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.
Article PDF
Similar content being viewed by others
References
Abbondandolo A.: A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces. Topol. Methods Nonlinear Anal. 9, 325–382 (1997)
M. Aguilar, S. Gitler and C. Prieto, Algebraic Topology from a Homotopical Viewpoint. Springer-Verlag, New York, 2002.
Crabb M. C., Jaworowski J.: Aspects of the Borsuk-Ulam theorem. J. Fixed Point Theory Appl. 13, 459–488 (2013)
K.-C. Chang, Methods in Nonlinear Analysis. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
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.
Gęba K.: Fredholm σ-proper maps of Banach spaces. Fund. Math. 64, 341–373 (1969)
Gęba K., Granas A.: Infinite dimensional cohomology theories. J. Math. Pures Appl. 9(52), 145–270 (1973)
Gęba K., Izydorek M., Pruszko A.: The Conley index in Hilbert spaces and its applications. Studia Math. 134, 217–233 (1999)
A. Granas and J. Dugundji, Fixed Point Theory. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
Izydorek M.: A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems. J. Differential Equations 170, 22–50 (2001)
Kryszewski W., Szulkin A.: An infinite-dimensional Morse theory with applications. Trans. Amer. Math. Soc. 349, 3181–3234 (1997)
Manolescu C.: Seiberg-Witten-Floer stable homotopy type of three-manifolds with b 1 = 0. Geom. Topol. 7, 889–932 (2003)
McCord C.: The connection map for attractor-repeller pairs. Trans. Amer. Math. Soc. 307, 195–203 (1988)
Salamon D.: Morse theory, the Conley index and Floer homology. Bull. Lond. Math. Soc. 22, 113–140 (1990)
J. Smoller, Shock Waves and Reaction-Diffusion Equations. 2nd ed., Grundlehren Math. Wiss. 258. Springer-Verlag, New York, 1994.
Szulkin A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375–418 (1992)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This 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.
About this article
Cite this article
Starostka, M. Morse cohomology in a Hilbert space via the Conley index. J. Fixed Point Theory Appl. 17, 425–438 (2015). https://doi.org/10.1007/s11784-015-0216-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0216-5