Equivariant Bordism and Thom Spectra

Speaker: Emma Brink – University of Bonn

Abstract: For a compact Lie group $G$, we describe equivariant versions of geometric bordism with stable tangential structures. These define $\mathbb Z$-graded equivariant homology theories which admit a Thom-Pontryagin comparison map to the homology theory represented by associated Thom spectra.

If $G$ has non-abelian connected identity component (e.g. $G=SU(2)$), the geometric bordism theory is not represented by a genuine $G$-spectrum as it fails to admit certain Wirthmüller isomorphisms. In particular, the Thom-Pontryagin map can not be an isomorphism. We show that if $G$ has abelian identity component, the Thom-Pontryagin map is an isomorphism. This follows from the non-equivariant statement and a comparison of geometric fixed points; analogous to the statement for unoriented bordism proven in [tom Dieck, 1972] and [Schwede, 2018].

If time permits, we will explain that for suitable multiplicative tangential structures, the Thom-Pontryagin map becomes an isomorphism after localization at a family of inverse Thom classes.