Quaternionic algebraic cycles and reality

Abstract

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group G a l ( C / R ) Gal({\mathbb C} / {\mathbb R}) . Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic K K -theory, and the other one classifies an equivariant cohomology theory Z ∗ ( − ) {\mathfrak Z}^*(-) which is a natural recipient of characteristic classes K H ∗ ( X ) → Z ∗ ( X ) KH^*(X) \to {\mathfrak Z}^*(X) for quaternionic bundles over Real spaces X X .