Introduction to Surgery Theory - William Gollinger

Описание: Sorry for the poor video quality; the conference room camera was broken for some reason, and this was the best we could do on the spot.

Abstract: Surgery Theory is a foundational approach to the classification of high-dimensional manifolds. It seeks to answer questions like "If M is a topological manifold does it admit a smooth structure? And if so, how many?" In this talk we will introduce the geometric notion of "surgery" and describe its effects on algebraic invariants using John Milnor's 1961 paper "A Procedure For Killing Homotopy Groups of Differentiable Manifolds" as the primary source. This paper provided fundamental algebraic results that determine how much topology we can eliminate from a manifold by the process of surgery, and identifies obstructions to being able to make your manifold as simple as possible. These results motivated the systematic study of Exotic Spheres in the landmark paper "Groups of Homotopy Spheres I" by Milnor and Michel Kervaire, and led to an almost complete description of manifolds that are homeomorphic to the standard sphere but not diffeomorphic. (The remaining piece comes from the J-homomorphism, which will have to wait for another time.)

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