## Geometry & Topology

### Algebraic cycles and the classical groups II: Quaternionic cycles

#### 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.

#### Article information

Source
Geom. Topol., Volume 9, Number 3 (2005), 1187-1220.

Dates
Revised: 28 April 2005
Accepted: 6 June 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799634

Digital Object Identifier
doi:10.2140/gt.2005.9.1187

Mathematical Reviews number (MathSciNet)
MR2174264

Zentralblatt MATH identifier
1081.14013

#### Citation

Lawson, H Blaine; Lima-Filho, Paulo; Michelsohn, Marie-Louise. Algebraic cycles and the classical groups II: Quaternionic cycles. Geom. Topol. 9 (2005), no. 3, 1187--1220. doi:10.2140/gt.2005.9.1187. https://projecteuclid.org/euclid.gt/1513799634

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