Topology of Closed One-Forms

http://dx.doi.org/10.1090/surv/108

Mathematical Surveys

and Monographs

Volume 108

t ^ E M ^ /

Topology of Closed One-Forms

Michael Farber

American Mathematical Society

E D I T O R I A L C O M M I T T E E

Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu

J. T. Stafford

2000 Mathematics Subject Classification. Primary 58E05, 57R70; Secondary 57R30.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-108

Library of Congress Cataloging-in-Publication D a t a Farber, Michael, 1951-

Topology of closed one-forms / Michael Farber. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 108) Includes bibliographical references and index. ISBN 0-8218-3531-9 (alk. paper) 1. Critical point theory (Mathematical analysis) 2. Differential topology. 3. Foliations (Math­

ematics) I. Title. II. Mathematical surveys and monographs ; no. 108.

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10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04

Contents

Preface

Chapter 1. The Novikov Numbers 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

Homological algebra of Morse inequalities The Novikov ring Nov(T) The rational subring 7Z(T) Homology of local coefficient systems The Novikov numbers Further properties of the Novikov numbers Novikov numbers and Betti numbers of flat line bundles

Chapter 2. The Novikov Inequalities 2.1. 2.2. 2.3.

Closed 1-forms Geometry of Novikov theory The Novikov inequalities

Chapter 3. The Universal Complex 3.1. 3.2. 3.3. 3.4.

The Main Theorem Line bundles and algebraic integers Generic flat vector bundles Examples

Chapter 4. Construction of the Universal Complex 4.1. 4.2. 4.3. 4.4.

Chain collapse Proof of Theorem 3.1 in the rank 1 case Proof of Theorem 3.1 in the general case Refined universal complex and deformation complex

Chapter 5. Bott-type Inequalities 5.1. 5.2.

Topology of the set of zeros Proofs of Theorems 5.1 and 5.5

Chapter 6. Inequalities with Von Neumann Betti Numbers

Chapter 7. Equivariant Theory 7.1. 7.2. 7.3. 7.4.

Basic 1-forms Equivariant Novikov inequalities Application: Fixed points of a symplectic circle action Signature via Novikov numbers

Chapter 8. Exactness of the Novikov Inequalities 8.1. Exactness Theorem

vii

1 1 6 9

12 17 21 30

35 35 38 45

49 49 54 56 58

61 61 63 66 75

81 81 86

91

99 100 101 104 108

113 113

V

CONTENTS

8.2. Finiteness theorem for codimension two knots 8.3. Surgery on codimension one submanifolds 8.4. Algebra of minimal lattices 8.5. Proof of The Exactness Theorem

Chapter 9. Morse Theory of Harmonic Forms 9.1. Topology of singular foliations of closed 1-forms 9.2. Intrinsically harmonic 1-forms 9.3. Examples of singular foliations 9.4. Proof of Calabi's Theorem 9.5. Morse numbers of harmonic 1-forms

Chapter 10. Lusternik-Schnirelman Theory, Closed 1-Forms, and Dynamics

10.1. Colliding the critical points 10.2. Closed 1-forms on topological spaces 10.3. Category of a space with respect to a cohomology class 10.4. Estimate of the number of zeros 10.5. Gradient-convex neighborhoods 10.6. Movable homology classes 10.7. Cohomological lower bound for cat(X, £) 10.8. Deformations and their spectral sequences 10.9. Families of flat bundles and higher Massey products 10.10. Estimate for cat(X,£) in terms of ^-survivors 10.11. Flows, Lyapunov 1-forms and asymptotic cycles

Appendix A. Manifolds with Corners

Appendix B. Morse-Bott Functions on Manifolds with Corners

Appendix C. Morse-Bott Inequalities

Appendix D. Relative Morse Theory

Bibliography

114 115 119 122

125 125 131 137 140 147

159 160 162 165 170 177 179 181 184 190 194 197

205

213

227

233

239

Index 245

Preface

This book studies fascinating geometrical, topological and dynamical properties of closed 1-forms on manifolds. Given a closed 1-form UJ, we are interested in the number of its zeros, in the geometry of the singular foliation UJ = 0, and in the dynamical properties of the gradient-like flows of UJ.

