gave a proof in the case of manifolds that fiber over the circle, and  and  gave proofs for the generic case of manifolds that do not fiber over the circle. Thurston's geometrization theorem also follows from Perelman's proof using Ricci flow of the more general Thurston geometrization conjecture.
Offered a demonstration in the case of manifolds that can be "fired" over the circle,  and  provided demonstrations for the typical case of manifolds that cannot be "fired" over the circle. Thurston's theorem on geometric structure also arises from Perelman's proof using Ricci flow to prove the more general conjecture on Thurston's geometrization.