$C_2$ equivariant characteristic classes over the rational Burnside ring

TL;DR

This paper provides minimal presentations for the rational $C_2$-equivariant cohomology of classifying spaces, enabling efficient computation of equivariant characteristic classes and their relations.

Contribution

It introduces minimal presentations for the $RO(C_2)$-graded Bredon cohomology of key classifying spaces with rational Burnside coefficients, advancing the understanding of equivariant characteristic classes.

Findings

Minimal presentations for cohomology of $B_{C_2}U(n), B_{C_2}SO(n), B_{C_2}Sp(n)$

Efficient description of rational $C_2$-equivariant characteristic classes

Relations between classes via maximal tori inclusions

Abstract

We give minimal presentations for the $RO(C_2)$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_2}U(n), B_{C_2}SO(n)$ and $B_{C_2}Sp(n)$ with coefficients in the rational Burnside Green functor $A_{\mathbf Q}$. This results in an efficient description of rational $C_2$ equivariant Chern, Pontryagin and symplectic characteristic classes. These classes are then related to each other using the inclusions of maximal tori.