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
Email: blaine@math.sunysb.edu, plfilho@math.tamu.edu, mlm@math.sunysb.edu
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