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Topology of Closed One-Forms Mathematical Surveys and Monographs http://dx.doi.org/10.1090/surv/108 Topology of Closed One-Forms Mathematical Surveys and Monographs Volume 108 Topology of Closed One-Forms Michael Farber t^EM^/ American Mathematical Society EDITORIAL COMMITTEE 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 Data 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. QA614.7.F37 2003 514/.74—dc22 2003062825 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2004 by the author. All rights reserved. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04 Contents Preface vii Chapter 1. The Novikov Numbers 1 1.1. Homological algebra of Morse inequalities 1 1.2. The Novikov ring Nov(T) 6 1.3. The rational subring 7Z(T) 9 1.4. Homology of local coefficient systems 12 1.5. The Novikov numbers 17 1.6. Further properties of the Novikov numbers 21 1.7. Novikov numbers and Betti numbers of flat line bundles 30 Chapter 2. The Novikov Inequalities 35 2.1. Closed 1-forms 35 2.2. Geometry of Novikov theory 38 2.3. The Novikov inequalities 45 Chapter 3. The Universal Complex 49 3.1. The Main Theorem 49 3.2. Line bundles and algebraic integers 54 3.3. Generic flat vector bundles 56 3.4. Examples 58 Chapter 4. Construction of the Universal Complex 61 4.1. Chain collapse 61 4.2. Proof of Theorem 3.1 in the rank 1 case 63 4.3. Proof of Theorem 3.1 in the general case 66 4.4. Refined universal complex and deformation complex 75 Chapter 5. Bott-type Inequalities 81 5.1. Topology of the set of zeros 81 5.2. Proofs of Theorems 5.1 and 5.5 86 Chapter 6. Inequalities with Von Neumann Betti Numbers 91 Chapter 7. Equivariant Theory 99 7.1. Basic 1-forms 100 7.2. Equivariant Novikov inequalities 101 7.3. Application: Fixed points of a symplectic circle action 104 7.4. Signature via Novikov numbers 108 Chapter 8. Exactness of the Novikov Inequalities 113 8.1. Exactness Theorem 113 V CONTENTS 8.2. Finiteness theorem for codimension two knots 114 8.3. Surgery on codimension one submanifolds 115 8.4. Algebra of minimal lattices 119 8.5. Proof of The Exactness Theorem 122 Chapter 9. Morse Theory of Harmonic Forms 125 9.1. Topology of singular foliations of closed 1-forms 125 9.2. Intrinsically harmonic 1-forms 131 9.3. Examples of singular foliations 137 9.4. Proof of Calabi's Theorem 140 9.5. Morse numbers of harmonic 1-forms 147 Chapter 10. Lusternik-Schnirelman Theory, Closed 1-Forms, and Dynamics 159 10.1. Colliding the critical points 160 10.2. Closed 1-forms on topological spaces 162 10.3. Category of a space with respect to a cohomology class 165 10.4. Estimate of the number of zeros 170 10.5. Gradient-convex neighborhoods 177 10.6. Movable homology classes 179 10.7. Cohomological lower bound for cat(X, £) 181 10.8. Deformations and their spectral sequences 184 10.9. Families of flat bundles and higher Massey products 190 10.10. Estimate for cat(X,£) in terms of ^-survivors 194 10.11. Flows, Lyapunov 1-forms and asymptotic cycles 197 Appendix A. Manifolds with Corners 205 Appendix B. Morse-Bott Functions on Manifolds with Corners 213 Appendix C. Morse-Bott Inequalities 227 Appendix D. Relative Morse Theory 233 Bibliography 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 Vlll 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 that 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 true but it requires a different meaning for the word "dominates". It turns out that 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] pub­ lished in 1985 to use a localization instead of the completion. I showed that the lo­ calization 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. Witten [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.
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