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arXiv · 1612.05524

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

Marek Izydorek·Thomas O. Rot·Maciej Starostka·Marcin Styborski·Robert C. A. M. Vandervorst
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Abstract

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 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.

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Marek Izydorek, Thomas O. Rot, Maciej Starostka, Marcin Styborski, Robert C. A. M. Vandervorst. 2016-12-16. Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces. https://arxiv.org/abs/1612.05524

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