Edward Witten came up with a related construction in the early 1980s sometimes known as Morse-Witten theory. Morse homology can be extended to finite dimensional non-compact or infinite-dimensional manifolds where the index remains finite, the metric is complete and the function satisfies the Palais-Smale condition, such as the energy functional for geodesics on a Riemannian manifold. The generalization to situations in which both index and coindex are infinite, but the relative index of any pair of critical points is finite, is known as Floer homology. Sergei Novikov generalized this construction to a homology theory associated to a closed one-form on a manifold. Morse homology is a special case for the one-form df. A special case of Novikov's theory is circle-valued Morse theory, which Michael Hutchings and Yi-Jen Lee have connected to Reidemeister torsion and Seiberg-Witten theory.
