Thurston elliptization conjecture

William Thurston's Elliptization Conjecture states that a closed 3-manifold with finite fundamental group has a spherical geometry, i.e. has a Riemannian metric of constant positive sectional curvature. Any 3-manifold with such a metric is covered by 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.

The Elliptization Conjecture is a special case of Thurston's Geometrization Conjecture.