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The Homotopy Index and Partial Differential Equations

  • Krzysztof P. Rybakowski

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Krzysztof P. Rybakowski
    Pages 1-71
  3. Krzysztof P. Rybakowski
    Pages 72-139
  4. Krzysztof P. Rybakowski
    Pages 140-194
  5. Back Matter
    Pages 195-208

About this book

Introduction

The homotopy index theory was developed by Charles Conley for two­ sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi­ cal measure of an isolated invariant set, is defined to be the ho­ motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the "exit ramp" of N . 1 It is shown that the index is independent of the choice of the in­ dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde­ generate critical point p with respect to a gradient flow on a com­ pact manifold. In fact if the Morse index of p is k, then the homo­ topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.

Keywords

convergence differential equation differential operator manifold metric space partial differential equation

Authors and affiliations

  • Krzysztof P. Rybakowski
    • 1
  1. 1.Institut für Angewandte MathematikAlbert-Ludwigs-UniversitätFreiburg i. Br.Germany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-72833-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1987
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-18067-8
  • Online ISBN 978-3-642-72833-4
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site