The Homotopy Index and Partial Differential Equations

1. Front Matter

   Pages I-XII

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2. The homotopy index theory

   • Krzysztof P. Rybakowski

   Pages 1-71

3. Applications to partial differential equations

   • Krzysztof P. Rybakowski

   Pages 72-139

4. Selected topics

   • Krzysztof P. Rybakowski

   Pages 140-194

5. Back Matter

   Pages 195-208

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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.

• convergence
• differential equation
• differential operator
• manifold
• metric space
• partial differential equation

• Institut für Angewandte Mathematik, Albert-Ludwigs-Universität, Freiburg i. Br., Germany

  Krzysztof P. Rybakowski

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