Thurston's Elliptization Conjecture for closed 3-manifolds

Let $M$ be a closed $3$-manifold with finite fundamental group. The spherical geometry is the model geometry $(S^3,SO(4))$.

Thurston's Elliptization Conjecture. Every closed $3$-manifold with finite fundamental group can be modeled on the spherical geometry $(S^3,SO(4))$.

The conjecture implies the Poincaré Conjecture because such a manifold has a universal cover that is a homotopy sphere. The source reports that a proof had been announced by Perelman; the conjecture is therefore recorded as solved.