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.

$\blacksquare$

Source of Name

This entry was named for Henri Poincaré.

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