Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-15T12:55:19.700Z Has data issue: false hasContentIssue false
  • English
  • Français

4-Torsion classes in the integral cohomology of oriented Grassmannians

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

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])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M. F. and Hirzebruch, F.. Cohomologie-Operationen und charakteristische Klassen. Math. Z. 77 (1961), 149–187.Google Scholar
Ayyer, A., Prasad, A. and Spallone, S.. Odd partitions in Young’s lattice. Sém. Lothar. Combin. 75 (2016), .Google Scholar
Basu, S. and Chakraborty, P.. On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes. J. Homotopy Relat. Struct. 15 (2020), 27–60.Google Scholar
Baum, P. F.. On the cohomology of homogeneous spaces. Topology 7 (1968), 15–38.Google Scholar
Borel, A.. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2) 57 (1953), 115–207.Google Scholar
Borel, A. and Haefliger, A.. La classe d’homologie fondamentale d’un espace analytique. Bull. Soc. Math. France 89 (1961), 461–513.Google Scholar
Brown, E. H. J.. The cohomology of $B{{\rm} SO} {n}$ and $B{{\rm} O} {n}$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), 283–288.Google Scholar
Čadek, M.. The cohomology of ${{\rm} BO}(n)$ with twisted integer coefficients. J. Math. Kyoto Univ. 39 (1999), 277–286.Google Scholar
Carlson., J. D. A ring structure on Tor. arXiv:2306.04860.Google Scholar
Cartan, H.. La transgression dans un groupe de Lie et dans un espace fibré principal, In Colloque de Topologie (Espace fibrés) Bruxelles 1950, 57–71 (Georges Thone, Liège, 1951).Google Scholar
Casian, L. and Kodama, Y.. On the cohomology of real Grassmann manifolds . (2013), arXiv preprint:1309.5520.Google Scholar
Ehresmann, C.. Sur la topologie de certaines variétés algébriques réelles. Journal de Mathématiques Pures et Appliquées (1) 16 (1937), , 9e série.Google Scholar
Fei Lai, H.. On the topology of the even-dimensional complex quadrics. Proc. Amer. Math. Soc. 46 (1974), 419–425.Google Scholar
Franz, M.. The cohomology rings of homogeneous spaces. J. Topol. 14 (2021), 1396–1447.Google Scholar
Fukaya, T.. Gröbner bases of oriented Grassmann manifolds. Homology Homotopy Appl. 10 (2008), 195–209.Google Scholar
Fulton, W.. With applications to representation theory and geometry, Of London Mathematical Society Student Texts, Vol. 35 (Cambridge University Press, Cambridge, 1997).Google Scholar
Hennig, J.. A cellular Milnor-Witt (co)homology computation for the moduli space of stable, genus 0 curves with marked points, submitted September 2024.Google Scholar
Hornbostel, J., Wendt, M., Xie, H. and Zibrowius, M.. The real cycle class map. Ann. K-Theory 6 (2021), 239–317.Google Scholar
Hudson, T., Matszangosz, Á. K. and Wendt, M.. Chow–Witt rings and topology of flag varieties. J. Topol. 17 (2024), arXiv:2302.11003.Google Scholar
Husemoller, D., Moore, J. C. and Stasheff, J.. Differential homological algebra and homogeneous spaces. J. Pure Appl. Algebra 5 (1974), 113–185.Google Scholar
James, G. and Kerber, A.. The representation theory of the symmetric group, Of Encyclopedia of Mathematics and its Applications, Vol. 16 (Addison-Wesley Publishing Co, Reading, MA, 1981) With a foreword by P. M. Cohn, With an introduction by Gilbert de B. RobinsonGoogle Scholar
Jacobson, J. A.. Real cohomology and the powers of the fundamental ideal in the Witt ring. Ann. K-Theory 2 (2017), 357–385.Google Scholar
Jovanović, M.. On integral cohomology algebra of some oriented Grassmann manifolds. Indag. Math. (N.S.) 35 (2024), 1–13.Google Scholar
Korbaš, J.. The characteristic rank and cup-length in oriented Grassmann manifolds. Osaka J. Math. 52 (2015), 1163–1172.Google Scholar
Lenart, C.. The combinatorics of Steenrod operations on the cohomology of Grassmannians. Adv. Math. 136 (1998), 251–283.Google Scholar
Matszangosz, Á. K.. On the cohomology rings of real flag manifolds: Schubert cycles. Math. Ann. 381 (2021), 1537–1588.Google Scholar
Matszangosz, Á. K. and Wendt, M.. The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes. Math. Z. 308 (2024), arXiv:2310.11129.Google Scholar
McCrory, C. and Parusiński, A.. The weight filtration for real algebraic varieties, Topology of stratified spaces, 121–160, Math. Sci. Res. Inst. Publ., Vol. 58 (Cambridge University Press, Cambridge, 2011).Google Scholar
Munkholm, H. J.. The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps. J. Pure Appl. Algebra. 5 (1974), 1–50.Google Scholar
Olsson, J. B.. Combinatorics and representations of finite groups, Volume 20 of Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen [Lecture Notes in Mathematics at the University of Essen], Vol. 20 (Universität Essen, Fachbereich Mathematik, Essen, 1993).Google Scholar
Prvulović, B. I. and Radovanović, M.. On the characteristic rank of vector bundles over oriented Grassmannians. Fund. Math. 244 (2019), 167–190.Google Scholar
Quillen, D.. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2) 94 (1971), 573–602, ibid.Google Scholar
Stong, R. E.. Cup products in Grassmannians. Topology Appl. 13 (1982), 103–113.Google Scholar
Scheiderer, C.. Real and étale cohomology, Lecture Notes in Mathematics 1588 (Springer-Verlag, Berlin, 1994).Google Scholar
Wendt, M.. Oriented Schubert calculus in Chow–Witt rings of Grassmannians. In: Motivic homotopy theory and refined enumerative geometry. Amer. Math. Soc. 745 (2020), 217–267, Contemp. Math.Google Scholar
Wendt, M.. Chow–Witt rings of Grassmannians. Algebr. Geom. Topol. 24 (2024), 1–48.Google Scholar
Wolf, J.. The cohomology of homogeneous spaces. Amer. J. Math. 99 (1977), 312–340.Google Scholar
Yang, N.. Split Milnor-Witt motives and its applications to fiber bundles. Camb. J. Math. 10 (2022), 935–1004.Google Scholar