Poincaré Conjecture/Dimension 2

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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 $\Bbb S^2$.

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Proof

Follows from the Classification of Compact Two-Manifolds.

$\Box$