**Mathematical Surveys and Monographs**

Volume: 108;
2004;
246 pp;
Hardcover

MSC: Primary 58; 57;

Print ISBN: 978-0-8218-3531-9

Product Code: SURV/108

List Price: $80.00

Individual Member Price: $64.00

**Electronic ISBN: 978-1-4704-1335-4
Product Code: SURV/108.E**

List Price: $80.00

Individual Member Price: $64.00

#### Supplemental Materials

# Topology of Closed One-Forms

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*Michael Farber*

This monograph is an introduction to the fascinating field of the topology, geometry and dynamics of closed one-forms.

The subject was initiated by S. P. Novikov in 1981 as a study of Morse type zeros of closed one-forms. The first two chapters of the book, written in textbook style, give a detailed exposition of Novikov theory, which plays a fundamental role in geometry and topology.

Subsequent chapters of the book present a variety of topics where closed one-forms play a central role. The most significant results are the following:

The solution of the problem of exactness of the Novikov inequalities for manifolds with the infinite cyclic fundamental group.

The solution of a problem raised by E. Calabi about intrinsically harmonic closed one-forms and their Morse numbers.

The construction of a universal chain complex which bridges the topology of the underlying manifold with information about zeros of closed one-forms. This complex implies many interesting inequalities including Bott-type inequalities, equivariant inequalities, and inequalities involving von Neumann Betti numbers.

The construction of a novel Lusternik-Schnirelman-type theory for dynamical systems. Closed one-forms appear in dynamics through the concept of a Lyapunov one-form of a flow. As is shown in the book, homotopy theory may be used to predict the existence of homoclinic orbits and homoclinic cycles in dynamical systems ("focusing effect").

#### Table of Contents

# Table of Contents

## Topology of Closed One-Forms

- Contents v6 free
- Preface vii8 free
- Chapter 1. The Novikov Numbers 114 free
- 1.1. Homological algebra of Morse inequalities 114
- 1.2. The Novikov ring Nov(Γ) 619
- 1.3. The rational subring R(Γ) 922
- 1.4. Homology of local coefficient systems 1225
- 1.5. The Novikov numbers 1730
- 1.6. Further properties of the Novikov numbers 2134
- 1.7. Novikov numbers and Betti numbers of flat line bundles 3043

- Chapter 2. The Novikov Inequalities 3548
- Chapter 3. The Universal Complex 4962
- Chapter 4. Construction of the Universal Complex 6174
- Chapter 5. Bott-type Inequalities 8194
- Chapter 6. Inequalities with Von Neumann Betti Numbers 91104
- Chapter 7. Equivariant Theory 99112
- Chapter 8. Exactness of the Novikov Inequalities 113126
- Chapter 9. Morse Theory of Harmonic Forms 125138
- Chapter 10. Lusternik-Schnirelman Theory, Closed 1-Forms,and Dynamics 159172
- 10.1. Colliding the critical points 160173
- 10.2. Closed 1-forms on topological spaces 162175
- 10.3. Category of a space with respect to a cohomology class 165178
- 10.4. Estimate of the number of zeros 170183
- 10.5. Gradient-convex neighborhoods 177190
- 10.6. Movable homology classes 179192
- 10.7. Cohomological lower bound for cat(X, ξ) 181194
- 10.8. Deformations and their spectral sequences 184197
- 10.9. Families of flat bundles and higher Massey products 190203
- 10.10. Estimate for cat(X, ξ) in terms of ξ-survivors 194207
- 10.11. Flows, Lyapunov 1-forms and asymptotic cycles 197210

- Appendix A. Manifolds with Corners 205218
- Appendix B. Morse-Bott Functions on Manifolds with Corners 213226
- Appendix C. Morse-Bott Inequalities 227240
- Appendix D. Relative Morse Theory 233246
- Bibliography 239252
- Index 245258

#### Readership

Graduate students and research mathematicians interested in geometry and topology.