Topology Seminar

Given a smooth structure on a manifold M, one obtains a smooth structure on the product M x R. If M is closed and has dimension at least 5, then Kirby--Siebenmann's product structure theorem states that this assignment is a bijection from the set of smooth structures on M to the set of smooth structures on M x R. In the equivariant setting, the product structure theorem does not hold; the analogous map is neither injective nor surjective. In this talk, I will describe a way to construct infinitely many equivariant smooth structures of a G-manifold M which become equivalent after taking a product with R. This construction uses algebraic K-theory and a small amount of number theory.