Geometrization Conjecture - The 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. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds 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 sol 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 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 planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure so this gives a minor extra complication.

In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.