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.
$\Box$