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.