Geometrization

Geometrization is a program in differential geometry and topology that seeks to understand the global structure of manifolds by decomposing them into pieces on which a uniform geometric model can be imposed. It provides a bridge between topology and geometry, replacing purely topological questions with a geometric classification.

In three dimensions, the Geometrization Conjecture, formulated by William Thurston, asserts that every compact 3-manifold can be cut along a collection of incompressible tori into submanifolds, each admitting one of eight model geometries. These geometries are S^3, E^3, H^3, S^2×R, H^2×R, Nil, Sol, and SL(2,R). The conjecture implies that the large-scale topology of a 3-manifold is governed by these geometric pieces, which can then be analyzed via their intrinsic geometry.

A key technical tool in Thurston’s program is the torus (JSJ) decomposition, which separates a 3-manifold into prime pieces along incompressible tori, after which the geometric structure on each piece can be studied independently. The full Geometrization Conjecture also implies the Poincaré Conjecture as a consequence.

Perelman’s proof, announced in 2003 and completed by 2006, established Geometrization using Ricci flow with surgery, confirming that every closed 3-manifold decomposes into pieces carrying the eight Thurston geometries. The result has profoundly shaped our understanding of 3-manifolds and continues to influence geometric topology and related fields.