Algebraic cycles and the classical groups. I: Real cycles

The groups of algebraic cycles on complex projective space ℙ(V) are known to have beautiful and surprising properties. Therefore, when V carries a real structure, it is natural to ask for the properties of the groups of real algebraic cycles on ℙ(V). Similarly, if V carries a quaternionic structure, one can define quaternionic algebraic cycles and ask the same question. In this paper and its sequel the homotopy structure of these cycle groups is completely determined. It turns out to be quite simple and to bear a direct relationship to characteristic classes for the classical groups. It is shown, moreover, that certain functors in K-theory extend directly to these groups. It is also shown that, after taking colimits over dimension and codimension, the groups of real and quaternionic cycles carry E∞-ring structures, and that the maps extending the K-theory functors are E∞-ring maps. This gives a wide generalization of the results in (Boyer et al. Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993) 373.) 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 simple quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. These calculations yield 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 related to the ℤ₂-equivariant Chern class on Atiyah's KR-theory.