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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In particular: Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_4$
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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.

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