GEOMETRIZATION CONJECTURE PDF

Is this a fair wording? How do we avoid bloat? Instead of allowing long lists of examples, we should give each of the geometries their own page, and put most of the information there. Do we have to use bold face?

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The conjecture[ edit ] 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 3-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. 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 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 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 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.

The eight Thurston geometries[ edit ] 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. 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.

The corresponding manifolds are exactly the closed 3-manifolds with finite fundamental group. 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.

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. 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.

There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the Weeks manifold. 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. 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. The first two are mapping tori of the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible.

The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a Seifert fiber space often in several ways.

Under normalized Ricci flow manifolds with this geometry converge to a 1-dimensional manifold. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface.

Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable. If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori. This geometry can be modeled as a left invariant metric on the Bianchi group of type III.

Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.

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Thurston's Geometrization Conjecture

Thurston announced a proof in the s and since then several complete proofs have appeared in print. Grigori Perelman sketched a proof of the full geometrization conjecture in using Ricci flow with surgery. There are now several different manuscripts see below with details of the proof. 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 3-manifolds this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds.

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Geometrization conjecture

The Geometrization Conjecture This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov-Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces. In the course of proving the geometrization conjecture, the authors provide an overview of the main results about Ricci flows with surgery on 3-dimensional manifolds, introducing the reader to this difficult material. The book also includes an elementary introduction to Gromov-Hausdorff limits and to the basics of the theory of Alexandrov spaces.

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The Geometrization Conjecture

The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov-Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces.

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