A closed 1-form, viewed locally, is a smooth function up to an additive constant. All local properties of smooth functions can be translated into the language of closed 1-forms. For example, the notion of a critical point of a function corresponds to the notion of a zero of a closed 1-form. The global structure of a closed 1-form UJ depends on its de Rham cohomology class £ = [UJ] G H1(M; R). The main subject of this book is to reveal the relations between the global and local features of closed 1-forms.

S.P. Novikov [Nl], [N2] initiated a generalization of Morse theory in which instead of critical points of smooth functions one deals with closed 1-forms and their zeros. He introduced the numbers bj(£) and qj(£) depending on a real cohomology class £ G Hl(M] R). We call bj(£) the Novikov Betti number and qj(£) the Novikov torsion number. In the special case £ = 0 (which corresponds to the classical Morse theory of functions) the number fy(£) equals bj(M), the Betti number of M, and the number qj (£) coincides with the minimal number of generators of the torsion subgroup of Hj(M;7i). The famous Novikov inequalities state that any closed 1-form UJ with Morse-type zeros has at least 6j(£) + qj(€) + Qj-i(0 zeros of Morse index j , for any j , where £ = [UJ] G Hl(M\ R) is the de Rham cohomology class of UJ. Nowadays, the Novikov theory is widely known and has numerous applications in geometry, topology, analysis, and dynamics.

This book starts with a detailed introduction into Novikov theory written in textbook style (Chapters 1 and 2). We hope that this material will be useful to readers who wish to apply Novikov theory. The first chapter studies the Novikov numbers 6j(£) and <Zj(£). We describe their main properties and compute them explicitly in some examples. The main issue here is to clarify the character of the dependence of these numbers on the cohomology class £. In the second chapter we describe the geometric ideas which led to the discovery of Novikov theory. Here we also give a rigorous proof of the Novikov inequalities.

Subsequent chapters are written in the style of a research monograph. The material described in chapters 3-10 is based mainly on my work; some of these results were obtained jointly with my collaborators, Maxim Braverman, Gabriel Katz, Jerome Levine and Andrew Ranicki, in alphabetical order. The last section of chapter 10 represents a joint work with Thomas Kappeler, Janko Latschev and Eduard Zehnder.

Chapters 3 and 4 describe the universal chain complex. The exposition follows my paper [Far 13] which develops our joint work with A. Ranicki [FR]. These

vii

V l l l PREFACE

two chapters, playing a central role in this book, give a very general answer to the problem of constructing the CiNovikov complexes" over different extensions of the group ring of the manifold.

A general well-known intuitive principle of Morse theory says tha t the topology of the set of critical points of a function dominates (in some sense) the topology of the underlying manifold. This principle, when applied to the Morse theory of closed 1-forms, remains t rue but it requires a different meaning for the word "dominates". It turns out tha t one has to apply to the homotopy type of the manifold a suitable noncommutative localization which appears in the construction of the universal complex.

S.P. Novikov in his work always used a suitable completion to construct the "Novikov complexes". As an alternative it was suggested in my paper [Far5] pub­lished in 1985 to use a localization instead of the completion. I showed tha t the lo­calization leads to a smaller ring having many advantages compared to the Novikov completions.

The initial fundamental idea of S.P. Novikov [N2] was based on a plan to construct the Novikov complex using dynamics of the gradient flow in the abelian covering associated with the given cohomology class. The dynamics of the gradient flows is used traditionally in Morse theory providing a bridge between the critical set of a function and the global ambient topology. A completely different approach to prove the Morse inequalities was first suggested by E. Wi t ten [Wi2]; it is based on the spectral theory of the Laplace operator deformed by the given Morse function.

The construction of the universal complex described in Chapters 3 and 4 uses a new method of algebraic collapse suggested originally in our work with A. Ranicki [FR]. This technique, combinatorial and algebraic in nature, is quite simple and powerful. It allows one to avoid heavy analytic problems arising when dealing with the two approaches mentioned above (dynamics and spectral theory).

