Thurston’s geometrization conjecture

Thurston’s geometrization conjecture, also known simply as the geometrization conjecture, states that compact 3-manifolds can be decomposed into pieces with geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces. It was proposed by William Thurston in the late 1970s, and implies several other conjectures, such as the Poincaré conjecture and Thurston’s elliptization conjecture.

Grigori Perelman sketched a proof of the geometrization conjecture in 2003 using Ricci flow with surgery, which (as of 2006) appears to be essentially correct.

1 The conjecture

A 3-manifold is called closed if it is compact and has no boundary.

Every closed 3-manifold has a prime decomposition: this means it is the connected sum of an essentially unique collection of prime three-manifolds. This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.

Every prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

There are 8 possible geometric structures in 3 dimensions, described in the next section. Cutting a prime 3-manifold along tori into pieces that are Seifert manifolds or atoroidal is called the JSJ decomposition: there is a minimal way of doing this, which is essentially unique.

There are similar statements for compact manifolds with boundary without $S^2$ boundary components.

In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.

2 The eight Thurston geometries

A model geometry is a simply connected smooth manifold $M$ acted on by a Lie group $G$, such that $G$ is maximal among groups acting smoothly and transitively on $M$ with compact stabilizers, and there is at least one compact manifold modeled on $M$.

A geometric structure on a manifold is an isomorphism of the manifold with $M\mathrm{\Gamma }$ for some model geometry $M$ where $\mathrm{\Gamma }$ is a discrete subgroup of $G$ acting freely on $M$.

$S^2$ is the round 2-sphere and ${ℍ}^2$ is the hyperbolic plane.

Seven of the eight geometries (all except hyperbolic) are now clearly understood and known to correspond to Seifert manifolds and torus bundles. Using information about Seifert manifolds, we can restate the conjecture more tersely as:

Every prime, 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 free abelian group on two generators (the fundamental group of a torus).

• It has $S^2\times ℝ$ geometry.

3 History

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

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 collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes ”almost round” just before the collapse. He later developed a program to prove the Geometrization Conjecture by Ricci flow.

In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by extending Hamilton’s Ricci flow program to include surgery whenever the Ricci flow produces singularities. As of 2006 the consensus among those who have checked his work is that it is essentially correct, and the details have now been filled in. According to the Clay Mathematics Institute, he may be eligible for the Institute’s Millennium Prize Problems, although he has not submitted his work to a peer-reviewed journal.

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Thurston's_geometrization_conjectureGeometrization conjecture as of November 10, 2006.

Title	Thurston’s geometrization conjecture
Canonical name	ThurstonsGeometrizationConjecture
Date of creation	2013-03-22 16:23:28
Last modified on	2013-03-22 16:23:28
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	12
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 57R60
Synonym	geometrization conjecture