Circle-valued Morse theory / Andrei V. Pajitnov.

Format:

Book

Author/Creator:

Pajitnov, Andrei V.

Series:

Gruyter studies in mathematics ; 32.

De Gruyter studies in mathematics, 0179-0986 ; 32

Language:

English

Subjects (All):

Morse theory.

Manifolds (Mathematics).

Physical Description:

1 online resource (464 pages)

Edition:

1st ed.

Place of Publication:

Berlin ; New York : De Gruyter, c2006.

Language Note:

English

Summary:

In the early 1920's M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology. Circle-valued Morse theory originated from a problem in hydrodynamics studied by S. P. Novikov in the early 1980's. Nowadays, it is a constantly growing field of contemporary mathematics with applications and connections to many geometrical problems such as Arnold's conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in the three-dimensional sphere. The aim of the book is to give a systematic treatment of geometric foundations of the subject and recent research results. The book is accessible to first year graduate students specializing in geometry and topology.

Contents:

Front matter

Contents

Preface

Introduction

Part 1. Morse functions and vector fields on manifolds

CHAPTER 1. Vector fields and C0 topology

CHAPTER 2. Morse functions and their gradients

CHAPTER 3. Gradient flows of real-valued Morse functions

Part 2. Transversality, handles, Morse complexes

CHAPTER 4. The Kupka-Smale transversality theory for gradient flows

CHAPTER 5. Handles

CHAPTER 6. The Morse complex of a Morse function

Part 3. Cellular gradients

CHAPTER 7. Condition (C)

CHAPTER 8. Cellular gradients are C0-generic

CHAPTER 9. Properties of cellular gradients

Part 4. Circle-valued Morse maps and Novikov complexes

CHAPTER 10. Completions of rings, modules and complexes

CHAPTER 11. The Novikov complex of a circle-valued Morse map

CHAPTER 12. Cellular gradients of circle-valued Morse functions and the Rationality Theorem

CHAPTER 13. Counting closed orbits of the gradient flow

CHAPTER 14. Selected topics in the Morse-Novikov theory

Backmatter

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [437]-444) and index.

ISBN:

9786612194269

9781282194267

1282194267

9783110197976

3110197979

OCLC:

236337992