# Thurston's Geometrization Conjecture

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## 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

## 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. http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html