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Morse Theory and Floer Homology

  • Michèle Audin
  • Mihai Damian

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Morse Theory

    1. Front Matter
      Pages 1-6
    2. Michèle Audin, Mihai Damian
      Pages 7-21
    3. Michèle Audin, Mihai Damian
      Pages 23-52
    4. Michèle Audin, Mihai Damian
      Pages 53-78
    5. Michèle Audin, Mihai Damian
      Pages 79-123
  3. The Arnold Conjecture, Floer Homology

    1. Front Matter
      Pages 125-128
    2. Michèle Audin, Mihai Damian
      Pages 129-149
    3. Michèle Audin, Mihai Damian
      Pages 151-188
    4. Michèle Audin, Mihai Damian
      Pages 189-219
    5. Michèle Audin, Mihai Damian
      Pages 221-303
    6. Michèle Audin, Mihai Damian
      Pages 305-357
    7. Michèle Audin, Mihai Damian
      Pages 359-381
    8. Michèle Audin, Mihai Damian
      Pages 383-451
    9. Michèle Audin, Mihai Damian
      Pages 453-476
    10. Michèle Audin, Mihai Damian
      Pages 515-531
  4. Back Matter
    Pages 533-596

About this book

Introduction

This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold.

The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications.

Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part.

The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis.

The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.

Keywords

Arnold Conjecture Floer Complex Floer Homology Gluing Hamiltonian System Maslov Index Morse Complex Morse Homology Morse Inequalities Morse Theory Symplectic Group Symplectic Manifold

Authors and affiliations

  • Michèle Audin
    • 1
  • Mihai Damian
    • 2
  1. 1.IRMA Université Louis PasteurStrasbourg CedexFrance
  2. 2.IRMA Université Louis PasteurStrasbourg CedexFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4471-5496-9
  • Copyright Information Springer-Verlag London 2014
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4471-5495-2
  • Online ISBN 978-1-4471-5496-9
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site