Algebraic cycles and the classical groups Part II: Quaternionic cycles

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space ℙ(V), 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 (K H-theory), in analogy with Atiyah's real spaces and K R,-theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.