#### Volume 9, issue 3 (2005)

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Algebraic cycles and the classical groups II: Quaternionic cycles

### H Blaine Lawson, Jr, Paulo Lima-Filho and Marie-Louise Michelsohn

Geometry & Topology 9 (2005) 1187–1220
 arXiv: math.AT/0507451
##### Abstract

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space $ℙ\left(V\right)$, 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.

##### Keywords
quaternionic algebraic cycles, characteristic classes, equivariant infinite loop spaces, quaternionic $K$–theory
##### Mathematical Subject Classification 2000
Primary: 14C25
Secondary: 55P43, 14P99, 19L99, 55P47, 55P91