The universal complex uses the notion of noncommutative localization in the sense of P. Cohn [Co]. We find the algebraic condition on the ring which implies the validity of the Novikov Principle. The universal complex gives many different "Novikov complexes" and many different inequalities comparing the numbers of ze­ros to the Bett i numbers of certain local coefficient systems. It is shown by example tha t these new inequalities are sometimes stronger than the Novikov inequalities.

In Chapter 5 we present several different generalizations of the Novikov in­equalities. First, we remove the Morse nondegeneracy assumption replacing it by nondegeneracy in the sense of Bott . First inequalities of this kind were obtained jointly with Maxim Braverman [BF1], [BF3]. They relate the Poincare polynomi­als of different connected components of the set of zeros to the Novikov counting polynomial of the manifold. A typical inequality (see (5.3)) claims

X>i-ind(Z)(S) > MO-Z

Here Z runs over all connected components of the set of zeros of CJ, ind(Z) denotes the index of Z , bj (Z) s tands for the Betti number of Z and £ denotes the cohomology class of UJ. Several theorems of this chapter are new; among them Theorems 5.5, 5.6 and 5.7. For example, Theorem 5.7 gives the inequality

J>-ind(Z)(Z) > &*(£) + 9 i (0+9i - l (0 z

PREFACE IX

(see (5.12)) which obviously generalizes the Novikov inequality. It is obtained under the assumption tha t the negative normal bundle of the set of zeros is orientable and the integral homology of the set of zeros has no torsion.

We describe in Chapter 6 Novikov-type inequalities where one uses the von Neu­mann Bett i numbers instead of the Novikov numbers; these results were originally obtained in [Far9].

Chapter 7 suggests an equivariant version of the critical point theory for closed 1-forms. Although this material originates from a joint work with Maxim Braver-man [BF2], [BF4], the exposition here is quite different and contains some new results. In this chapter we describe relations in an equivariant setting between the topology of the set of zeros of a closed equivariant basic 1-form and suitable equivariant cohomological invariants of the manifold. One defines integers (which are called equivariant Novikov numbers) playing a key role in this problem. As an application it is shown how these results (i.e., the equivariant generalization of Novikov theory) help to compute the cohomology of the fixed point set of a sym-plectic circle action. Finally, we present a formula expressing the signature of a symplectic manifold with a symplectic circle action through the Novikov numbers. This result was originally published in [Far8]; it generalizes a theorem of J.D.S. Jones and J.H. Rawnsley [JR], who studied the special case of Hamiltonian circle actions.

Next Chapter 8 describes the main theorem of the paper [Far5] about the exactness of the Novikov inequalities for manifolds with an infinite cyclic funda­mental group. Roughly, it states tha t in any nonzero cohomology class one may find a closed 1-form for which the Novikov inequalities become equalities. This result is in the spirit of Smale's theorem [Sm2] about the existence of minimal Morse functions on simply connected manifolds. It solves a problem raised by S.P. Novikov [N2]. The well-known result of W. Browder and J. Levine [BL] giving conditions for fibering a manifold over a circle is a consequence of this theorem. This chapter also contains a finiteness theorem for codimension two stable knots: it states tha t such knots are determined up to a finite ambiguity by their Alexander modules and Milnor form.

E. Calabi [Ca] raised the problem of whether it is possible to improve the inequalities for closed 1-forms with Morse-type zeros if one additionally assumes tha t the 1-form is harmonic with respect to a Riemannian metric. This problem is discussed in Chapter 9, representing the results of a joint work [FKL] with Gabriel Katz and Jerome Levine and also the subsequent work of K. Honda [Ho]. We prove in this chapter tha t the harmonicity imposes no further Morse restrictions on the number of zeros. This chapter also contains a detailed study of the geometric properties of singular foliations of closed 1-forms.

