Thurston elliptization conjecture
| Field | Geometric topology |
|---|---|
| Conjectured by | William Thurston |
| Conjectured in | 1980 |
| First proof by | Grigori Perelman |
| First proof in | 2006 |
| Implied by | Geometrization conjecture |
| Equivalent to | Poincaré conjecture Spherical space form conjecture |
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
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
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. 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.
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