Foundations of Morse Theory: From Sard's Theorem to Morse Inequalities

Abstract

The main goal of this work is to provide formal and descriptive proofs of the fundamental results of Morse theory. Morse theory is a branch of differential topology that studies the topological structure of smooth manifolds by analysing smooth real valued functions defined on them, known as Morse functions. The first part of the project is devoted to the study of these functions. Specifically, we present a detailed proof of Sard’s Theorem and use it to show that Morse functions are abundant and form a dense subset in the topological space of smooth functions on a manifold. In the second part, the foundational theorems of Morse theory are proved, including Morse’s Lemma and results that describe the homotopy types of the level sets of a Morse function based on its non-degenerate critical points. The final part of the work establishes the Morse inequalities, which relate key topological invariants of compact manifolds–namely, the Betti numbers and the Euler characteristic–to the number of critical points of a Morse function defined on the manifold.