Topology of Closed One-Forms
About this Title
Michael Farber, Tel Aviv University, Tel Aviv, Israel
Publication: Mathematical Surveys and Monographs
Publication Year 2004: Volume 108
ISBNs: 978-0-8218-3531-9 (print); 978-1-4704-1335-4 (online)
MathSciNet review: MR2034601
MSC (2000): Primary 58E05; Secondary 57R20, 57R70, 58A10
Viewed locally, a closed 1-form on a manifold is a smooth function up to an additive constant. The global structure of a closed 1-form is mainly determined by its de Rham cohomology class. This book studies fascinating geometrical, topological, and dynamical properties of closed 1-forms and, in particular, reveals the relations between their global and local features.
In 1981, S. P. Novikov initiated a generalization of Morse theory in which instead of critical points of smooth functions one deals with closed 1-forms and their zeros. The first two chapters of the book, written in textbook style, give a detailed exposition of Novikov theory, which now plays a fundamental role in geometry and topology.
In the following chapters we describe the universal chain complex which lives over a localization (in the sense of P. M. Cohn) of the group ring and relates the topology of the underlying manifold with information about zeros of closed 1-forms. Using this complex, many different variations and generalizations of the Novikov inequalities are obtained, including Bott-type inequalities for closed 1-forms, equivariant inequalities, and inequalities involving von Neumann Betti numbers. Another significant result in the book is a solution of the problem about exactness of the Novikov inequalities for manifolds with the infinite cyclic fundamental group.
One of the chapters deals with the problem raised by E. Calabi about intrinsically harmonic closed 1-forms and their Morse numbers. The solution of this problem and a detailed study of topological properties of singular foliations of closed 1-forms are presented.
The last chapter suggests a completely new Lusternik-Schnirelman-type theory for dynamical systems. Closed 1-forms appear in dynamics through the concept of a Lyapunov 1-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”).
Graduate students and research mathematicians interested in geometry and topology.
Table of Contents
- 1. The Novikov numbers
- 2. The Novikov inequalities
- 3. The universal complex
- 4. Construction of the universal complex
- 5. Bott-type inequalities
- 6. Inequalities with Von Neumann Betti numbers
- 7. Equivariant theory
- 8. Exactness of the Novikov inequalities
- 9. Morse theory of harmonic forms
- 10. Lusternik-Schnirelman theory, closed 1-forms, and dynamics
- Appendix A. Manifolds with corners
- Appendix B. Morse-Bott functions on manifolds with corners
- Appendix C. Morse-Bott inequalities
- Appendix D. Relative Morse theory