Thurston's geometrization conjecture for closed orientable irreducible 3-manifolds

Let $\mathcal{M}$ be a closed, orientable and irreducible 3-manifold. Its geometric decomposition is a decomposition in which every resulting component is geometric.

Thurston's geometrization conjecture. Each component resulting from the geometric decomposition of $\mathcal{M}$ is geometric.

This conjecture asserts that the geometric decomposition provides a standard decomposition of every closed, orientable, irreducible 3-manifold into geometric pieces. It is the central goal of geometrization and is known to be solved by the proof of Thurston's geometrization conjecture via Ricci flow.