Geometrization conjecture
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Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces. It was proposed by William Thurston in 1982, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's geometrization theorem, or hyperbolization theorem, states that Haken manifolds satisfy the conclusion of geometrization conjecture. Thurston announced a proof in the 1980s and since then s...
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces. It was proposed by William Thurston in 1982, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's geometrization theorem, or hyperbolization theorem, states that Haken manifolds satisfy the conclusion of geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print. Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery. There are now four different manuscripts with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that donot lead to the geometrization conjecture.
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