Geometrizable three-manifold

A closed orientable three-manifold is geometrizable if its prime summands can be split along finitely many disjoint essential tori into pieces whose interiors carry complete geometric structures modeled on the eight Thurston geometries. Prime decomposition and capping sphere boundaries precede the torus cuts; a single homogeneous geometry on the original connected sum is not required.

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.

The eight simply connected three-dimensional homogeneous model geometries are $S^3$, ${\mathbb{R}}^3$, ${\mathbb{H}}^3$, $S^2\times \mathbb{R}$, ${\mathbb{H}}^2\times \mathbb{R}$, ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$, $\mathrm{Nil}$ and $\mathrm{Sol}$. A geometric structure is a locally homogeneous structure obtained as a quotient of a model by a discrete freely acting isometry group, equivalently with compatible model charts.