The geometrisation conjecture of 3-manifolds

Abstract

[en] This thesis aims to be a first approach to Thurston’s geometrisation conjecture, which states that each 3-manifold decomposes canonically into pieces admitting geometric structures. Starting from the definition of a model geometry, we will see first that the only three model geometries in dimension 2 are the Euclidean, the elliptic and the hyperbolic. Then we will show how Thurston’s theorem asserts that there are a total of eight model geometries in dimension 3, and we will classify six of them as Seifert spaces. We will finish by explaining the geometrisation conjecture through a historical perspective, from the first results on sphere and torus decompositions to Perelman’s proof. We will also sketch a proof of the Poincaré conjecture as an immediate corollary.