Geometrization conjecture

Thurston's Conjecture (also known as Thurston's Geometrization Conjecture concerns the geometric structure of compact 3-dimensional manifolds.

It was proposed by William Thurston.

It 'includes' other conjectures, such as the Poincaré Conjecture and the Spherical Space-Form Conjecture.

Here are some essential concepts used in the conjecture:

3D manifolds exhibit a phenomenon called a standard two-level decomposition.

1. a connected sum decomposition, where every compact 3-manifold is the connected sum of a unique collection of prime three-manifolds
2. the Jaco-Shalen-Johannson torus decomposition

The Jaco-Shalen-Johannson torus decomposition is defined as follows:

"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"

Here is a formulation of Thurston's conjecture:

Separate a 3-manifold into:

its connected sum

And

The Jaco-Shalen-Johannson torus decomposition

Each remaining component can then be described using one particular geometry from the following list:

1. Euclidean geometry
2. Hyperbolic geometry
3. Spherical geometry]
4. The geometry of S² x R
5. The geometry of H² x R
6. The geometry of SL₂R
7. Nil geometry, or
8. Sol geometry.

In the list of geometries above, S² is the 2-sphere (in a topological sense) and H² is the hyperbolic plane.

If Thurston's conjecture is correct, then so is the Poincaré Conjecture.

The Fields Medal was awarded to Thurston in 1982 for proving that his conjecture was valid in some of these cases.

Six of the eight geometries above are now relatively clearly appreciated and hyperbolic geometry has seen significant progreess.

This is not the case with the geometry of constant positive curvature (see Riemann) which is not yet well understood.

It is in this geometry that the Thurston elliptization conjecture can be seen to extend the Poincaré conjecture.