Poincaré Conjecture/Dimension 3

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 $\Sigma^3$ be a smooth $3$-manifold.

Let $\Sigma^3$ satisfy:

$H_0 \struct {\Sigma; \Z} = 0$

and:

$H_3 \struct {\Sigma; \Z} = \Z$

Then $\Sigma^3$ is homeomorphic to the $3$-sphere $\Bbb S^3$.

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 $H_3$ 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

Follows from Thurston's Geometrization Conjecture.

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$

Historical Note

The Poincaré Conjecture for a long time was settled for all dimensions but $3$.

Grigori Perelman finally provided the missing piece by demonstrating the truth of Thurston's Geometrization Conjecture in $2003$.