Development of Morse Theory

In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined on the manifolds. We first prove the Morse lemma, which says that, near critical points, such functions can be written in a useful way that gives us topological information. We then show how the homotopy type of the manifold is related to the information obtained from the function. Next we show that functions satisfying the required conditions exist for every manifold. Finally, as an application of the theory, we find the homotopy type of complex projective space.