that

cq 2 H2q .B/:

19 Exercises

199 nC1q H .B/Œ =. nC1  †nC1 cq  nC1q / !H .E/ qD1 .1/

is an injection.) (Note: The classes cq 2 H2q .B; Z/ are Chern classes as explained in Exercise (30) below. This definition, due to Grothendieck, is the easiest.) 27. A final problem on spectral sequences. Let Un be the unitary group of n  n matrices A satisfying Atr A D I. Prove that the cohomology ring H .Un / Š ^.x1 ; x3 ; : : : ; x2n1 / is an exterior algebra with generators in dimensions 1, 3, : : : ; 2n  1. Thus, there is an isomorphism of rings H .Un / Š H .S1  S3  : : :  S2n1 /: Hint: Writing A D .e1 ; : : : ; en / where the ei are column vectors in Cn , the unitary condition is that e1 ; : : : ; en should give a unitary frame. The map Un ! S2n1 A 7! e1 gives a fibration Un1 !

Un ? ? y S2n1

(why?). By induction we have H .Un1 / Š ^.x1 ; : : : ; x2n3 /. Now use the Wang sequence (Exercise 23). Exercises on obstruction theory. In the following exercises on obstruction theory, we assume that F ! E ! B is a fibration where  0 .B/ D 0 and  1 .B/ acts trivially on the cohomology of the fiber.   28. Euler classes and Euler numbers. Given F ! E ! B, assume that  i .F/ D 0 for i < n  1. Denote by B.k/ the kskeleton of B (always assuming that B is a CW complex). Show that ¢

(a) there exists a cross-section B.n1/ ! E (recall that a cross-section ¢ of E ! B is a map ¢ W B ! E such that  ¢ D identity. Think of this as ¢.x/ is contained in Fx , the fiber over x, for all x 2 B). ¢1 (b) Any two sections Bn2 ! E are homotopic. ¢2

200

19 Exercises ¢

(c) The obstruction to finding B.n/ ! E is given by a cohomology class c.E/ 2 Hn .B; Hn1 .F//. Remark. (a), (b), (c) imply that in a spherical fibration Sn1 ! E ! B, there exists a unique class e.E / 2 Hn .B; Z/, the Euler class, such that Sn1 ! E ! B has a section over B.n/ , e.E / D 0. In case B is an oriented n-manifold, we may define the Euler number of the spherical fibration by e.E/ D< e.E/; B > : In this case Sn1 ! E ! B has a section if and only if the Euler number e.E/ is zero. In this exercise, use that for each n-cell en . on B, Ejen  en  F: 29. Let B be a compact, oriented n-manifold and Rn ! E0 ! B an oriented vector bundle (Exercise: Give the definition). Choose a metric in Rn ! E0 ! B and let Sn1 ! E ! B be the unit sphere bundle. Let e(E) be the Euler number as defined in the previous exercise, so that we have a nowhere zero section of Rn ! E0 ! B , e.E / D 0. (a) Verify that to compute e.E/, you do the following: ¢ Find a nonzero section B.n1/ ! E .E D E  zero section) (here we are assuming that B has been triangulated). For each n-cell en’ , we have Ejen’ ¢ en’  Rn and so ¢ gives @en’ ! Rn  f0g, or equivalently ¢’

@en’ ! Sn1 then e.E/ D †’ degree.¢’ /: Why is it sufficient to use any section ¢? (b) Consider the universal line bundle

C

1

/ E

 

/ P1

where

  Œz0 ; z1 is the complex line through .z0 ; z1 / in C . Using the Euclidean metric on C2 , the associated sphere bundle is the Hopf fibration 2

S1 ! S3 ! P1 : Over P1  f1g, where 1 is the point with homogeneous coordinates Œ0; 1 /, choose the cross-section

19 Exercises

201

¢.Œz0 ; z1 / D .1; z1 =z0 /: Using this, show that e.E/ D 1: 30. Definition of Chern classes using obstruction theory. (a) Recall the Stiefel manifold S.n  k C 1; n/ of n  k C 1 frames in Cn . We computed (cf. Exercise 16)  i .S.n  k C 1; n/ D 0;

i < 2k  1

 2k1 .S.n  k C 1; n// Š Z: Let Cn ! E ! B be a complex vector bundle over a CW complex B. Using Exercise (28), deduce that there is a class ck .E/ 2 H2k .B; Z / such that Cn ! E ! B.2k/ has a field of n  k C 1 frames , ck .E/ D 0. Show that if we have a diagram of mappings E ! F ? ? y

