Geometrization conjecture

The geometrization conjecture, also known as Thurston's geometrization conjecture, concerns the geometric structure of compact 3-dimensional manifolds. It was proposed by William Thurston in the late 1970s. It 'includes' other conjectures, such as the Poincaré conjecture and the spherical space-form conjecture. Here are some essential concepts used in the conjecture:

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

the prime decomposition, where every compact 3-manifold is the connected sum of a unique collection of prime three-manifolds
the JSJ decomposition

Here is a formulation of Thurston's conjecture:

Separate a 3-manifold into its prime decomposition, and then each irreducible summand is reduced by its JSJ decomposition.

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

Euclidean geometry
Hyperbolic geometry
Spherical geometry
The geometry of S² × R
The geometry of H² × R
The geometry of SL₂R
Nil geometry, i.e. geometry of group of upper triangular 3-by-3 matrices with units on diagonal
Sol geometry, i.e. geometry of group of upper triangular 2-by-2 matrices.

In the list of geometries above, S² is the 2-sphere (in a topological sense) and H² is the hyperbolic plane. Six of the eight geometries above (all except hyperbolic and spherical) are now clearly understood and known to correspond to Seifert manifolds and certain torus bundles. Using information about Seifert manifolds, we can restate the conjecture more tersely as:

Every irreducible, compact 3-manifold falls into exactly one of the following categories:

it has a spherical geometry
it has a hyperbolic geometry
The fundamental group contains a subgroup isomorphic to the free abelian group on two generators (this is the fundamental group of a torus).

If Thurston's conjecture is correct, then so is the Poincaré Conjecture (via Thurston elliptization conjecture). The Fields Medal was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds.

Progress has been made in proving that 3-manifolds that should be hyperbolic are in fact so. Mainly this progress has been limited to checking examples and reduction to more seemingly tractable conjectures, e.g. Virtually Haken Conjecture.

The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would smooth out any bumps in the metric, resulting in a metric of constant positive curvature, i.e. a spherical metric. He later developed a program to prove the Geometrization Conjecture by Ricci flow.

Grigori Perelman may have now solved the Geometrization conjecture (and thus also the Poincaré Conjecture) and there seems to be a consensus among experts that the proof is correct. Note that Perelman is eligible for a million dollar Millennium Prize Problems but his work will need to survive two years of systematic scrutiny after publication, which he seems uninterested in, before the Clay Mathematical Institute can deem the conjecture to have been solved.