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Journal of Fixed Point Theory and Applications

, Volume 17, Issue 4, pp 753–773 | Cite as

On the stable Conley index in Hilbert spaces

  • Tirasan KhandhawitEmail author
Article

Abstract

In this paper, we study Conley theory of flows on a Hilbert space. Our approach is to apply finite-dimensional approximation which is a slight refinement of the construction developed by Gęba, Izydorek, and Pruszko (1999). For instance, we include subspaces other than invariant subspaces in the construction. As a main result, we define a stable Conley index as an object in the stable homotopy category and show that it does not depend on choices in the construction.

Keywords

Conley index in Hilbert spaces Conley index finite-dimensional approximation 

Mathematics Subject Classification

37B30 

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References

  1. 1.
    C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Reg. Conf. Ser. Math. 38, American Mathematical Society, Providence, RI, 1978.Google Scholar
  2. 2.
    C. L. Douglas, Twisted parametrized stable homotopy theory. Preprint, arXiv:math/0508070, 2005.
  3. 3.
    A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets. Ergodic Theory Dynam. Systems 7 (1987), 93-103.Google Scholar
  4. 4.
    K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications. Studia Math. 134 (1999), 217–233.Google Scholar
  5. 5.
    T. Khandhawit, Twisted Manolescu-Floer spectra for Seiberg-Witten. Ph.D. thesis, Massachusetts Institute of Technology, 2013.Google Scholar
  6. 6.
    P. Kronheimer and C. Manolescu, Periodic Floer pro-spectra from the Seiberg-Witten equations. Preprint, arXiv:math/0203243, 2002.
  7. 7.
    C. Manolescu, Seiberg-Witten-Floer stable homotopy type of three-manifolds with b 1 = 0. Geom. Topol. 7 (2003), 889–932.Google Scholar
  8. 8.
    C. Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., to appear.Google Scholar
  9. 9.
    J. P. May, Equivariant Homotopy and Cohomology Theory. CBMS Reg. Conf. Ser. Math. 91, American Mathematical Society, Providence, RI, 1996.Google Scholar
  10. 10.
    P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, RI, 1986.Google Scholar
  11. 11.
    Rybakowski K. P.: On the homotopy index for infinite-dimensional semiflows. Trans. Amer. Math. Soc. 269, 351–382 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Salamon D.: Connected simple systems and the Conley index of isolated invariant sets. Trans. Amer. Math. Soc 291, 1–41 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation. Comm. Partial Differential Equations 21 (1996), 1431–1449.Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced StudyThe University of TokyoKashiwaJapan

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