E0 ? ? y

B ! B0 f

0

with F a linear isomorphism on the fibers, then f ck .E / D ck .E/. The classes c0 .E/ D 1; c1 .E/; : : : ; cn .E/ are the Chern classes of Cn ! E ! (b) Using Exercise 29(b), show that for the universal line bundle C ! E ! P1 the class c1 .E/ is equal to minus the standard generator in H2 .P1 ; Z/ coming from the complex orientation of P1 . (c) Let C ! En ! Pn be the universal line bundle over Pn . 1 .z0 ; : : : ; zn / D line (tz0 ; : : : ; tzn / in CnC1 ). Using Exercise 30(a) above, show that c1 .En / D g where g 2 H2 .Pn ; Z/ is the generator that restricts to the natural generator on P1 . Remark. Given a vector bundle Cn ! E ! B, let P.E/ be the bundle of lines in E; thus, the fiber P.E/x D P.Ex / is the set of lines through the origin in Ex Š Cn . Representing points in P.E/ as pairs .x; /, where  is a line in Ex , there is   a line bundle C ! L.E/ ! P.E/ where  1 .x;  / D fline g  Ex . Consider the fibration Pn1 ! P.E/ ! B. The first Chern class c1 .L.E// 2 H2 .P.E/; Z/

202

19 Exercises

restricts to -{ generator of H2 .Pn1 I Z/g by Exercise 30(a) and (c). From this, we deduce that Pn1 ! P.E/ ! B satisfies the condition in Exercise 25, and so for  D c1 .L.E//, H .P.E// Š H .B/Œ =. n  †nqD1 .1/nq cq .E/ nq /: The classes cq .E/ 2 H2q .B/ are the same Chern classes as those defined by obstruction theory (this is not trivial). The Chern classes are the fundamental invariants for measuring the nontriviality of a vector bundle Cn ! E ! B. 31. Hopf theorem on singularities of a vector field. Given a vector field ™ D †i ™i @=@xi in a neighborhood of the origin in Rn and having an isolated zero at x D 0, there is associated an integer, the index ind0 .„/ of ™ at x D 0, defined as follows:

Q In a small sphere kxk D ", the function ™.x/ D .™1 ; : : : ; ™n / is nonzero, and so Q Q f.x/ D ™.x/=k ™.x/k gives a map f

Sn1 ! Sn1 : Then, ind0 .™ / D degree (f). Let B be a compact, oriented n-manifold and ™ a vector field having only isolated zeros.

We may view ™ as a section of the tangent bundle T ! B. Prove the formula e.T/ D †™.x/D0 indX .™/ Remark. You may assume that B has a CW decomposition with only cells of dimension  n (why?) and that ™ has zeros only at interior points of the n-cells. Now try to apply Exercise 29(a).

19 Exercises

203

This proves that †™.x/D0 indX .™/ is independent of the vector field ™. To actually calculate †™.x/D0 indX .™/ for a particular vector field ™, you make a smooth triangulation of B and use the second barycentric subdivision to find a particularly nice vector field. There is a beautiful intuitive discussion of this on pp. 201–203 of [25]. For this vector field ™, you find the formula †™.x/D0 indX .™/ D †.1/p f# of p-cells in cell decomposition of Bg Putting these formulas together and using Exercise 21 gives the final result e.T/ D ¦ .B/ D †™.x/D0 indX .™/ which is a famous theorem of Poincaré-Hopf. 32. Two lemmas. 0

(a) Let X D X [f enC1 be obtained by attaching an .n C 1/-cell to X by f

@enC1 ! X. Using the definition, show that the obstruction to extending id

0

X ! X to X ! X is [f] 2  n .X/. (b) Given A  X, a cohomology class ’ 2 Hq .X/, and a cocycle ’Q 2 ZQ q .A/ 0 which represents the restriction ’jA, show that there is a cocycle ’Q 2 ZQ q .X/ 0 such that ’Q jA D ’. Q (Here we are using cellular cochains.) 33. Cellular approximation theorem. Suppose that X, Y are CW complexes and g

f

X ! Y is a continuous map. Show that f is homotopic to a map X ! Y which is cellular in the sense that g.X.n/ /  Y.n/ for the respective n-skeleton. (Hint: For simplicity, assume first that dim X < fn

