Algebraic cycles and the classical groups II: Quaternionic cycles

Abstract

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space $\mathbb{P}\left(V\right)$, where $V$ carries a real structure, and completely determined their homotopy type. We also extended some functors in $K$–theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel–Whitney classes. In this sequel, we establish corresponding results in the case where $V$ has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a similarly simple description. The stabilized space of quaternionic algebraic cycles admits a nontrivial infinite loop space structure yielding, in particular, a delooping of the total Pontrjagin class map. This stabilized space is directly related to an extended notion of quaternionic spaces and bundles ($KH$–theory), in analogy with Atiyah’s real spaces and $KR$–theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.