Canonical chain complexes for Morse-Smale vector fields

TL;DR

This paper constructs canonical chain complexes for Morse-Smale vector fields, extending Morse theory to include closed orbits, and proves their invariance and utility in deriving Morse inequalities.

Contribution

It introduces a method to build canonical chain complexes for Morse-Smale vector fields using cech homology spectral sequences, improving upon previous non-canonical approaches.

Findings

Constructed canonical chain complexes for Morse-Smale vector fields.

Demonstrated invariance of these complexes under topological equivalence.

Provided examples illustrating the construction and applications.

Abstract

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that, similarly to the Morse chain complex of a gradient-like vector field, even in the presence of closed orbits, Morse-Smale vector fields admit canonical chain complexes, invariant under topological equivalence, from which one can algebraically derive Morse inequalities. In this paper we show that this is actually the case, improving the state of the art that only offers non-canonical chain complexes. Technically, we achieve this result considering the \v{C}ech homology spectral sequence of the unstable manifolds filtration. In particular, we turn bounded exact couples into chain complexes such that the limit page of the spectral sequence associated with an exact couple gives the homology of the chain complex. We showcase our construction with examples.