Every closed orientable three-manifold admits a prime and torus decomposition into pieces modeled on the eight Thurston geometries. In particular, a closed irreducible orientable three-manifold with fundamental group carries a flat metric. The Bieberbach theorem then identifies it with the three-torus because an abelian crystallographic group has trivial linear holonomy. Simply connected closed three-manifolds are three-spheres by the Poincaré conjecture.
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The Geometrization Conjecture is a fundamental concept in the field of 3-manifold topology, proposed by mathematician William Thurston in the late 20th century. It asserts that every closed, orientable 3-manifold can be decomposed into pieces that each have one of a specific set of geometric structures. These structures correspond to eight possible geometries that can be assigned to a manifold.