Chapter 10 suggests a Lusternik-Schnirelman-type critical point theory for closed 1-forms. The main distinction from Novikov theory is tha t here one makes no additional requirements about the nature of the zeros of a closed 1-form. Recall tha t Novikov theory is based on the assumption of nondegeneracy of zeros which plays an important role there. Chapter 10 gives a generalization of the notion of the Lusternik-Schnirelman category. For any pair (A, £) consisting of a polyhedron X and a real cohomology class £ G i ^ 1 ( X ; R ) , we define a nonnegative integer cat (AT, £), the category of X with respect to the cohomology class £. The number cat(AT,£) depends only on the homotopy type of (AT, £) and coincides with ca t (A)

X PREFACE

in the case £ = 0. If £ ^ 0, then cat(X, £) < cat(X). We show by example that the difference cat(X) — cat(X, £) may be arbitrarily large. The main theorem of this chapter states that any smooth closed 1-form LU on a smooth closed manifold M must have at least cat(M, £) geometrically distinct zeros, where £ = [LU] G Hl(M] R) denotes the cohomology class of a;, assuming that LU admits a gradient-like vector field with no homoclinic cycles. Viewed differently, the main theorem of Chapter 10 claims that any gradient-like vector field of a closed 1-form LU has a homoclinic cycle if the number of zeros of LU is less than cat(M, £).

Let us rephrase this surprising new "focusing" phenomenon: when the number of zeros of a closed 1-form LU becomes less than cat(M,£) (which is a homotopy invariant!), any gradient-like vector field for LU has a homoclinic cycle. This result is a manifestation of a deep interaction between homotopy theory and dynamics.

Chapter 10 mainly follows my paper [Far 16]. A slightly different version of the Lusternik-Schnirelman theory for closed 1-forms was suggested in [Far 17] and in a more general form in [FK]. The results of this chapter correct some of my earlier statements made in [Far 11] and [Far 12].

The last section of Chapter 10 describes the notion of a Lyapunov 1-form of a flow and gives necessary and sufficient conditions for the existence of a Lyapunov 1-form in a prescribed cohomology class £ G Hl(M\ R). Here we use the notion of an asymptotic cycle introduced by S. Schwartzman [Sch]. The exposition is based on a joint work with T. Kappeler, J. Latschev and E. Zehnder [FKLZ], [FKLZ1].

In a series of appendices we give an exposition of Morse-Bott theory for mani­folds with corners. This subject belongs to the mathematical folklore and is known to experts although no systematic treatment of these topics seems to exist in the literature.

This book is not designed to be an encyclopedia on the theory of closed 1-forms. It does not cover all results where the topology of closed 1-forms plays a role in mathematics. Unfortunately several important topics were left outside the scope of the book and the interested reader is invited to complete the picture by reading the original journal articles. We will mention briefly some of these topics.

In 1999 M. Hutchings and Y.-J. Lee [HL1], [HL2] made the fascinating discov­ery that the Lefschetz ("-function counting the closed orbits of the gradient flow of a Morse closed 1-form can be computed in terms of the Reidemeister torsion of the Novikov complex. This result was later generalized by several authors; see [P8], [Schul], [Schu2].

The methods of Novikov theory play an important role in group theory in studying finiteness properties of discrete groups. This research was initiated by J.-CI. Sikorav in his thesis [Sil] written in 1987. J.-Cl. Sikorav proved that the vanishing of the Novikov-Sikorav homology in dimension one with respect to a co­homology class £ is equivalent to the kernel ker£ being finitely generated. Some further results and references can be found in the paper of M. Damian [Da] where the relations with the invariants of Bieri-Neumann-Strebel [BNS] and Bieri-Renz [BR] are explained. Here I would like to mention the related work of A. Ranicki [Ran] which proves that the vanishing of the Novikov-Sikorav homology is equiva­lent to the finite domination of the space of infinite cyclic covering.

A few words on the terminology. The terms "Novikov homology", "Novikov ring" and "Novikov complex" are used too often in the mathematical literature and the meaning of these terms varies in different papers. This may lead to ambiguity

P R E F A C E xi

and misunderstanding. For example, the term "Novikov ring" denotes both the commutative ring Nov of the formal power series (see §1.2) and also the completion Z-7T£ of the group ring Zn determined by a cohomology class £ : TT —» R (see §3.1.5). The latter noncommutative ring was first introduced by J.-CI. Sikorav [Sil]. I suggest to resolve this ambiguity by calling the ring ZTT^ the Novikov-Sikorav completion and the corresponding homology the Novikov-Sikorav homology.

