For three-manifolds, the proof follows from the categorization given by Thurston geometrization.
Its generalized version can in general be realized as a result of the h-cobordism theorem, but, as Milnor's exotic sphere demonstrates, it only applies to piecewise linear and topological manifolds in these cases. For three-manifolds, the proof follows from the categorization given by Thurston geometrization. This proof was recently finished when the finite time extinction of the Ricci flow was established. FTP, name this recently resolved problem in mathematics which states that every simply connected three-manifold is a sphere.
Its generalized version can in general be realized as a result of the h-cobordism theorem, but, as Milnor's exotic sphere demonstrates, it only applies to piecewise linear and topological manifolds in these cases.
{Poincaré} conjecture