Algebraic Cycles and the Classical Groups - Part I, Real Cycles

Abstract

Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are ring maps. The stabilized space of cycles is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed by the representations R^{n,n} for n > 0. This gives a wide generalization of the results in Boyer, Lawson, Lima-Filho, Mann and Michelsohn on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. This yields an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory.