Every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure (…)
The proof of the geometrization conjecture implies that of
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The conjecture was stated by William Thurston in 1982. The Ricci flow-strategy for the proof was devised by Richard Hamilton and the remaining technical problem of controlling the pinching was finally solved by Grigori Perelman.
Wikipedia, Geometrization conjecture
Laurent Bessieres, Gerard Besson, Michel Boileau, Sylvain Maillot, Joan Porti, Geometrisation of 3-manifolds (pdf)