Morse cohomology in a Hilbert space via the Conley index

Open Access
Article

Abstract

The main theorem of this paper states thatMorse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed.

Keywords

Morse homology Morse–Witten–Floer complex Conley index Seiberg–Witten–Floer homology 

Mathematics Subject Classification

Primary 53D40 Secondary 55N20 

References

  1. Abb97.
    Abbondandolo A.: A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces. Topol. Methods Nonlinear Anal. 9, 325–382 (1997)MATHMathSciNetGoogle Scholar
  2. AGP.
    M. Aguilar, S. Gitler and C. Prieto, Algebraic Topology from a Homotopical Viewpoint. Springer-Verlag, New York, 2002.Google Scholar
  3. C-J.
    Crabb M. C., Jaworowski J.: Aspects of the Borsuk-Ulam theorem. J. Fixed Point Theory Appl. 13, 459–488 (2013)MATHMathSciNetCrossRefGoogle Scholar
  4. Ch.
    K.-C. Chang, Methods in Nonlinear Analysis. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
  5. FOOO.
    K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I. AMS/IP Stud. Adv. Math. 46, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2009.Google Scholar
  6. G.
    Gęba K.: Fredholm σ-proper maps of Banach spaces. Fund. Math. 64, 341–373 (1969)MathSciNetGoogle Scholar
  7. G-G.
    Gęba K., Granas A.: Infinite dimensional cohomology theories. J. Math. Pures Appl. 9(52), 145–270 (1973)MathSciNetGoogle Scholar
  8. GIP.
    Gęba K., Izydorek M., Pruszko A.: The Conley index in Hilbert spaces and its applications. Studia Math. 134, 217–233 (1999)MathSciNetGoogle Scholar
  9. G-D.
    A. Granas and J. Dugundji, Fixed Point Theory. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.Google Scholar
  10. Izy.
    Izydorek M.: A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems. J. Differential Equations 170, 22–50 (2001)MATHMathSciNetCrossRefGoogle Scholar
  11. K-Sz.
    Kryszewski W., Szulkin A.: An infinite-dimensional Morse theory with applications. Trans. Amer. Math. Soc. 349, 3181–3234 (1997)MATHMathSciNetCrossRefGoogle Scholar
  12. Man.
    Manolescu C.: Seiberg-Witten-Floer stable homotopy type of three-manifolds with b 1 =  0. Geom. Topol. 7, 889–932 (2003)MATHMathSciNetCrossRefGoogle Scholar
  13. McC.
    McCord C.: The connection map for attractor-repeller pairs. Trans. Amer. Math. Soc. 307, 195–203 (1988)MATHMathSciNetCrossRefGoogle Scholar
  14. Sal.
    Salamon D.: Morse theory, the Conley index and Floer homology. Bull. Lond. Math. Soc. 22, 113–140 (1990)MATHMathSciNetCrossRefGoogle Scholar
  15. Smol.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations. 2nd ed., Grundlehren Math. Wiss. 258. Springer-Verlag, New York, 1994.Google Scholar
  16. Sz.
    Szulkin A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375–418 (1992)MATHMathSciNetCrossRefGoogle Scholar
  17. Tau.
    Taubes C. F.: The Seiberg-Witten equations and the Weinstein conjecture. II. More closed integral curves of the Reeb vector field. Geom. Topol. 13, 1337–1417 (2009)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Mathematical InstitutePolish Academy of SciencesWarsawPoland
  2. 2.Faculty of Technical Physics and Applied MathematicsGdansk University of TechnologyGdanskPoland

Personalised recommendations