1 and suppose by induction that we have a map X ! Y homotopic to f with fn .X.n/ /  Y.n/ . Show now that we have  nC1 .Y.nC1/ ; Y.n/ / ! nC1 .Y; Y.n/ / ! 0 and that  nC1 .Y.nC1/ ; Y.n/ / is a free abelian group on the n C 1 cells in X (use relative Hurewicz together with Hq .Y; Y.n/ / D 0 for 0  q  n). Now let enC1 be an n C 1 cell of X. Then, fn W .enC1 ; @enC1 / ! .Y; Y.n/ / is an element in  nC1 .Y; Y.n/ /. By the above remark, fn may be deformed into Qfn W .enC1 ; @enC1 / ! .Y.nC1/ ; Y.n// . We will be done if we show that, during the homotopy fn  Qfn , the map may be kept constant on @enC1 (why is this necessary?). Thus, we must prove the

204

19 Exercises ’

Lemma. Given A  B  Y, if .enC1 ; Sn / !.Y; A/ is homotopic to “

”

.enC1 ; Sn / !.B; A/, then ’ is homotopic to .enC1 ; Sn / !.B; A/ where all maps in the homotopy are constant on Sn D @enC1 . Proof. Given

and a homotopy ’t of ’ with ’0 D ’ ’1 D “ ’t .Sn /  A; put a “collar” around Sn ,! Sn  I and define ”t by the picture

Proceeding inductively over the .n C 1/-cells, we now have fnC1 W X.nC1/ ! Y.nC1/ fnC1 jX.n/ D fn fnC1  f on X.nC1/ Apply the homotopy extension property to extend fnC1 to all of X.



What modification is necessary in case dim X D 1‹ 34. Whitehead products. (Preliminary exercises on wedges and products of spheres.) (a) Let Sp _ Sq  Sp  Sq be the usual embedding. Considering the n-sphere as Sn D In =@In ; show that: (i) The quotient Sp  Sq =Sp _ Sq is homeomorphic to SpCq . (ii) H` .Sp  Sq ; Sp _ Sq I G/ Š H` .SpCq I G/; ` > 0 for any coefficient group G.

19 Exercises

205

(Hint: For (i), Sp  Sq =Sp _ Sq D .Ip =@Ip  Iq =@Iq /=.Ip =@Ip  / [ .  Iq =@Iq /  IpCq =@IqCq

.why‹/:

For (ii) use the previous exercise about excision.) (Note: This show that Sp  Sq D .Sp _ Sq / [f epCq where fW SpCq1 ! p S _ Sq , for some map f representing an element Œf 2  pCq1 .Sp _ Sq /:/ (b) Recall the definition of the Whitehead product: Given Sp _ Sq  Sp  Sq , i

the obstruction to extending the identity map Sp _Sq ! Sp _Sq to Sp Sq ! Sp _ Sq is a cohomology class „.i/ 2 HpCq .Sp  Sq ; Sp _ Sq I  pCq1 .Sp _ Sq // Š  pCq1 .Sp _ Sq /. By definition the Whitehead product Wp;q 2  pCq1 .Sp _ Sq / corresponds to ™.i/ under this isomorphism. Using Exercise 32(a) and Part (i) above, we have Wp;q is the class of the f

attaching map @epCq ! Sp _ Sq used in constructing Sp  Sq by attaching a cell to Sp _ Sq . By considering the cohomology rings of Sp  Sq and Sp _ Sq _ SpCq , show that Wp;q ¤ 0. If ’ 2  p .X/ and “ 2  q .X/ are represented by maps ’W Q Sp ! X and Q“W Sq ! X, then the Whitehead product Œ’; “ 2  pCq1 .X/ is represented by the composite mapping SpCq1 ! Sp _ Sq ! X ’ Q _“Q

f

where f is the above attaching map. Thus Œ’; “ D .’ _ “/ .Wp;q / where Wp;q 2  pCq1 .Sp _ Sq / was defined above. Show that the Whitehead product satisfies the relations: Œ’; “ D .1/p;q Œ“; a

.symmetry/

Œ’; Œ“; ” D ŒŒ’; “ ; ” C .1/ Œ“; Œ’; ” pq

.Jacobi identity/;

where ’ 2  p .X/ and “ 2  q .X/. 35. Transgression. (The general case discussed here is not often used. The most   important case is the one dealt with in Exercise 36.) Let F ! E ! B be

206

19 Exercises

a fibration with  0 .B/ D f0g;  1 .B/ D f0g;  1 .F/ D f0g, so that the Serre spectral sequence is applicable. Consider the cohomology sequences (with q > 0) Hq .E/ ! Hq .F/ ! HqC1 .E; F/ ! HqC1 .E/ • x x ?  ?  ?  ?  HqC1 .B; x0 / ! HqC1 .B/ Š

