Mathematics

In 1960, and according to the myth, while relaxing on the beaches of Rio, Smale proved (in a way fitting the beauty of those beaches) the Poincaré Conjecture for dimensions higher than or equal to 5 as a corollary of another result: The h-Cobordism Theorem, which translates some homotopy conditions on a cobordism to it being smoothly trivial.  The main ingredient is handle decompositions (think of a CW decomposition where lower dimensional cells are thickened to fit the manifold’s dimension), which, in higher dimensions, can be related nicely to homology.   In this talk (that should be accessible to everyone) we will introduce these notions and give the main ideas of the proof.