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Geometrization conjecture

** Thurston's geometrization conjecture ** states that compact [3-manifold] s can be decomposed into [submanifold] s 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 [PoincarĂ© conjecture] and Thurston's [elliptization conjecture] .

Thurston's ** geometrization theorem ** , or ** hyperbolization theorem ** , states that [Haken manifold] s 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 (see below) with details of the proof. The PoincarĂ© conjecture and the [spherical space form conjecture] are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

** 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 prime three-manifolds (this decomposition is essentially unique except for a small problem in the case of non-orientable 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.

Here is a statement of Thurston's conjecture:

:Every oriented 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.

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