Geometrization conjecture

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 , 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. 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).