Poincaré Conjecture/Dimension 6 or Greater

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Theorem

Let $\Sigma^m$ be a smooth $m$-manifold where $m \ge 6$.

Let $\Sigma^m$ satisfy:

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

and:

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

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

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Proof

We can cut two small $m$-disks $D', D$ from $\Sigma$.

The remaining manifold, $\Sigma \setminus \paren {D' \cup D}$ is an h-cobordism between $\partial D'$ and $\partial D$.

These are just two copies of $\Bbb S^{m-1}$.

By the $h$-cobordism theorem, there exists a diffeomorphism:

$\phi: \Sigma \setminus \paren {D' \cup D} \to \Bbb S^{m - 1} \times \closedint 0 1$

which can be chosen to restrict to the identity on one of the $\Bbb S^{m - 1}$.

Let $\Xi$ denote this $\Bbb S^{m - 1}$ such that $\phi$ restricts to the identity.

Since $\psi \vert_\Xi = Id$, we can extend $\psi$ across $D$, the interior of $\Xi$ to obtain a diffeomorphism $\phi': \Sigma \setminus D \to \Bbb S^{m - 1} \cup D'$.

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Let $\Bbb D^m$ denote this latter manifold, which is merely an $m$-disk.

Our diffeomorphism $\phi': \Sigma \setminus D \to \Bbb D^m$ induces a diffeomorphism on the boundary spheres $\Bbb S^{m - 1}$.

Any diffeomorphism of the boundary sphere $\Bbb S^{m - 1}$ can be extended radially to the whole disk:

$\map {\operatorname {int} } {\Bbb S^{m - 1} } = D$

but only as a homeomorphism of D.

Hence the extended function $\phi: \Sigma \to \Bbb S^m$ is a homeomorphism.

$\blacksquare$