Homotopy Type and Critical Values

Take M , a finite-dimensional differentiable manifold, and f : M → R a smooth function. Such a function f is called a Morse function if it has no degenerate critical points. Morse theory allows us to connect the topology, in particular the homotopy type, of M with the behavior of f on M . In the following sections, we will state and prove two important theorems in Morse theory. Using these two theorems we can observe a poweful result in topology which states that any differential manifold is homotopy equivalent to a CW complex with an n-cell for each critical point of index n. But, first we recall some definitions and the Morse lemma.