Thurston's 8 Model Geometries
Overview
Every closed 3-manifold decomposes into pieces admitting one of eight geometries: S³, E³, H³, S²×ℝ, H²×ℝ, Nil, SL(2,ℝ)̃, Sol. Thurston conjectured; Perelman proved via Ricci flow.
Era: 1982–2003
Proposed at: Princeton, USA / Saint Petersburg, Russia
Formal structure
Domain: 3-manifold M; geometry (X, G) with X homogeneous, G transitive
Eight: S³, E³, H³, S²×ℝ, H²×ℝ, Nil, SL(2,ℝ)̃, Sol
Constraints: M closed; each piece has finite volume; geometrization
Objective: Classify 3-manifolds; Poincaré conjecture (S³ case)