Topology of Closed One-forms: (Mathematical Surveys and Monographs)

Viewed locally, a closed one-form on a manifold is a smooth function up to an additive constant. The global structure of a closed one-form is mainly determined by its de Rham cohomology class. In this book, Michael Farber studies fascinating geometrical, topological, and dynamical properties of closed one-forms. In particular, he 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 one-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 the author describes the universal chain complex that 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 one-forms. Using this complex, many different variations and generalizations of the Novikov inequalities are obtained, including Bott-type inequalities for closed one-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 one-forms and their Morse numbers. Presented are the solution of this problem and a detailed study of topological properties of singular foliations of closed one-forms. The last chapter suggests a new Lusternik-Schnirelman-type theory for closed one-forms. The dynamics of the gradient-like flows play a crucial role in this theory. As is shown here, homotopy theory may be used to predict the existence of homoclinic orbits and homoclinic cycles in dynamical systems ("focusing effect"). The book is suitable for graduate students and researchers interested in geometry and topology.

Table of Contents:
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.