#### 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

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**