The transgression £ is the map from the subgroup Tq .F/ D •1   HqC1 .B/  Hq .F/ to the quotient group GqC1 .F/ D HqC1 .B/=ker   v defined by £ .•1   ’/ D ’: This map is of fundamental importance in the cohomology theory of bundles. Using the proof of the Serre spectral sequence, show that for q  1 0;q

0;q

(i) Tq .F/ EqC1  E2 Hq .F/. (ii) (iii)

qC1; 0 GqC1 .F/ Š EqC1 is a quotient 0; q qC1; 0 £ D dqC1 : EqC1 ! EqC1 .

qC1; 0

of E2

Š HqC1 .B/.

(Hint: This requires that you look carefully into the construction of the spectral sequence. To get some idea, try the case q D 1.  

q D 1: To prove (i) and (ii) in this case, you must show that H2 .B/ ! H2 .E; F/ is an isomorphism. This follows from H2 .E; F/ H2 .B/ (use the relative Hurewicz isomorphism and  i .E; F/  i .B// and the fact that H2 .B/, H2 .E; F/ are torsion free (why? use  1 .B/ D 0 D  1 .E; F/.) The fact that £ D d2 in this case follows from the definition of d2 as a coboundary in an exact cohomology sequence of pairs .B.n/ ; B.n2/ / where B.q/ D  1 .X.q/ /. 36. Some examples of transgression. (This exercise depends on Exercise 35, which you should in any case read over carefully first; cf. also the discussion at the beginning of Chap. 12.) (a) Suppose that F ! E ! B is a fibration and Hq .E/ D 0 for q > 0 (e.g., if E is contractible). Show that fHi .F/ D 0 for 0 < i < qg

19 Exercises

207

, fHj .B/ D 0 for 0 < j < q C 1g and that the transgression is an isomorphism 

£W Hq .F/ ! HqC1 .B/: (Hint: It will suffice to assume that Hi .F/ D 0 D Hj .B/ for 0 < i < q; 0 < j < q C 1 and show that 0;q

0;q

(i) E2 D : : : D EqC1 qC1;0

(ii) E2

qC1;0

D : : : D EqC1 0;q

qC1;0

Q qD1 is an isomorphism. (iii) dqC1 W EqC1 !E The E2 -term of the Serre spectral sequence looks like Hq .FI I Z/ FF FF FF FF FF FF  FF FF FF FF FF dqC1 FF  FF FF FF FF FF FF FF  FF FF FF FF " Z 0  0 HqC1 .BI Z/ 0; q

From this, it follows that dr D 0 on E2 for 2  r  q, and this gives (i) and (ii). 0; q qC1;0 Now then dqC1 W EqC1 ! EqC1 must be an isomorphism, since 0; q

qC1; 0

otherwise we would get a nontrivial element in E1 or E1 since H .E/ D 0:/ (b) Let

, but E1 D 0

K . ; n  1/ ! P # K. ; n/ be the path fibration .n  2/. Show that in the Serre spectral sequence,

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19 Exercises

E20; n1 D : : : D En0; n1 Š Hn1 .K. ; n  1// 0 n; 0 n En; 2 D : : : D En Š H .K. ; n  1// dn

0 En0; n1 ! En; n is an isomorphism:

(Hint: Use part (a) of this exercise above.) (c) Show that if we have a map of fiber spaces F0 ! ? ? y

F ? ? y

E0 ! Qf ? ? y

E ? ? y

X0 ! X f

then transgression is natural. (This follows either from the definition or from the spectral sequence interpretation, both of which are given in Exercise 35.) 37. Chern classes and transgression. Recall the Stiefel manifold S.n; N/ of n-frames in CN and that  q .S.n; N// D 0;

0 < q < 2N  2n C 1:

Using the inclusions CN  CNC1 , we have S.n; N/  S.n; N C 1/ and if we let S.n/ D n-frames in C1 be the infinite Stiefel manifold, then (i)  q .S.n// D 0 for q > 0 (If you don’t like 00 100 here, take S.n; N/ for N arbitrarily large.) Next, recall the Grassmann manifold G(n) of n-planes in C1 and the fibration U.n/ ! S.n/ ! G.n/ where S.n is the Stiefel manifold of unitary n-frames in C1 and the map S.n/ ! G.n/ assigns to each frame the subspace that it spans. Recall that (ii) H .G.n// D CŒx1 ; x2 ; : : : ; xn is a polynomial algebra with generators

