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 ,
- 5. the Geometry of ,
- 6. the Geometry of ,
- 7. Nil Geometry, or
- 8. Sol Geometry.

© 1996-9

1999-05-26