Thurston elliptization conjecture
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature. A 3-manifold with such a metric is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. Note that this means that 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 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
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002
- G. Perelman,Ricci flow with surgery on three-manifolds, 2003
- G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003
- Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006) (fills in the details of Perelman's proof of the geometrization conjecture).
- Cao, Huai-Dong (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelman theory of the Ricci flow" (PDF). 10 (2): 165–498. Retrieved 2006-07-31.
{{cite journal}}: Cite journal requires|journal=(help); Unknown parameter|Journal=ignored (|journal=suggested) (help) - John W. Morgan. Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57-78 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the 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. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture.