This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with. This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e. Thurston’s Geometrization Conjecture (now, a theorem of Perelman) aims to answer the question: How could you describe possible shapes of our universe?.

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A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. His death is a great loss for mathematics. This geometry fibers over the line with fiber the plane, and is the geometry of the identity component of the group G.

Geomtrization 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.

Mathematics > Differential Geometry

On the other hand, the geometrisation conjecture will be rather visibly lurking beneath the surface in the discussion of this lecture. Terence Tao on C, Notes 2: In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition conjefture it.

There are comjecture possible geometric structures in 3 dimensions, described in the next section. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. Thurston’s hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

[] Completion of the Proof of the Geometrization Conjecture

Anonymous on Polymath15, eleventh thread: The complete list of such manifolds is given in the article on Spherical 3-manifolds. The geometry of6. InHamilton showed that given a closed 3-manifold with a metric of positive Ricci curvaturethe 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.


Thurston’s conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. All of these important topics are of independent interest. All finite volume manifolds with solv geometry are compact. This geometry can be modeled as a left invariant metric on the Bianchi group of type V.

Contact the MathWorld Team. Walk through homework problems step-by-step from beginning to end. A convergence theorem in the geometry of Alexandrov spaces. The method is to understand the limits as time goes to infinity of Ricci flow with surgery.

Infinite volume manifolds can have many different types of geometric structure: The second decomposition is the Jaco-Shalen-Johannson torus decompositionwhich states that irreducible orientable compact 3- manifolds have a canonical up to isotopy minimal collection of disjointly embedded incompressible tori such that each component of the 3- geometrizatino removed by the tori is either “atoroidal” or “Seifert-fibered.

Ordering on the AMS Bookstore is limited to individuals for personal use only. I will take it for granted that this result is of interest, but you can read the Notices article of Milnorthe Bulletin article of Morganor the Clay Mathematical Institute description of the problem also by Milnor for background and motivation for this conjecture.

The Geometrization Conjecture

The third is the only example geometrizatkon a non-trivial connected sum with a geometric structure. Examples are the 3-torusand more generally the mapping torus of a finite order automorphism of the 2-torus; see torus bundle. Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space sometimes in two ways.

The gepmetrization is that the Ricci flow cnojecture 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.

Under normalized Ricci flow compact manifolds with this geometry converge rather slowly to R 1. 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. The point stabilizer is the dihedral group of order 8.


Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight xonjecture of geometric structure. Also containing proofs of Perelman’s Theorem 7. By geometfization to use this website, you agree to their use. Anonymous on C, Notes 2: Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. To find out more, including how to control cookies, see here: Publication Month and Year: Press, Boston, MA, Collection of teaching and learning tools built by Wolfram education experts: Every closed 3-manifold has a prime decomposition: Finite volume manifolds with this geometry are all compact and have conjscture structure of a Seifert fiber space often in several ways.

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. Under Ricci flow manifolds with Euclidean geometry remain invariant. Unlimited random practice problems and answers with built-in Step-by-step solutions. This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.

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.

The Fourier… Anonymous on Jean Bourgain. Writing up the results, and exploring negative t Career advice The uncertainty principle A: Before stating Thurston’s geometrization conjecture in detail, some background information is useful. Three-dimensional geojetrization possess what is known as a standard two-level decomposition.