On the h-cobordism theorem and applications

Abstract

[en] H-cobordism is a notion developed by John Milnor, for the case of smooth manifolds, which involves a combination of homotopy theory and the theory of cobordisms. Stephen Smale used this concept to prove the h-cobordism theorem, which applies to compact smooth manifolds of dimension greater than five, with boundary. Later on, Milnor proved another version of the theorem in which he replaced the h-cobordism condition by a purely topological one, though he still used differential topology to prove the theorem. We will prove this second version of the theorem and, afterwards, will compare the two of them and will see that they are equivalent indeed. Finally, we will see that one of the corollaries of h-cobordism theorem proves a version of the so called generalized Poincaré conjecture in dimensions greater than four.