Geometrization conjecture

The Geometrization conjecture is the theorem, proved by Grigori Perelman in 2003, that every closed 3-manifold can be decomposed into pieces each of which admits one of eight homogeneous geometric structures. Proposed by William Thurston in the 1980s, it subsumed the Poincaré conjecture as a special case — the 3-sphere is the unique closed 3-manifold with spherical geometry and trivial fundamental group. The proof employed Ricci flow with surgery, adapting Richard Hamilton's geometric evolution equations to handle the topological singularities that arise when the flow collapses regions of the manifold. The eight geometries are not merely metric possibilities; they represent a complete classification of the ways that three-dimensional space can be homogeneous, and the conjecture asserts that the infinite variety of 3-manifold topologies is secretly organized by this finite geometric alphabet.