Algebra/Topology seminar

Speaker: Emma Brink

Title: Equivariant Bordism and Thom spectra.

Abstract: In this talk, we explore the relation between bordism and Thom spectra in equivariant homotopy theory. For a compact Lie group G, we define geometric bordism of G-manifolds with a given stable tangential structure. This admits a Thom-Pontryagin comparison map to the homology theory represented by an associated Thom spectrum which we construct using parametrized category theory. The main result is that the Thom-Pontryagin map is an isomorphism when the connected component of G is central - for example if G is a product of a finite group and a torus. And when G is not of this type, geometric bordism is not represented by a genuine G-spectrum as it fails to admit certain Wirthmüller isomorphisms. This extends work of tom Dieck and Schwede for unoriented bordism to very general stable tangential structures. We will also see that many classical Thom spectra admit multiple (global) equivariant refinements.