Conjugation spaces and equivariant Chern classes

TL;DR

This paper develops a new method to define equivariant Chern classes for Real bundles using conjugation space structures, linking these classes to classical Chern and Stiefel-Whitney classes.

Contribution

It introduces a construction of equivariant Chern classes in Z/2-equivariant cohomology leveraging conjugation space structures, connecting equivariant and classical characteristic classes.

Findings

Equivariant Chern classes determine classical Chern classes.

They also determine Stiefel-Whitney classes of fixed point bundles.

The approach uses the conjugation space structure of BU.

Abstract

Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in the Z/2-equivariant cohomology of X with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of h, forgetting the involution on X, and the Stiefel-Whitney classes of the real bundle of fixed points.