What is "h-cobordism"

In geometric topology and differential topology, an (n+1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps

$M \hookrightarrow W \quad\mbox{and}\quad N \hookrightarrow W$

are homotopy equivalences.

The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.

The theorem was first proved by Stephen Smale for which he received the Fields Medal and is the fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the Generalized Poincaré Conjecture.