Algebraic cycles and the classical groups 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.

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

References

Forward citations

Publication

Received: 24 April 2002

Revised: 28 April 2005

Accepted: 6 June 2005

Published: 1 July 2005

Proposed: Ralph Cohen

Seconded: Gunnar Carlsson, Haynes Miller

Authors

H Blaine Lawson, Jr
Department of Mathematics Stony Brook University Stony Brook New York 11794 USA

Paulo Lima-Filho
Department of Mathematics Texas A&M University College Station Texas 77843 USA

Marie-Louise Michelsohn
Department of Mathematics Stony Brook University Stony Brook New York 11794 USA

Volume 9, issue 3 (2005)

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

Abstract