19 Exercises

209

xq 2 H2q .G.n//: Finally, recall that for the unitary group, the cohomology (iii) H .U.n// D ^.y1 ; : : : ; yn / is an exterior algebra with generators yq 2 H2q1 .U.n//: Prove that each yq transgresses and £ .yq /  xq mod .x1 ; : : : ; xq1 / (Hint: From equation (i) it follows that Hq .S.n// D 0 for q > 0. The E2 of the spectral sequence looks like y3 y2 y1 y2 y1

Z Z Z 0 Z Z

0

Z

0

x1

Z˚Z

0

fx21 ; x2 g

Z˚Z˚Z fx31 ; x1 x2 :x3 g

with yq 2 H2q1 .U.n// and xq 2 H2q .G.n//. Now argue as follows: d2 y1 D x1

since H2 .S.n// D 0

d2 y2 D 0 D d3 y3

because of the 00 s in the spectral sequence

) y2 is transgressive and d4 y2  x2 mod .x21 / d2 y3 D 0 since d2 y1 y2 x1 D x21 y2 C x1 x2 y2 ¤ 0 and d3 y3 D d4 y3 D d5 y3 D 0 because of 00 s in the spectral sequence ) y3 is transgressive and d6 y3  x3 mod .x1 x2 /; etc:/ Remark. It can be proved that xq 2 H2q .G.n// is the qth Chern class of the universal vector bundle Cn !

E ? ? y G.n/

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19 Exercises

Taking into account Exercise 30 above, this gives a third definition of Chern classes (these definitions are all the same up to sign!) Roughly speaking, these various definitions have the following advantages: 8 9 < Useful for “classical” = Definition 1 using ! algebraic topology, such as : ; obstruction theory existence of vector fields, etc. 8 9 8 9 Quickest definition and the > ˆ ˆ > Definition 2 using < = < = easiest one to prove the ! the projective bundle : ; ˆ duality theorem, also it is > ˆ > : ; P.E/ useful in algebraic geometry 9 8 > ˆ This definition is useful in > ˆ 8 > 9 ˆ > ˆ > ˆ Defn. 3 using algebraic geometry and also ˆ > > ˆ ˆ > > ˆ < = = < transgression and is the definition which shows ! ˆ ˆ cohomology of the > how to compute Chern classes > ˆ > > ˆ : > ; ˆ > ˆ > ˆ Grassmannians using curvature via deRham’s > ˆ > ˆ ; : theorem





38. Cohomology of K.Z; 3/. Prove that 8 q D 0; 3 0/ .why‹/ 

H .K.Z; 2// D H .CP1 / D Z Œx ;

x 2 H2 .K.Z; 2//:

The E2 term has H .K.Z; 2//, which is a polynomial algebra on a generator x in degree 2, up the left-hand side and H .K.Z; 3// along the bottom row. Thus, in the range, we are considering d2 D 0 and d3 looks like

19 Exercises

211

Z I II II II II II II 3 II 0 II II II II II I$ Z I 0 0 II II II II II II 2 II 0 0 0 II II II II II I$ 0 0 Z I II II II II II II Š II 0 0 0 II II II II II I$ Z 0 0

Z

0

Z

0

Z

where y 2 H3 .K.Z; 3// is a generator, by Hurewicz. Then, show that d2 D 0 for low values and .why‹/ d3X D y .why‹/ d3 x2 D 2xy d3 y D 0 .obvious/: 0; 4 =d E Š Z=2Z with generator xy: ) E3;2 3 3 3 3;2 But E3;2 4 D 0 (why? use E1 D 0). d3

d3

6;0 0;4 3;2 ) kerfE3;2 3 ! E3 g D i:mfE3 ! E3 g 4 ) E6;0 3 Š Z=2Z .to prove this you must show that H .K.Z; 3// D 0

) H6 .K.Z; 3// Š Z=2Z: 39. Homotopy groups of spheres. Prove that  4 .S2 / Š Z=2Z:

212

19 Exercises

(Hint: To do this, use the Postnikov tower (= P. T.) for the 2-sphere S2 . Letting .S2 /n be the nth stage of the P. T., recall that there is a map S2 !fn .S2 /n such that  i .S2 / Š  i ..S2 /n / i  n and 

Hi .S2 / ! Hi ..S2 /n /

0  i  n C 1:

.fn /

Step one: Step two:

