Topology of manifolds determined by general functions

Abstract : We will begin with function on smooth compact Manifolds and its implications on topology. We show that for a general smooth function the critical points are non degenerate (by Sard's theorem) and each critical point has an "index" which determines how the manifold changing while passing such a point. We will illustrate this phenomenon by explicit example and indicate a proof of how the manifold is "reconstructed" (up to homotopy) by such a general smooth function.