Hamilton vs Smale's Poincare Program

 The Poincare conjecture was one of the seven Millenium problems that was solved in 2003 by Grisha Perelman in his papers [math/0211159], [math/0303109] and [math/0307245], which showed that when performing Ricci flow on a closed $N$-manifold, it reduces to a type II singularity in finite-time, in the sense that under Ricci flow any closed $N$-manifold is homeomorphic (and diffeomorphic) to $\mathbb{S}^{N}$. This is following the Ricci flow program, initiated by Hamilton (who Perelman stated deserved the Millenium prize, declining upon being told that (1) Hamilton could not be awarded for the complete general proof of Poincare, and (2) since the papers were solely arXived in math.DG and not published, the prize required that he publish the papers if Perelman accepted). While Ricci flow now seems to be the generic approach to Poincare conjecture, in 1961 Smale announced his results that under the h-cobordism theorem, in $N\geq 6$ one can show that for certain kinds of manifolds, using the Alexandrov trick one can show that in general, any closed $N$-manifold is homeomorphic to $\mathbb{S}^{N}$. See my (highly naive) notes on this result, linked below. It must be said that, while the h-cobordism theorem holds for topological and smooth manifolds alike (from Rourke and Sanderson), the Poincare conjecture in Smale's proof need not hold for smooth manifolds. Despite several other reasons why this proof does not hold for all kinds of manifolds (and why this proof fails for certain dimensional cases in $N\geq 6$), I believe that Smale's result was a prime motivator for all the other works leading up to the proof of the Poincare conjecture, whether directly or indirectly. As of the superiority of Hamilton over Smale, I choose to believe that Hamilton's works proved to be far more fundamental and intrinsic to Poincare conjecture (and math.DG) than Smale's work. However, this is only my belief. 

Notes on Poincare conjecture and the h-cobordism theorem