Cohomology Group. Morse, And Witten Method

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 main idea comes from dynamic system. The second part of the paper concentrates on Witten's alternative and powerful approach by using Hodge theory. Witten extended these ideas of Morse homology to the manifolds of infinite dimension.