  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].
