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

TL;DR

This paper develops equivariant real cycle class maps linking Witt-sheaf cohomology of classifying spaces to singular cohomology of real points, with applications to invariants of spin groups.

Contribution

It introduces equivariant real cycle class maps for real schemes, connecting Witt-sheaf cohomology with singular cohomology of classifying spaces, and computes invariants for spin groups.

Findings

Established a relation between Witt-sheaf cohomology and singular cohomology of classifying spaces.

Computed Witt-sheaf cohomological invariants for spin groups over the reals.

Provided examples illustrating the connection between geometric classifying spaces and real forms.

Abstract

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.