Stable foliations and semi-flow Morse homology

Abstract

In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration using Conley pairs and their pre-images under the time-$T$-map of the heat flow. A crucial step is to contract each Conley pair onto its part in the unstable manifold. To achieve this we construct stable foliations for Conley pairs using the recently found backward $\lambda$-Lemma [31]. These foliations are of independent interest [23].

54 pages, 14 figures

Cited by in corpus (3)

• Conley pairs in geometry - Lusternik-Schnirelmann theory and more
• Classical Morse theory revisited I -- Backward $\lambda$-Lemma and homotopy type
• The regularized free fall II -- Homology computation via heat flow