Poincaré Conjecture

Theorem
If a smooth m-manifold $\Sigma^m$ satisfies $H_0(\Sigma;\Z)=0$ and $H_m(\Sigma;\Z)=\Z$, then $\Sigma^m$ is homeomorphic to the m-sphere $\Bbb S^m$.

Proof
The proof procedes on several dimensional-cases. Note that the case $m=3$ is incredibly intricate, and that a full proof would be impractical to produce here. An outline of the $m=3$ case will be given instead.


 * m=1

Follows from the Classification of Compact One-Manifolds.


 * m=2

Follows from the Classification of Compact Two-Manifolds.


 * m=3

Follows from Thurston's Geometrization Conjecture, proved by Grigori Perelman.


 * m=4

Follows from 4-dimensional Topological h-Cobordism Theorem of A. Casson and M. Freedman. Proof in progress


 * m=5

Temporary summary: Any $\Sigma^5$ bounds a contractible 6-manifold $Z$. If $\Bbb D^6$ is a 6-disk (AKA 6-ball), then $Z-\Bbb D^6$ is an h-cobordism between $\Sigma$ and $\partial \Bbb D^6 = \Bbb S^5$, and hence $\Sigma$ is differomorphic to $\Bbb S^5$ by the h-Cobordism Theorem.


 * m$\ge$6

We can cut two small m-disks $D', D''$ from $\Sigma$. The remaining manifold, $\Sigma - (D' \cup D)$ is an h-cobordism between $\partial D'$ and $\partial D$, which are just two copies of $\Bbb S^{m-1}$. By the h-cobordism theorem, $\exists$ a diffeomorphism $\phi:\Sigma - (D' \cup D'') \rightarrow \Bbb S^{m-1} \times [0,1]$, which can be chosen to restrict to the identity on one of the $\Bbb S^{m-1}$. This $\Bbb S^{m-1}$ such that $\phi$ restricts to the identity, we'll call $\Xi$.

Since $\psi |_\Xi = Id$, we can extend $\psi$ across $D$, the interior of $\Xi$ to obtain a diffeomorphism $\phi': \Sigma - D \to \Bbb S^{m-1} \cup D'$. Note this latter manifold is merely an m-disk; we'll call it $\Bbb D^m$ to distinguish it from our $D', D''$.

Now our diffeomorphism $\phi': \Sigma - D'' \to \Bbb D^m$ 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 $int(\Bbb S^{m-1})=D$, but only as a homeomorphism of D.

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

It was first posed in 1904, and was finally solved by the work of, who solved Thurston's Geometrization Conjecture in 2003.