Editing Geometrization conjecture (section)

{{Short description|Three dimensional analogue of uniformization conjecture}}
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{{Infobox mathematical statement
| name = Geometrization theorem
| image = 
| caption = 
| field = [[Geometric topology]]
| conjectured by = [[William Thurston]]
| conjecture date = 1982
| first proof by = [[Grigori Perelman]]
| first proof date = 2006
| implied by =
| generalizations =
| consequences = [[Poincaré conjecture]]<br>[[Thurston elliptization conjecture]]
}}

[[File:William_Thurston.jpg | thumb | right | alt=A man with dark hair, large square glasses, and a white collared shirt is shown with a small smile | William Thurston (pictured) proposed this conjecture in 1982]]
In mathematics, '''Thurston's geometrization conjecture''' (now a [[theorem]]) states that each of certain three-dimensional [[topological space]]s has a unique [[geometry|geometric]] structure that can be associated with it. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surface (topology)|surface]]s, which states that every [[simply connected space|simply connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|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 {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}} as part of his [[Thurston's 24 questions|24 questions]], and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].

Thurston's  [[hyperbolization theorem]] implies that [[Haken manifold]]s 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 theory|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 Poincaré conjecture, though Perelman declined both awards.    

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.