# 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.

## Source of Name

This entry was named for William Paul Thurston.