It is a pleasure for me to thank S.P. Novikov, who initiated the theory described in this book, for his great inspiration, encouragement and support.

I would also like to thank many colleagues for discussions which were helpful in writing this book. Among them I would like to thank my friends Jerome Levine, Thomas Kappeler, Vladimir Turaev and Shmuel Weinberger.

Finally I would like to thank the anonymous referee for generously sharing his ideas and for his highly professional and helpful comments.

Michael Farber

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Index

Alexander polynomial, 58 ^-Algebraic integer, 33 Associated prime of a flat line bundle, 83 Asymptotic cycle of a flow, 199

Basic 1-form, 100 p-Betti number, 3 Bott-type nondegeneracy condition, 81

Calabi graph of a closed 1-form, 141 Calabi Theorem, 132 Category of a space with respect to a coho­

mology class, 160, 165 Cech cohomology class of a closed 1-form,

165 Chain collapse, 61 Chain recurrent set R^, 201 Cohn localization, 52 Collaring theorem, 208 Cone of tangent directions, 206 Connected sum constructions for closed 1-

forms, 138 Continuous closed 1-form, 163 Critical point of a function on manifold with

corners, 213 (6, T)-Cycle, 202

Deformation complex, 47, 79 Deformation of a chain complex, 184 Deformation of a closed 1-form, 148 Domination relation y, 1

Equivariant incidence coefficients, 44 Equivariant Novikov numbers, 102 Euler-Poincare-Morse Theorem, 3 Exactness theorem, 113

Finiteness theorem for codimension two knots, 114

Flat vector bundle, 15 Focusing effect, 171

Generalized moment map, 93 ^-Generic flat vector bundle, 56 Generic closed 1-form, 126

Generic flat vector bundle, 56 Gradient-like flow, 39 Gradient-like vector field, 170

Hessian, 217 Higher Massey products, 192 Homoclinic, 41 Homoclinic cycle, 159 Homoclinic orbit, 159, 171 Homologically n-connected, 164 Homology class movable to ±oo, 179 Homomorphism of periods, 37

Intrinsically harmonic closed 1-form, 131

Lifting property, 182 Local coefficient system, 12 Local system a^, 16 Locally path connected, 164 Lyapunov 1-form of a flow, 198 Lyapunov function of a flow, 198

i-Manifold, 109 Manifold with corners, 205 Monodromy representation, 13 Morse counting polynomial, 228 Morse critical point, 214 Morse index of a zero, 36 Morse Lacunary Principle, 2, 92

Neat submanifold of a manifold with cor­ners, 209

^-Negative, 50 Newton diagram, 19 Nondegenerate critical submanifold, 219 Novikov Betti number, 17 Novikov complex, 44 Novikov homology, 17 Novikov inequalities, 45 Novikov Principle, 44 Novikov ring, 7 Novikov torsion number, 17 Novikov-Sikorav completion, 52 Novikov-Sikorav homology, 52 Novikov-Taimanov Principle, 236

245

246 INDEX

Orientation bundle, 81

p-Poincare polynomial, 4

Quadrant structure, 209 Quasi-regular point, 203

Rank of a cohomology class, 25 Rational part of Novikov ring, 9 Rearrangements of closed 1-forms, 148 Regular manifold with corners, 207

Separator, 233 Signature, 108 Simple chain collapse, 63 Singular cohomology class determined by a

closed 1-form, 164 Singular foliation, 125 Singular leaf component, 126 Spectral sequence of a deformation of a chain

complex, 185 Stable disk, 40 Star operator, 140 Stratification of a manifold with corners, 206 ^-Survivor, 193 Symplectic circle actions, 92 Symplectic vector field, 38

Tangent space of a manifold with corners, 206

Torsion number, 5 Transitive closed 1-form, 132

Universal chain complex, 53 Unstable disk, 40

Vanishing Theorem, 92 Von Neumann Betti number, 91

Zero of a closed 1-form, 36

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