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Algebraic cycles and the
classical groups II: Quaternionic cycles
H Blaine Lawson, Jr, Paulo Lima-Filho and Marie-Louise Michelsohn |
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Geometry & Topology 9 (2005) 1187–1220
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arXiv: math.AT/0507451 |
Abstract |
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In part I of this work we studied the spaces of real algebraic cycles on a complex projective space , where carries a real structure, and completely determined their homotopy type. We also extended some functors in –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 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 (–theory), in analogy with Atiyah’s real spaces and –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
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Mathematical Subject Classification 2000
Primary: 14C25
Secondary: 55P43, 14P99, 19L99, 55P47, 55P91
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References |
Forward citations |
Publication
Received: 24 April 2002
Revised: 28 April 2005
Accepted: 6 June 2005
Published: 1 July 2005
Proposed: Ralph Cohen
Seconded: Gunnar Carlsson, Haynes Miller
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Authors |
| H Blaine Lawson, Jr | |
| Department of Mathematics Stony Brook University Stony Brook New York 11794 USA |
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| Paulo Lima-Filho | |
| Department of Mathematics Texas A&M University College Station Texas 77843 USA |
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| Marie-Louise Michelsohn | |
| Department of Mathematics Stony Brook University Stony Brook New York 11794 USA |
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