Geometrization conjecture

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server.

Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments.

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.

In 2 dimensions, every closed surface has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.

[1] 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 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 ; this is a special case of a complete (G,X)-structure.

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).

Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways).

Other examples are given by the Seifert–Weber space, or "sufficiently complicated" Dehn surgeries 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.

Finite volume manifolds with this geometry are all compact and have the structure of a Seifert fiber space (often in several ways).

Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable.

This geometry can be modeled as a left invariant metric on the Bianchi group of type VIII or III.

Finite volume manifolds with this geometry are orientable and have the structure of a Seifert fiber space.

This geometry can be modeled as a left invariant metric on the Bianchi group of type II.

Finite volume manifolds with this geometry are compact and orientable and have the structure of a Seifert fiber space.

Under normalized Ricci flow, compact manifolds with this geometry converge to R2 with the flat metric.

(Nevertheless, 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.)

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.

The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.

In 1982, Richard S. 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.

In 2003, Grigori Perelman announced 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.

Perelman did not release any details on the proof of this result (Theorem 7.4 in the preprint 'Ricci flow with surgery on three-manifolds').

Beginning with Shioya and Yamaguchi, there are now several different proofs of Perelman's collapsing theorem, or variants thereof.

[8] A second route to the last part of Perelman's proof of geometrization is the method of Laurent Bessières and co-authors,[9][10] which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm for 3-manifolds.

[11][12] A book by the same authors with complete details of their version of the proof has been published by the European Mathematical Society.