Thurston's Geometrization Conjecture
Contents
Theorem
When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:
- $(1): \quad$ Euclidean geometry
- $(2): \quad$ Hyperbolic geometry
- $(3): \quad$ Spherical geometry
- $(4): \quad$ The geometry of $\mathbb S^2 \times \R$
- $(5): \quad$ The geometry of $\mathbb H^2 \times \R$
- $(6): \quad$ The geometry of the universal cover $S L_2 \R^~$ of the Lie group $S L_2 \R$
- $(7): \quad$ Nil geometry
- $(8): \quad$ Sol geometry
where:
- $\mathbb S^2$ is the 2-sphere
- $\mathbb H^2$ is the hyperbolic plane.
Proof
Discuss Grigori Perelman's work -- it is generally believed his proof of this is valid, in which case this needs to be renamed to Theorem.
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Source of Name
This entry was named for William Paul Thurston.
Sources
- Weisstein, Eric W. "Thurston's Geometrization Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html
