Applications of the h-Cobordism Theorem

Abstract

We provide an exposition of J. Milnor's proof of the h-Cobordism Theorem. This theorem states that a smooth, compact, simply connected n-dimensional manifold W with n greater than or equal to 6, whose boundary boundaryW consists of a pair of closed simply connected (n-1)-dimensional manifolds M0 and M1 and whose relative integral homology groups H(W,M0) are all trivial, is diffeomorphic to the cylinder M0 x [0, 1]. The proof makes heavy use of Morse Theory and in particular the cancellation of certain pairs of Morse critical points of a smooth function. We pay special attention to this cancellation and provide some explicit examples. An important application of this theorem concerns the generalized Poincare conjecture, which states that a closed simply connected n-dimensional manifold with the integral homology of the n-dimensional sphere is homeomorphic to the sphere. We discuss the proof of this conjecture in dimension n greater than or equal to 6, which is a consequence of the h-Cobordism Theorem.