Poincaré Conjecture/Dimension 4
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Theorem
Let $\Sigma^4$ be a smooth $4$-manifold.
Let $\Sigma^4$ satisfy:
- $H_0 \struct {\Sigma; \Z} = 0$
and:
- $H_4 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^4$ is homeomorphic to the $4$-sphere $\Bbb S^4$.
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Proof
Follows from $4$-dimensional Topological $h$-Cobordism Theorem.
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Historical Note
The Poincaré Conjecture for dimension $4$ is dependent upon the truth of the Topological $h$-Cobordism Theorem.
This was proved by Andrew John Casson and Michael Hartley Freedman.