.S2 /2 D K.Z; 2/ D CP1 since  2 .S2 / D Z. There is a fibration K.Z; 3/ ! .S2 /3 # K.Z; 2/ D .S2 /2

since  3 .S2 / Š Z. Moreover, H4 ..S2 /3 / D 0. Look in the spectral sequence of this fibering, whose E2 is, using the previous exercise, Z=2Z

0

0

Z

0

0

Z

0

Z

0

Z

Show then that d4 y D x2 , and from this, deduce that for 0  q  6

19 Exercises

213

8 q D 0; 2 n. Proof. Recall the fibration Ur ! S.r; N/ ! G.r; N/, where S.r; N/ is the Stiefel manifold of r-frames in CN and also that  i .S.r; N// D 0 for i < 2N  2r C 1 )  i .S.r; N// D 0 for i < n C 1 under the conditions of the theorem. Let’s first show that every vector bundle Cr ! E ! X is of the form f Ur for some map X ! G.r; N/. Letting X.k/ be the k-skeleton, suppose we have f.k/

X.k/ ! G.r; N/ f.k/ Ur Š EjX.k/ : Now the restriction of E to each .k C 1/-cell ekC1 is trivial (why?), so that we have EjekC1 Š ekC1  Cr f.k/

@ekC1 ! G.r; N/ . /

) f.k/ Ej@ekC1 Š @ekC1  Cr : f.k/

This gives us a lifting of @ekC1 ! G.r; N/ to

19 Exercises

215

S.r; N/ ; v v vv vv v  vv f.k/ / G.r; N/ g

@ekC1

by taking the r-frame to be the coordinate frame in Cr under the isomorphism . /. Since  k .S.r; N// D 0, the map g extends over ekC1 , and in this way, we may extend f.k/ over the .k C 1/ -skeleton (why?). Thus, ŒX; G.r; N/ ! Vectr .X/ is onto, and now a relative version of the same argument shows that it is oneto-one.)  Remark. A special case of this theorem is the isomorphism Vectr .Sn / Š  n .G.r; N// Š  n .BUr / .N large relative to r, n/ 0

We call two vector bundles E, E on a space X stably isomorphic if E ˚ t` E0 ˚ t`0 where t` is the trivial bundle of rank `. The equivalence classes of stable vector bundles will be denoted by K(X), and then the above theorem gives K.X/ Š ŒX; BU : Returning to the n-sphere, we have then K.Sn / Š  n .BU/; and so Bott periodicity computes the stable vector bundles over spheres. 41. Axioms for homology. Suppose we are given a covariant functor X!H .X/ D ˚p0 Hp .X/ from finite CW complexes to abelian groups such that f

f

(i) X ! Y induces H .X/ ! H .Y/ which depends only on the homotopy class of f.

216

19 Exercises



Z D0 0 >0 (iii) Given X  Y, if we define (ii) H .pt/ D

H .Y; X/ Š H .Y=X/ (this property forces excision to be true), then the exact homology sequence for a pair holds. Show that H .X/ D H .XI Z/ is ordinary homology. (Hint: By (ii), this is true if dim X D 0. Suppose by induction it is true when Z; D 0 dim X < n. If Dn is the n-disk with @Dn D Sn1 , then H .Dn / D f g 0; n > 0 Z; D 0; n  1 by (i) and H .Sn1 / Š f g by induction. The exact homology 0; otherwise sequence then gives . /

H .Dn ; @Dn / Š H .Dn ; @Dn /:

Suppose now that X D Y [f en where dim Y  n  1. Then, we have @

Hp .Y/ ! Hp .X/ ! Hp .X; Y/ ! Hp1 .Y/ oo Hp .Y/ ! Hp .X/ ! Hp .X; Y/ ! Hp1 .Y/ where the isomorphism Hp .X; Y/ Hp .X; Y/ follows by . / above. Taking p D n, we get a map Hn .X/ ! Hn .X/ which is an Š. For p D n  1, we have Hn1 .X/ Š Hn1 .Y/=Hn .X; Y/ and Hn1 .X/ Š Hn1 .Y/=Hn .X; Y/: This gives a map Hn1 .X/ ! Hn1 .X/ which is an isomorphism. The remaining Hp ; Hp for p < n  1 are already isomorphic (why?). Now complete the proof by induction on the number of n-cells in X. Remark. The analogous theorem for cohomology is also true with the same proof. In general, an extraordinary cohomology theory is a contravariant functor X!K .X/

