Topology of Closed One-forms

ISBN-13: 9780821835319 / Angielski / Twarda / 2004

Michael Farber

Opis

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 i