Editing Geometrization conjecture (section)

==The conjecture==
A 3-manifold is called [[Closed manifold|closed]] if it is [[compact space|compact]] – without "punctures" or "missing endpoints" – and has no [[Manifold#Manifold with boundary|boundary]] ("edge").

Every closed 3-manifold has a [[prime decomposition (3-manifold)|prime decomposition]]: this means it is the [[connected sum]] ("a gluing together") of [[prime manifold|prime 3-manifold]]s.{{efn|This decomposition is essentially unique except for a small problem in the case of  [[Orientability#Orientability of manifolds|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 manifold|oriented]] prime closed [[3-manifold]] can be cut along [[torus|tori]], so that the [[interior (topology)|interior]] of each of the resulting manifolds has a geometric structure with finite volume.

There are [[Geometrization conjecture#The eight Thurston geometries|8 possible geometric structures]] in 3 dimensions. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are [[Seifert manifold]]s or [[atoroidal]]  called the [[JSJ decomposition]], which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an [[Anosov map]] of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)

For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the [[Orientable double cover|oriented double cover]]. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along [[projective plane]]s and [[Klein bottle]]s as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.

In 2 dimensions, every closed surface has a geometric structure consisting of a [[metric (mathematics)|metric]] with constant curvature; it is not necessary to cut the manifold up first. Specifically, every closed surface is diffeomorphic to a quotient of '''S'''<sup>2</sup>, '''E'''<sup>2</sup>, or '''H'''<sup>2</sup>.<ref name=geng/>