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4-Torsion classes in the integral cohomology of oriented Grassmannians

Part of: Special varieties Applied homological algebra and category theory Homology and homotopy of topological groups and related structures

Published online by Cambridge University Press:  15 January 2025

Ákos K. Matszangosz  and
Matthias Wendt
Ákos K. Matszangosz
Affiliation:
HUN-REN Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary ([email protected]) (corresponding author)
Matthias Wendt
Affiliation:
Fachgruppe Mathematik und Informatik, Bergische Universität Wuppertal, Gaussstrasse 20, 42119 Wuppertal, Germany ([email protected])
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Abstract

We investigate the existence of 4-torsion in the integral cohomology of oriented Grassmannians. We establish bounds on the characteristic rank of oriented Grassmannians and prove some cases of our previous conjecture on the characteristic rank. We also discuss the relation between the characteristic rank and a result of Stong on the height of w1 in the cohomology of Grassmannians. The existence of 4-torsion classes follows from the results on the characteristic rank via Steenrod square considerations. We thus exhibit infinitely many examples of 4-torsion classes for oriented Grassmannians. We also prove bounds on torsion exponents of oriented flag manifolds. The article also discusses consequences of our results for a more general perspective on the relation between the torsion exponent and deficiency for homogeneous spaces.

Keywords

Bockstein and Steenrod squarescharacteristic rankintegral cohomology with local coefficientsoriented Grassmannianpartitions

MSC classification

Primary: 57T15: Homology and cohomology of homogeneous spaces of Lie groups
Secondary: 55U20: Universal coefficient theorems, Bockstein operator 14M15: Grassmannians, Schubert varieties, flag manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 43
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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