h-Cobordism Theorem

If $W$ is a Simply Connected, Compact Manifold with a boundary that has two components, $M_1$ and $M_2$, such that inclusion of each is a Homotopy equivalence, then $W$ is Diffeomorphic to the product $M_1\times [0,1]$ for $\mathop{\rm dim}(M_1)\geq 5$. In other words, if $M$ and $M'$ are two simply connected Manifolds of Dimension $\geq 5$ and there exists an h-Cobordism $W$ between them, then $W$ is a product $M\times I$ and $M$ is Diffeomorphic to $M'$.

The proof of the $h$-cobordism theorem can be accomplished using Surgery. A particular case of the $h$-cobordism theorem is the Poincaré Conjecture in dimension $n\geq 5$. Smale proved this theorem in 1961.

See also Diffeomorphism, Poincaré Conjecture, Surgery

References

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