19 Exercises

217

from spaces X to abelian groups which satisfy the cohomology axioms corresponding to (i) and (iii), but not necessarily the axiom specifying the value on the one-point space. Using Bott periodicity and denoting by Sn X the n-fold suspension of X, one obtains K-theory with Kn .X/ D K .Sn X/ defn:

is an extraordinary cohomology theory. 42. Little problems on filtrations. (i) Show that if A D A0 A1 : : : Am D f0g and B D B0 B1 : : : Bn D f0g are filtered abelian groups and if f: A ! B is a filtration preserving  homomorphism inducing isomorphisms Ap =ApC1 ! Bp =BpC1 , then f itself is an isomorphism. (ii) Using (i) show that if a map of Serre spectral sequences induces an isomorphism on E1 , then the abutments are isomorphic. (iii) As an example of the failure of the converse of (ii), let X be a CW complex id

with base point x0 2 X. Consider the trivial fibrations X ! x0 and X ! X. The identity map X ! X induces a map of these fibrations which is an isomorphism on the abutment of the spectral sequences (which is H .X/ in both cases). Show from the definitions that this map is not an isomorphism on E1 unless H .X/ D 0 for > 0. 43. The comparison theorem for spectral sequences. Let Erp;q E0 rp;q be two homology spectral sequences (same as cohomology spectral sequences with upper and lower indices interchanged and with arrows reversed; also the proof of the Serre spectral sequence for homology is the same as for cohomology). Suppose that we have a map Erp;q !E0 p;q r

.for all r and the map commutes with dr 0 s/

such that E20;q Š E0 0;q 2

and 1

0 E1 p;q Š E p;q

for all p; q  0:

0

2 Show that E2p;0 Š Ep;0 for all p  0.

Remark. This may be interpreted as saying that if we have a map between fiber spaces

218

19 Exercises

F ! ? ? y

F0 ? ? y

E ! ? ? y

E0 ? ? y

B ! B0 



inducing isomorphisms on H .F/ ! H .F0 / and H .E/ ! H .E0 /, then H .B/ ! H .B0 / is also an isomorphism (compare the previous exercise). Step one: Show that E2p;0 Š E0 p;0 for a fixed p ) 2

E2p;q Š E0 p;q for all q: 2

Step two: Prove that E20;0 Š E0 20;0 and E21;0 Š E0 1;0 : 2



Step three: Check that E30;1 ! E0 30;1 (use E1 0;1 ); 



E20;1 ! E0 0;1 and E32;0 ! E0 2;0 2

3

.use E1 2;0 /:

Use the exact sequence 0 ! E32;0 ! E22;0 ! KerfE20;1 ! E30;1 g ! 0 to conclude that 

E22;0 ! E0 2;0 : 2

Step four: For general p, we attack Erp;0 by descending induction on r  2 until 

we find E2p;0 ! E0 2p;0 . pC1 



0 First Ep;0 ! E0 p;0 (why? use E1 p;0 ) and E0;p1 ! E 0;p1 (why?). Using pC1

pC1

pC1





E20;p1 ! E0 20;p1 and ascending induction on r, show that E0;p1 ! E0 0;p1 . p



Conclude that Ep;0 ! E0 p;0 : Step five: Using ascending induction on k, prove that p

p



Ekpr;s ! Ekpr;s for 2  k  r; s  0:

p

19 Exercises

219

In particular, 

Erpr;r1 ! E0 pr;r1 : r



0 rCk Using this and descending induction on k  1, prove that ErCk pr;r1 ! E pr;r1 . In particular, 

0 ErC1 pr;r1 ! E pr;r1 : rC1



Step six: Show that image fErp;0 ! Erp1;r1 g ! image fE0 rp;0 ! E0 rp1;r1 g n o  0rC1 r r using ErC1 ! E and the fact that ker E !E pr;r1 pr;r1 pr;r1 p2r;2r2 agree o n with ker E0rpr; r1 E0rp2r; 2r2 . 



0rC1 Conclude that Erp; 0 ! E0rp;0 using ErC1 p; 0 ! Ep; 0 .

