Thurston's Geometrization Theorem states that every closed three-dimensional manifold can be decomposed into pieces, each admitting one of eight standard geometric structures, so that a three-manifold's geometry determines much of its topology. Proposed by William Thurston in 1982 as a conjecture, it is a three-dimensional analogue of the uniformization theorem for surfaces, which shows that every simply connected Riemann surface can be given one of three geometries. Grigori Perelman proved the geometrization conjecture in 2003 using Richard Hamilton's Ricci flow technique, and the proof includes the Poincare Conjecture as a special case.
Facts
StatementEvery oriented prime closed 3-manifold can be cut along 2-tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Geometrization conjecture, Wikipedia
The conjecture
Every oriented prime closed 3-manifold can be cut along 2-tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
History
Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server.
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