Introduction to Morse theory and Morse homology - Leal, Julian David Espinel (2021)
In this work we present a study of Morse theory with the aim of introducing the Morse homology theorem as its natural extension. For this we prove the classic Morse theorem, which states that a Morse function defined on a manifold determines its topology as CW-complex through its critical points. Next, we introduce the Morse and Poincaré polynomials, their relations, and the perfect Morse functions that show when the number of non-degenerate critical points is equal to the k-th Betti number of the manifold. Finally, we present the stable and unstable manifolds given by the gradient flow of a Morse-Smale function and the Morse-Smale-Witten chain complex whose homology is isomorphic to the singular homology of the manifold.