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 $$\mathbb{S}^m$$.

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 $$\mathbb{D}^6$$ is a 6-disk (AKA 6-ball), then $$Z-\mathbb{D}^6$$ is an h-cobordism between $$\Sigma$$ and $$\partial \mathbb{D}^6 = \mathbb{S}^5$$, and hence $$\Sigma$$ is differomorphic to $$\mathbb{S}^5$$ 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 $$\mathbb{S}^{m-1}$$. By the h-cobordism theorem, $$\exists$$ a diffeomorphism $$\phi:\Sigma - (D' \cup D'') \rightarrow \mathbb{S}^{m-1} \times [0,1]$$, which can be chosen to restrict to the identity on one of the $$\mathbb{S}^{m-1}$$. This $$\mathbb{S}^{m-1}$$ 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 $$\phi': \Sigma - D \to \mathbb{S}^{m-1} \cup D'$$. Note this latter manifold is merely an m-disk; we'll call it $$\mathbb{D}^m$$ to distinguish it from our $$D', D''$$.

Now our diffeomorphism $$\phi': \Sigma - D'' \to \mathbb{D}^m$$ induces a diffeomorphism on the boundary spheres $$\mathbb{S}^{m-1}$$. Any diffeomorphism of the boundary sphere $$\mathbb{S}^{m-1}$$ can be extended radially to the whole disk $$int(\mathbb{S}^{m-1})=D$$, but only as a homeomorphism of D.

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

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