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<math>n \geq 5 \ </math> was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a [[Fields Medal]].   

<math>n \geq 5 \ </math> was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a [[Fields Medal]].   

<math>n=4 \ </math> was proved in 1981 by M. Freedman with his theorem on Casson handles, for which he shared a Fields medal with A Casson, who had pioneered study in this area.  Freedman's theorem was essentially a strengthening of the results of Smale's cobordism theorem, obtained at the expense of weakening the premises.   

<math>n=4 \ </math> was proved in 1981 by Michael Freedman with his theorem on Casson handles, for which he shared a Fields medal with Andrew Casson, who had pioneered study in this area.  Freedman's theorem was essentially a strengthening of the results of Smale's cobordism theorem, obtained at the expense of weakening the premises.   

Unfortunately, the so-called "Whitney trick" which allowed the cobordism theorems of the late 20th century to produce such breakthroughs in topology is not possible in three dimensions, leaving Poincare's original formulation of the conjecture open.

Unfortunately, the so-called "Whitney trick" which allowed the cobordism theorems of the late 20th century to produce such breakthroughs in topology is not possible in three dimensions, leaving Poincare's original formulation of the conjecture open.