A Brief History of Morse Homology

Morse theory was originally due to Marston Morse [5]. It gives us a method to study the topology of a manifold using the information of the critical points of a Morse function defined on the manifold. Based on the same idea, Morse homology was introduced by Thom, Smale, Milnor, and Witten in various forms. This paper is a survey of some work in this direction. The first part of the paper focuses on the classical flow line approach by Thom, Smale, and Milnor. The second part of the paper will concentrate on Witten’s alternative and powerful approach using Hodge theory. 1 Basic concepts in classical Morse theory In this paper, let M be a compact n dimensional manifold and f : M → R be a smooth function. A point p ∈ M is called a critical point if the induced map f∗ : TMp → TRf(p) has rank zero. In other words, p is a critical point of f if and only if in any local coordinate system around p one has ∂f ∂x1 (p) = ∂f ∂x2 (p) = · · · = ∂f ∂xn (p) = 0. The real value f(p) is then called a critical value of f . A critical point p is said to be non-degenerate if, in a local coordinate system around p, the Hessian ( ∂ 2f ∂xi∂xj (p)) of f at p is non-degenerate. For a non-degenerate critical point, the number of negative eigenvalues of the Hessian is its Morse index. If all critical