= Geometrization conjecture
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= Geometrization theorem
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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> $\mathbb Z^3$ 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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