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.