In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. 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 types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
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. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award.
The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
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In mathematics, Thurston's geometrizationconjecture states that each of certain three-dimensional topological spaces has a unique geometric structure...
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a conjecture has been proven, it is no longer a conjecture but a theorem. Many important theorems were once conjectures, such as the Geometrization theorem...
fields to 3-manifolds. Thurston was next led to formulate his geometrizationconjecture. This gave a conjectural picture of 3-manifolds which indicated...
manifolds and diffeomorphisms. A proof of this conjecture, together with the more powerful geometrizationconjecture, was given by Grigori Perelman in 2002 and...
conjectures, see the references in the articles on geometrizationconjecture or Poincaré conjecture. William Thurston. Three-dimensional geometry and topology...
William Thurston's geometrizationconjecture, Hamilton produced a number of results in the 1990s which were directed towards the conjecture's resolution. In...
generalization of the Poincaré conjecture to the non-simply connected case. The conjecture is implied by Thurston's geometrizationconjecture, which was proven by...
in Grigori Perelman's proof of the Geometrizationconjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem. Geometric...
Thurston's geometrization theorem also follows from Perelman's proof using Ricci flow of the more general Thurston geometrizationconjecture. Thurston's...
Poincaré Conjecture, Richard S. Hamilton's formulation of a strategy to prove the conjecture, and William Thurston's geometrizationconjecture. Yau's long...
latter is an outline of a proof of Thurston's geometrizationconjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman...
– (2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the GeometrizationConjecture". arXiv:math/0612069. do Carmo, Manfredo Perdigão (1992)...
spaces such as Jacob's ladder. In dimension 3, William Thurston's geometrizationconjecture, proven by Grigori Perelman, gives a partial classification of...
claiming a resolution of the renowned Poincaré conjecture, along with the more general geometrizationconjecture. His work contained a number of notable new...
theorem for 3-manifolds, the capstone of his proof of Thurston's geometrizationconjecture, can be understood as an extension of the implicit function theorem...
zero curvature/flat, and negative curvature/hyperbolic – and the geometrizationconjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into...
experience does not necessarily rule out other geometries. The geometrizationconjecture gives a complete list of eight possibilities for the fundamental...
After the proof of the geometrizationconjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed...
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