Redirected from Thurston's Geometrization Conjecture
Firstly, 3manifolds[?] exhibit a phenomenon called a standard twolevel decomposition[?].
Secondly, the JacoShalenJohannson torus decomposition is defined as follows:
"Irreducible orientable compact 3manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3manifold removed by the tori is either atoroidal[?] or a Seifert manifold[?]"
Here is a formulation of Thurston's conjecture:
Each remaining component can then be described using one particular geometry from the following list:
In the list of geometries above, S^{2} is the 2sphere[?] (in a topological sense) and H^{2} is the hyperbolic plane[?]. Six of the eight geometries above are now clearly understood and known to correspond to Seifert manifolds. Using information about Seifert manifolds, we can restate the conjecture more tersely as:
Every irreducible, compact 3manifold falls into exactly one of the following categories:
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 3manifolds 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 3manifolds that should be spherical has been slower, but provided the spark needed for Richard S. Hamilton[?] to develop his Ricci flow[?]. In 1982, Hamilton showed that given a closed 3manifold 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) but because this latter makes Perelman eligible for a million dollar Clay Millennium Prize[?] his work will need to survive two years of systematic scrutiny before the conjecture(s) will be deemed to have been solved.
Search Encyclopedia

Featured Article
