Poincaré Conjecture/Dimension 2

Theorem
Let $\Sigma^2$ be a smooth $2$-manifold.

Let $\Sigma^2$ satisfy:
 * $H_0 \struct {\Sigma; \Z} = 0$

and:
 * $H_2 \struct {\Sigma; \Z} = \Z$

Then $\Sigma^2$ is homeomorphic to the $2$-sphere $\Bbb S^2$.

Proof
Follows from the Classification of Compact Two-Manifolds.