Thurston's Geometrization Conjecture

Theorem

When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:

$(1): \quad$ Euclidean geometry

$(2): \quad$ Hyperbolic geometry

$(3): \quad$ Spherical geometry

$(4): \quad$ The geometry of $\mathbb S^2 \times \R$

$(5): \quad$ The geometry of $\mathbb H^2 \times \R$

$(6): \quad$ The geometry of the universal cover $S L_2 \R^~$ of the Lie group $S L_2 \R$

$(7): \quad$ Nil geometry

$(8): \quad$ Sol geometry

where:

$\mathbb S^2$ is the 2-sphere

$\mathbb H^2$ is the hyperbolic plane.

Proof

This theorem requires a proof. In particular: Discuss Grigori Perelman's work -- it is generally believed his proof of this is valid, in which case this needs to be renamed to Theorem. 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.

Source of Name

This entry was named for William Paul Thurston.

Sources

• Weisstein, Eric W. "Thurston's Geometrization Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html