Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

• Published in: Journal of Differential Equations Vol. 263; no. 11; pp. 7162 - 7186
• Main Authors: Izydorek, Marek, Rot, Thomas O., Starostka, Maciej, Styborski, Marcin, Vandervorst, Robert C.A.M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.12.2017
• Subjects: (Local) Morse homology; [formula omitted]-flows; Conley index; Morse–Conley–Floer homology; 58E05; Morse–Conley–Floer homology; 37B30; Conley index; 37C10; LS-flows; (Local) Morse homology
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2017.08.007

In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.