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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 Poincare conjecture and the linearization conjecture.
The Elliptization Conjecture is a special case of Thurston's Geometrization Conjecture.
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