Equivariant real cycle class map and Witt-sheaf cohomology of classifying spaces

In this paper, we study equivariant real cycle class maps for group actions on real schemes, with a view toward Witt-sheaf characteristic classes. The cycle class maps take values in singular cohomology of the real points of the quotient stack, which are identified with the homotopy fixed-points of complex conjugation on the complex points. This provides a strong relation between Witt-sheaf cohomology of the geometric classifying space of a real algebraic group and the singular cohomology of the classifying spaces of its strong real forms, which we discuss in a number of examples. As a sample application, we compute the number of Witt-sheaf cohomological invariants of spin groups over the reals.