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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


In part I of this work we studied the spaces of real algebraic cycles on a complex projective space (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 (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.

quaternionic algebraic cycles, characteristic classes, equivariant infinite loop spaces, quaternionic $K$–theory
Mathematical Subject Classification 2000
Primary: 14C25
Secondary: 55P43, 14P99, 19L99, 55P47, 55P91
Forward citations
Received: 24 April 2002
Revised: 28 April 2005
Accepted: 6 June 2005
Published: 1 July 2005
Proposed: Ralph Cohen
Seconded: Gunnar Carlsson, Haynes Miller
H Blaine Lawson, Jr
Department of Mathematics
Stony Brook University
Stony Brook
New York 11794
Paulo Lima-Filho
Department of Mathematics
Texas A&M University
College Station
Texas 77843
Marie-Louise Michelsohn
Department of Mathematics
Stony Brook University
Stony Brook
New York 11794