Thurston's Geometrization Conjecture

Thurston's conjecture has to do with geometric structures on 3-D Manifolds. Before stating Thurston's conjecture, some background information is useful. 3-dimensional Manifolds possess what is known as a standard 2-level Decomposition. First, there is the Connected Sum Decomposition, which says that every Compact 3-Manifold is the Connected Sum of a unique collection of Prime 3-Manifolds.

The second Decomposition is the Jaco-Shalen-Johannson Torus Decomposition, which states that irreducible orientable Compact 3-Manifolds have a canonical (up to Isotopy) minimal collection of disjointly Embedded incompressible Tori such that each component of the 3-Manifold removed by the Tori is either ``atoroidal'' or ``Seifert-fibered.''

Thurston's conjecture is that, after you split a 3-Manifold into its Connected Sum and then Jaco-Shalen-Johannson Torus Decomposition, the remaining components each admit exactly one of the following geometries:

: 1. Euclidean Geometry,
: 2. Hyperbolic Geometry,
: 3. Spherical Geometry,
: 4. the Geometry of $\Bbb{S}^2 \times \Bbb{R}$ ,
: 5. the Geometry of $\Bbb{H}^2 \times \Bbb{R}$ ,
: 6. the Geometry of ${\it SL}_2R$ ,
: 7. Nil Geometry, or
: 8. Sol Geometry.

Here, $\Bbb{S}^2$ is the 2-Sphere and $\Bbb{H}^2$ is the Hyperbolic Plane. If Thurston's conjecture is true, the truth of the Poincaré Conjecture immediately follows.