The eventual proof of this theorem was made possible by Richard Hamilton's work on Ricci flows.
The Whitehead link is a counterexample to an attempted proof of this theorem. A more general form of this theorem which describes a classification system based on decompositions was formulated by William Thurston. John Milnor showed that this theorem was valid in seven (*) dimensions, leading to work by Steven Smale which showed the validity of this theorem in dimensions five and higher, after which Michael Freedman pushed the upper dimension bound on this theorem down to four. The eventual proof of this theorem was made possible by Richard Hamilton's work on Ricci flows. For 10 points, identify this conjecture originated by its French namesake, which holds that every simply connected closed three-manifold is hemeomorphic to the 3-sphere, proven by Grigoriy Perelman.
The Whitehead link is a counterexample to an attempted proof of this theorem.
Poincare conjecture