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  • Textbook
  • © 2014

Morse Theory and Floer Homology

  • Translation of the popular French textbook

  • Provides a unified presentation of Morse theory and Floer homology that is unique in the English language

  • Explains all the required background on symplectic geometry, differential geometry, algebraic topology and analysis

  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • ISBN: 978-1-4471-5496-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 109.99
Price excludes VAT (USA)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XIV
  2. Morse Theory

    1. Front Matter

      Pages 1-6
    2. Morse Functions

      • Michèle Audin, Mihai Damian
      Pages 7-21
    3. Pseudo-Gradients

      • Michèle Audin, Mihai Damian
      Pages 23-52
    4. The Morse Complex

      • Michèle Audin, Mihai Damian
      Pages 53-78
    5. Morse Homology, Applications

      • Michèle Audin, Mihai Damian
      Pages 79-123
  3. The Arnold Conjecture, Floer Homology

    1. Front Matter

      Pages 125-128
    2. What You Need to Know About Symplectic Geometry

      • Michèle Audin, Mihai Damian
      Pages 129-149
    3. The Arnold Conjecture and the Floer Equation

      • Michèle Audin, Mihai Damian
      Pages 151-188
    4. The Geometry of the Symplectic Group, the Maslov Index

      • Michèle Audin, Mihai Damian
      Pages 189-219
    5. Linearization and Transversality

      • Michèle Audin, Mihai Damian
      Pages 221-303
    6. Floer Homology: Spaces of Trajectories

      • Michèle Audin, Mihai Damian
      Pages 305-357
    7. From Floer to Morse

      • Michèle Audin, Mihai Damian
      Pages 359-381
    8. Floer Homology: Invariance

      • Michèle Audin, Mihai Damian
      Pages 383-451
    9. The Elliptic Regularity of the Floer Operator

      • Michèle Audin, Mihai Damian
      Pages 453-476
    10. Exercises for the Second Part

      • Michèle Audin, Mihai Damian
      Pages 515-531
  4. Back Matter

    Pages 533-596

About this book

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

Reviews

From the book reviews:

“The present book is an excellent, detailed and self-contained introduction to Morse theory and Floer homology which makes both topics easily accessible to graduate or even advanced undergraduate students.” (Sonja Hohloch, Mathematical Reviews, August, 2014)

“Morse Theory and Floer Homology is a relatively high-level introduction to, and in fact a full account of, the extremely elegant and properly celebrated solution to the Arnol’d problem by the prodigious and tragic Andreas Floer … . the book is exceptionally well written. Indeed, this is a very good book on a beautiful and important subject and will richly repay those who take the time to work through it.” (Michael Berg, MAA Reviews, February, 2014)

Authors and Affiliations

  • IRMA Université Louis Pasteur, Strasbourg Cedex, France

    Michèle Audin, Mihai Damian

Bibliographic Information

  • Book Title: Morse Theory and Floer Homology

  • Authors: Michèle Audin, Mihai Damian

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4471-5496-9

  • Publisher: Springer London

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag London Ltd., part of Springer Nature 2014

  • Softcover ISBN: 978-1-4471-5495-2

  • eBook ISBN: 978-1-4471-5496-9

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XIV, 596

  • Number of Illustrations: 114 b/w illustrations

  • Additional Information: Original French edition published by EDP Sciences, Les Ulis Cedex A, France, 2010

  • Topics: Geometry, Differential Geometry, Algebraic Topology, Manifolds and Cell Complexes

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • ISBN: 978-1-4471-5496-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 109.99
Price excludes VAT (USA)