44. Homotopy of wedges of 2-spheres. Let X D S21 _ : : : _ S2m be a wedge of 2-spheres. Then the second homotopy group 2 m  2 .X/ Š ˚m iD1  2 .Si / Š Z ;

and there are Whitehead products  ij D ŒS2i ; S2j 2  3 .X/: Show that these products for i  j give a free Z-basis for  3 .X/. Thus,   2 m.mC1/=2  3 _m : iD1 Si Š Z (Hint: This problem is a good exercise in Postnikov towers. Since  2 .X/ Š 2 ˚m iD1  2 .Si /, the P.T. for X begins with X2 D … m iD1 K.Z; 2/: The next step is / X3 x< x x xx xx x  xx f2 / X2 X

K. ; 3/

f3

We want to determine what   is. From the exact cohomology sequence of .X2 ; X/, we have

220

19 Exercises

H3 .X;  / ! H4 .X2 ; XI  / ! H4 .X2 ;  / ! H4 .X;  / jj jj 0 0 Since  i .X2 ; X/ D 0 for 0  i  3 (why?), the pair .X2 ; X/ begins with cells in dimension 4, and so H4 .X2 ; XI  / Š Hom.H3 .X2 ; X/;  / Š Hom. ;  / .why‹/: Thus, we have 0 ! Hom. ;  / ! H4 .X2 ;  / ! 0: Since H .X2 I Z/ is free, we obtain then 

0 ! Hom. ;  / ! Hom.H4 .X2 /;  / ! 0: From this, we see that   is free abelian of finite rank and Hom. I Z/ Š H4 .X2 I Z/ Š Zm.mC1/=2 : Thus,   is a free abelian group of rank m.m C 1/=2. It remains to identify   Š Zm.mC1/=2 as having a basis given by the Whitehead products ŒS2i ; S2j .i  j/. To do this, replace K.Z; 2/ by CP2 in the P.T. for X. This is okay since  i .CP2 / Š  i .K.Z; 2// for 0  i  4. Thus, we have K. ; 3/

_S

v

2

v

v

v

/ X3 v:

 / Q CP2

(_P1 is the 2-skeleton of … CP2 ). The k-invariant is by definition the primary obstruction to finding a section to the inclusion _P1 ,! … CP2 : The primary obstruction is a class in H4 .… CP2 ; 3 ._P1 //:

19 Exercises

221

Recall that the Whitehead product of the generators of 2 .S20 / and 2 .S21 / is the obstruction to extending the identity S20 _ S21 ! S20 _ S21 to a map S2  S2 ! S2 _ S2 : This essentially completes the desired identification, once we show that the usual Hopf map S3 ! S2 represents the Whitehead product of the generator of 2 .S2 / with itself (why?) Consider the 4-skeleton of … CP2 .

References

1. Baumslag, G.: Lecture notes on nilpotent groups. In: CBMS Regional Conference Series in Mathematics, vol. 2. American Mathematical Society, Providence (2007) 2. Chen, K.T.: Iterated path integrals. Bull. Amer. Math. Soc. 83, 831–879 (1977) 3. Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Homotopy theory of compact Kahler manifolds. Inv. Math. 29, 245–274 (1974) 4. Donaldson, S.: An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18, 279–315 (1983) 5. Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom. 17, 357–453 (1982) 6. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory. AMS/IP Studies in Advanced Math, vol. 46. American Mathematical Society and International Press, Providence (2009) 7. Greenberg, M.: Lectures on Algebraic Topology. W.A. Benjamin, New York (1977) 8. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classic Series, Wiley, New York (1994) 9. Hu, S.T.: Homotopy Theory. Academic, New York (1959) 10. Kadeishvili, T.: Cohomology C1 -algebra and rational homotopy type. Banach Center Publ. 85, 225–240 (2009). Available at arXiv0811.1655 (Math.AT) 11. Kan, D.: A combinatorial definition of homotopy groups. Ann. Math. 67, 282–312 (1958) 12. Kirby, R., Siebenmann, L.: Foundational essays on topological manifolds, smoothings, and triangulations. In: Annals of Mathematics Studies vol. 88. Princeton Univeristy Press, Princeton (1977) 13. M. Kontsevich. Homological algebra of mirror symmetry. Available at arXiv:alggeom/9411018v1, 1994. 14. Mal’cev, A.: On a class of homogeneous spaces. Amer. Math. Soc. Transl. 9, 276–307 (1962) 15. Manolescu, C.: Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture (2013). Available at arXiv:1303.2354 16. Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics. In: Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence (2007) 17. Milnor, J.: On spaces having the homotopy type of a CW complex. Trans. Amer. Math. Soc. 50, 272–280 (1959) 18. Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. IHES 48, 137–204 (1978) 19. Quillen, D.: Rational homotopy theory. Ann. Math. 90, 205–295 (1969)

P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, DOI 10.1007/978-1-4614-8468-4, © Springer Science+Business Media New York 2013

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