Conjugation Spaces and Equivariant Chern Classes

Let η 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 has a canonical structure of a conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in certain equivariant cohomology groups of X with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of η, forgetting the involution on X, and the Stiefel-Whitney classes of the real bundle of fixed points.