Geometry & Topology

Algebraic cycles and the classical groups II: Quaternionic cycles

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

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In part I of this work we studied the spaces of real algebraic cycles on a complex projective space (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

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

Received: 24 April 2002
Revised: 28 April 2005
Accepted: 6 June 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C25: Algebraic cycles
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 14P99: None of the above, but in this section 19L99: None of the above, but in this section 55P47: Infinite loop spaces 55P91: Equivariant homotopy theory [See also 19L47]

quaternionic algebraic cycles characteristic classes equivariant infinite loop spaces quaternionic $K$–theory


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.

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