Thurston Elliptization Conjecture
Every closed three-manifold with finite fundamental group has a metric of constant positive scalar
curvature, and hence is homeomorphic to a quotient
, where 
is a finite group of rotations that acts freely on 
.
Since the trivial group is in particular a finite group, the elliptization conjecture implies the Poincaré conjecture.
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Thurston's geometrization
conjecture