Circle-Valued Morse Theory (de Gruyter Studies in Mathematics 32) (De Gruyter Studies in Mathematics)

In The Early 1920s 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 1980s. 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. Frontmatter -- 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 Andrei V. Pajitnov. Includes Bibliographical References (p. [437]-444) And Index. Contents......Page 8 Preface......Page 12 Introduction......Page 16 Part 1 Morse functions and vector fieldson manifolds......Page 26 CHAPTER 1 Vector fields and C0 topology......Page 28 CHAPTER 2 Morse functions and their gradients......Page 44 CHAPTER 3 Gradient flows of real-valued Morse functions......Page 78 Part 2 Transversality, handles, Morse complexes......Page 120 CHAPTER 4 The Kupka-Smale transversality theory forgradient flows......Page 122 CHAPTER 5 Handles......Page 174 CHAPTER 6 The Morse complex of a Morse function......Page 206 History and Sources......Page 238 Part 3 Cellular gradients.......Page 240 CHAPTER 7 Condition (C)......Page 242 CHAPTER 8 Cellular gradients are C0-generic......Page 254 CHAPTER 9 Properties of cellular gradients......Page 292 Sources......Page 332 Part 4 Circle-valued Morse maps andNovikov complexes......Page 334 CHAPTER 10 Completions of rings, modules and complexes......Page 336 CHAPTER 11 The Novikov complex of a circle-valued Morsemap......Page 346 CHAPTER 12 Cellular gradients of circle-valued Morse functions and the Rationality Theorem......Page 378 CHAPTER 13 Counting closed orbits of the gradient flow......Page 394 CHAPTER 14 Selected topics in the Morse-Novikov theory......Page 424 History and Sources......Page 446 Bibliography......Page 448 Selected Symbols and Abbreviations......Page 456 Subject Index......Page 460

In the early 1920s 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 1980s. 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.

In 1927, 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. This book aims to give a systematic treatment of the geometric foundations of a subfield of that topic, the circle-valued Morse functions Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. The aim of the present book is to give a systematic treatment of the geometric foundations of a subfield of that topic, the circle-valued Morse functions, a subfield of Morse theory.