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Topology of Closed One-Forms
Michael Farber, Tel Aviv University, Israel, and University of Durham, England
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Mathematical Surveys and Monographs
2004; 246 pp; hardcover
Volume: 108
ISBN-10: 0-8218-3531-9
ISBN-13: 978-0-8218-3531-9
List Price: US$76 Member Price: US$60.80
Order Code: SURV/108

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").

Graduate students and research mathematicians interested in geometry and topology.

• The Novikov numbers
• The Novikov inequalities
• The universal complex
• Construction of the universal complex
• Bott-type inequalities
• Inequalities with von Neumann Betti numbers
• Equivariant theory
• Exactness of the Novikov inequalities
• Morse theory of harmonic forms
• 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
• Bibliography
• Index