Poincaré Conjecture/Dimension 6 or Greater

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

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 $\Bbb S^m$.

This article, or a section of it, needs explaining. In particular: Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $\phi': \Sigma \setminus D' ' \to \Bbb S^{m - 1} \cup D'$.

This article, or a section of it, needs explaining. In particular: What does $\psi \vert_\Xi$ mean? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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.

$\blacksquare$