01 · The idea
In everyday language
A three-manifold looks locally like ordinary three-dimensional space, but its global connections may be very different. Thurston proposed studying it by suitable decompositions and asking which standard geometric structures fit the pieces. Hyperbolic geometry, with constant negative curvature, plays an unexpectedly large role. His hyperbolisation results established this viewpoint for important classes of manifolds and made geometric constructions central to topology. The broader geometrisation picture involves eight model geometries, not a claim that every manifold is hyperbolic. Local geometry and global gluing must be understood together. Its later full proof built on the landscape Thurston had revealed.
Why it matters
Thurston’s viewpoint reshaped low-dimensional topology, turning the search for geometric structures and decompositions into a central organising principle for understanding three-manifolds.
