Speaker: Thomas Brazelton

Abstract:

The million dollar proof of the Poincaré Conjecture provides a resolution to a statement about 3-manifolds. We can ask the same question about n-manifolds, and attain an analogous statement, called the Generalized Poincaré Conjecture. We will demonstrate how this is proved in cases n=1, n=2 and n≥6, using some simple facts from algebraic topology, and assuming a few famous results. In the process, we will touch on some major ideas in algebraic topology, namely homotopy theory and cobordism.