Geometrization conjecture

The geometrization conjecture says every compact 3-manifold can be cut along spheres and tori into pieces admitting one of eight homogeneous geometric structures. Grigori Perelman proved it using Ricci flow.

3-manifold: A topological space that locally resembles three-dimensional Euclidean space. The conjecture classifies compact examples by splitting them into geometrically structured pieces.
Ricci flow: An evolution equation that deforms a Riemannian metric according to its Ricci curvature. Perelman used it to evolve 3-manifold metrics toward geometrically meaningful forms.
William Thurston: An American mathematician whose work transformed the study of 3-manifolds and hyperbolic geometry. He formulated the geometrization program that became the conjecture.
Poincaré theorem: The theorem that every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Geometrization implies this landmark classification result as a special case.
Prime decomposition: The decomposition of a 3-manifold into prime manifolds, which cannot be split as nontrivial connected sums. It supplies the first stage of the conjecture’s structural decomposition.
Ricci flow with surgery: Ricci flow interrupted at high-curvature regions, which are cut out and capped before evolution resumes. Surgery controls singularities while preserving enough information to establish geometrization.
Thurston's geometrization conjecture: Thurston’s conjecture that compact 3-manifolds admit a canonical decomposition into geometrically modeled pieces. Its formulation established the classification problem addressed by the modern conjecture.
3-manifold topology: The study of the topological structure and classification of three-dimensional manifolds. Geometrization provides a central framework for classifying its compact spaces.
JSJ decomposition: The decomposition of an irreducible 3-manifold along tori into pieces that are Seifert-fibered or atoroidal. Its torus cuts isolate pieces to which the geometric classification applies.
Hamilton–Perelman theory: The program using Ricci flow and analytic estimates to prove the geometrization conjecture. It provides the proof strategy behind the conjecture’s resolution.

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