Homolog\'ia de Morse en variedades compactas

TL;DR

This paper proves that the homology derived from the Morse-Witten complex, constructed from a Morse function on a compact Riemannian manifold with Morse-Smale gradient flow, is isomorphic to the manifold's singular homology.

Contribution

It establishes the isomorphism between the Morse-Witten homology and the singular homology for compact Riemannian manifolds with Morse-Smale functions.

Findings

Homology of Morse-Witten complex is isomorphic to singular homology.

Construction of Morse-Witten complex for Morse functions.

Validation of Morse homology as a topological invariant.

Abstract

Given a compact Riemannian manifold $(M g)$ and Morse function $f:m\to \mathbb{R}$ whose gradient flow satisfies the Morse-Smale condition, (i.e. the stable and unstable manifolds of f intersect transversely) we construct a chain complex called the Morse-Witten Complex. Our goal on this paper is show that the homology of the Morse-Witten complex is isomorphic to the singular homology of $M$.