Poincaré Conjecture/Dimension 5

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Theorem

Let $\Sigma^5$ be a smooth $5$-manifold.

Let $\Sigma^m$ satisfy:

$H_0 \struct {\Sigma; \Z} = 0$

and:

$H_5 \struct {\Sigma; \Z} = \Z$

Then $\Sigma^5$ is homeomorphic to the $5$-sphere $\Bbb S^5$.

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Proof

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Summary:

Any $\Sigma^5$ bounds a contractible $6$-manifold $Z$.

Let $\Bbb D^6$ be a $6$-disk (AKA $6$-ball).

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Then $Z - \Bbb D^6$ is an $h$-cobordism between $\Sigma$ and $\partial \Bbb D^6 = \Bbb S^5$.

Hence $\Sigma$ is differomorphic to $\Bbb S^5$ by the $h$-Cobordism Theorem.