Poincaré Conjecture

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Theorem

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

Let $\Sigma^m$ satisfy:

$H_0 \left({\Sigma; \Z}\right) = 0$

and:

$H_m \left({\Sigma; \Z}\right) = \Z$



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.

Dimension $m = 1$

Follows from the Classification of Compact One-Manifolds.

$\Box$


Dimension $m = 2$

Follows from the Classification of Compact Two-Manifolds.

$\Box$


Dimension $m = 3$

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



$\Box$


Dimension $m = 4$

Follows from $4$-dimensional Topological h-Cobordism Theorem of Andrew John Casson and Michael Hartley Freedman.


$\Box$


Dimension $m = 5$


Summary:

Any $\Sigma^5$ bounds a contractible $6$-manifold $Z$.

Let $\Bbb D^6$ be a $6$-disk (AKA 6-ball).



Then $Z - \Bbb D^6$ is an $h$-cobordism between $\Sigma$ and $\partial \Bbb D^6 = \Bbb S^5$.

Hence $\Sigma$ is differomorphic to $\Bbb S^5$ by the h-Cobordism Theorem.

$\Box$


Dimension $m \ge 6$


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

The remaining manifold, $\Sigma - \left({D' \cup D''}\right)$ 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:

$\phi: \Sigma - \left({D' \cup D''}\right) \to \Bbb S^{m-1} \times \left[{0 \,.\,.\, 1}\right]$

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



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

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:

$\operatorname{int} \left({\Bbb S^{m - 1} }\right) = D''$

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 Jules Henri Poincaré.


Historical Note

The Poincaré Conjecture was first posed in $1904$ by Jules Henri Poincaré.

It was finally resolved by the work of Grigori Perelman, who solved Thurston's Geometrization Conjecture in $2003$.