Geometry & Topology, Vol. 9 (2005) Paper no. 27, pages 1187--1220.

Algebraic cycles and the classical groups II: Quaternionic cycles

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

Abstract. 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 (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

AMS subject classification. Primary: 14C25. Secondary: 55P43, 14P99, 19L99, 55P47, 55P91.

E-print: arXiv:math.AT/0507451

DOI: 10.2140/gt.2005.9.1187

Submitted to GT on 24 April 2002. (Revised 28 April 2005.) Paper accepted 6 June 2005. Paper published 1 July 2005.

Notes on file formats

H Blaine Lawson Jr, Paulo Lima-Filho, Marie-Louise Michelsohn
BL, MM: Department of Mathematics, Stony Brook University
Stony Brook, NY 11794, USA
PL: Department of Mathematics, Texas A&M University
College Station, TX 77843, USA


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