Exotic aspherical 4-manifolds (Kyle Hayden)

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: The Borel conjecture predicts that closed aspherical manifolds (i.e., those whose higher homotopy groups all vanish) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth analog of the Borel conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions $\leq 3$ but long known to be false in all dimensions $\geq 5$. In particular, I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds.

In the morning background talk (at 11:30am in Park 338), I will review some of the history of exotic smooth structures on manifolds (especially in dimension four), with an eye towards obstacles that arise in the aspherical setting.