h-Cobordism Theorem

If  is a simply connected, compact manifold with a boundary that has two components,  and , such that inclusion of each is a homotopy equivalence, then  is diffeomorphic to the product  for . In other words, if  and  are two simply connected manifolds of dimension  and there exists an h-cobordism  between them, then  is a product  and  is diffeomorphic to .

The proof of the -cobordism theorem can be accomplished using surgery. A particular case of the -cobordism theorem is the Poincaré conjecture in dimension . Smale proved this theorem in 1961.

REFERENCES:

Smale, S. "Generalized Poincaré's Conjecture in Dimensions Greater than Four." Ann. Math. 74, 391-406, 1961.