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.
Rezensionen / Stimmen
"Overall the book covers a lot of material in a style and detail that should be easily accessible to both graduate students and researchers wanting to learn the subject."Dirk Schutz in: Mathematical Reviews 2008 "The book under review is a very nice and valuable text on the Morse-Novikov theory. It can help anyone who wants to learn the basis of the theory as well as some more recent and advanced developments and applications."Davod Chataur in: Zentralblatt MATH 1118/2007
Reihe
Sprache
Verlagsort
Zielgruppe
Für Beruf und Forschung
US School Grade: College Graduate Student
Maße
Höhe: 246 mm
Breite: 175 mm
Dicke: 31 mm
Gewicht
ISBN-13
978-3-11-015807-6 (9783110158076)
Schweitzer Klassifikation
Andrei Pajitnov, University of Nantes, France.
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
