Geometrization conjecture

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

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

  1. the prime decomposition, where every compact 3-manifold is the connected sum of an essentially unique collection of prime three-manifolds
  2. the JSJ decomposition

Here is a formulation of Thurston's conjecture:

Separate a closed 3-manifold into its prime decomposition (capping off spherical boundaries with 3-balls), and then each irreducible summand is reduced by its JSJ decomposition. The interior of each of the resulting manifolds is covered by a simply-connected homogeneous space such that the group of covering transformations are isometries of the homogeneous space, thus endowing the manifold with the local geometry of the homogeneous space. In addition, each manifold (with its induced Riemannian metric) has finite volume.

There are exactly eight such simply-connected homogeneous spaces that admit finite volume quotients; they are called Thurston model geometries.

The following is a list of the eight geometries:

  1. Euclidean geometry
  2. Hyperbolic geometry
  3. Spherical geometry
  4. The geometry of S2 × R
  5. The geometry of H2 × R
  6. The geometry of SL2R
  7. Nil geometry, i.e. geometry of group of upper triangular 3-by-3 matrices with units on diagonal
  8. 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:

  1. it has a spherical geometry
  2. it has a hyperbolic geometry
  3. 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, although there are some notable results.

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, at least in the case of 3-manifolds with finite fundamental group. 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 von 3-Mannigfaltigkeiten


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