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, which (as of 2008) appears to be essentially correct. See the Solution of the Poincaré conjecture for a discussion of the proof.
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:
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.
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold M is a diffeomorphism from M to X/Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.)
There is some connection with the Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However S2×R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.O3(R), and the group G is the 6-dimensional Lie group O4(R), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite fundamental group. Examples include the 3-sphere, the Poincaré homology sphere, Lens spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type IX. Manifolds with this geometry are all compact, orientable, and have the structure of a Seifert fiber space (often in several ways). The complete list of such manifolds is given in the article on Spherical 3-manifolds. Under Ricci flow manifolds with this geometry collapse to a point in finite time. O3(R), and the group G is the 6-dimensional Lie group R3.O3(R), with 2 components. Examples are the 3-torus, and more generally the mapping torus of a finite order automorphism of the 2-torus; see torus bundle. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the Bianchi groups of type I or VII0. Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways). The complete list of such manifolds is given in the article on Seifert fiber spaces. Under Ricci flow manifolds with Euclidean geometry remain invariant. O3(R), and the group G is the 6-dimensional Lie group O1,3(R)+, with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest known volume is the Weeks manifold. Other examples are given by the Seifert-Weber space, or "sufficiently complicated" Dehn sugeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the Bianchi group of type V. Under Ricci flow manifolds with hyperbolic geometry expand.
A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (However a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π1(M):
Infinite volume manifolds can have many different types of geometric structure: for example, R3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.
There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:
It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by Ricci flow with surgery. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries S3 and S2×R, while what is left at large times should have a thick-thin decomposition into a "thick" piece with hyperbolic geometry and a "thin" graph manifold.
In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above. The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7.4 in the Ricci flow with surgery preprint. This theorem was stated by Perelman without proof. There are now three different proofs of Perelman's Theorem 7.4. There is the method of Shioya and Yamaguchi [T. Shioya and T. Yamaguchi, 'Volume collaped three-manifolds with a lower curvature bound,' Math. Ann. 333 (2005), no. 1, 131-155.] that uses Perelman's stability theorem [V. Kapovitch, 'Perelman's Stability Theorem', preprint arXiv:math/0703002, 2007.] and a fibration theorem for Alexandrov spaces [T. Yamaguchi. A convergence theorem in the geometry of Alexandrov spaces. In Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), volume 1 of Semin. Congr., pages 601-642. Soc. math. France, Paris, 1996.] . There is the method of Bessieres et.al. [L. Bessires, G. Besson, M. boileau, S. maillot, J. Porti, 'Weak collapsing and geometrization of aspherical 3-manifolds,' preprint arXiv:math/0706:2065, 2007.], which uses Thurston's hyperbolization theorem for Haken manifolds [J-P. Otal, 'Thurston's hyperbolization of Haken manifolds,'Surveys in differential geometry, Vol. III Cambridge, MA, 77-194, Int. Press, Boston, MA, 1998.] and Gromov's norm for 3-manifolds [M. Gromov. Volume and bounded cohomology. Inst. Hautes Etudes Sci. Publ. Math., (56):5-99 (1983), 1982.]. Finally there is the method of Morgan and Tian [J. Morgan and G. Tian, 'Completion of the Proof of the Geometrization Conjecture', arXiv:math/0809.4040, 2008.] that only uses Ricci flow. From Perelman's Theorem 7.4, the Geometrization conjecture "quickly" follows.