In part I of this work we studied the spaces of real algebraic cycles on a complex projective
space , where
carries a real
structure, and completely determined their homotopy type. We also extended some functors
in –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
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
(–theory), in analogy with
Atiyah’s real spaces and –theory,
and the characteristic classes that we introduce for these objects are nontrivial. The
paper ends with various examples and applications.