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

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