Editing Thurston elliptization conjecture

{{Infobox mathematical statement
| name = Thurston elliptization conjecture
| image = 
| caption = 
| field = [[Geometric topology]]
| conjectured by = [[William Thurston]]
| conjecture date = 1980
| first proof by = [[Grigori Perelman]]
| first proof date = 2006
| implied by = [[Geometrization conjecture]]
| equivalent to = [[Poincaré conjecture]]<br>[[Spherical space form conjecture]]
| generalizations =
| consequences = 
}}

[[William Thurston]]'s '''elliptization conjecture''' states that a closed [[manifold|3-manifold]] with finite [[fundamental group]] is [[spherical 3-manifold|spherical]], i.e. has a [[Riemannian metric]] of constant positive sectional curvature.

==Relation to other conjectures==
A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere.
If the original 3-manifold had in fact a trivial fundamental group, then it is [[homeomorphic]] to the [[3-sphere]] (via the [[covering map]]). Thus, proving the elliptization conjecture would prove the [[Poincaré conjecture]] as a corollary. In fact, the elliptization conjecture is [[Logical equivalence|logically equivalent]] to two simpler conjectures: the [[Poincaré conjecture]] and the [[spherical space form conjecture]].

The elliptization conjecture is a special case of Thurston's [[geometrization conjecture]], which was proved in 2003 by [[G. Perelman]].

==References==
For the proof of the conjectures, see the references in the articles on [[geometrization conjecture]] or [[Poincaré conjecture]].
* William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. {{ISBN|0-691-08304-5}}.
* William Thurston. [http://www.msri.org/publications/books/gt3m/ The Geometry and Topology of Three-Manifolds], 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3.

[[Category:Riemannian geometry]]
[[Category:3-manifolds]]
[[Category:Conjectures that have been proved]]


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