Poincaré Conjecture

Theorem
Let $\Sigma^m$ be a smooth $m$-manifold.

Let $\Sigma^m$ satisfy:
 * $H_0 \struct {\Sigma; \Z} = 0$

and:
 * $H_m \struct {\Sigma; \Z} = \Z$

Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$.

Proof
The proof proceeds on several dimensional-cases. Note that the case $m = 3$ is incredibly intricate, and that a full proof would be impractical to produce here. An outline of the $m = 3$ case will be given instead.