Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T05:19:11.941Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-torsion non-algebraic classes in the Brown–Peterson tower

Part of: Fiber spaces and bundles Cycles and subschemes

Published online by Cambridge University Press:  16 October 2020

MASAKI KAMEKO
MASAKI KAMEKO*
Affiliation:
Department of Mathematical Sciences, College of Systems Engineering and Science, Shibaura Institute of Technology, Saitama-City337-8570, Japan. e-mail: kameko@shibaura-it.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalising the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown–Peterson tower. We construct non-torsion non-algebraic elements in the Brown–Peterson tower for the prime number 2.

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 55R35: Classifying spaces of groups and $H$-spaces 55R40: Homology of classifying spaces, characteristic classes
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 113 - 132
Copyright
© Cambridge Philosophical Society 2020

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.)

Footnotes

†

This work was supported by JSPS KAKENHI Grant Numbers JP25400097 and JP17K05263.

References

Atiyah, M. F. and Hirzebruch, F.. Analytic cycles on complex manifolds. Topology 1 (1962), 25–45.10.1016/0040-9383(62)90094-0CrossRefGoogle Scholar
Antieau, B.. On the integral Tate conjecture for finite fields and representation theory. Algebr. Geom. 3 (2016), no. 2, 138–149.CrossRefGoogle Scholar
Colliot Thélène, J.-L. and Szamuely, T.. Autour de la conjecture de Tate à coefficients Zl pour les variétés sur les corps finis. The geometry of algebraic cycles. Clay Math. Proc., vol. 9 (Amer. Math. Soc., Providence, RI, 2010), pp. 83–98 (French).Google Scholar
Kameko, M.. On the integral Tate conjecture over finite fields. Math. Proc. Camb. Phil. Soc. 158 (2015), no. 3, 531–546.Google Scholar
Kono, A. and Yagita, N.. Brown–Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups. Trans. Amer. Math. Soc. 339 (1993), no. 2, 781–798.10.1090/S0002-9947-1993-1139493-4CrossRefGoogle Scholar
Landweber, P. S.. Elements of infinite filtration in complex cobordism. Math. Scand. 30 (1972), 223–226.CrossRefGoogle Scholar
Pirutka, A. and Yagita, N.. Note on the counterexamples for the integral Tate conjecture over finite fields. Doc. Math. Extra vol: Alexander S. Merkurjev’s sixtieth birthday (2015), 501–511.Google Scholar
Quick, G.. Examples of non-algebraic classes in the Brown–Peterson tower. Math. Z. 293 (2019), no. 1-2, 25–37.10.1007/s00209-018-2164-4CrossRefGoogle Scholar
Soule, C. and Voisin, C.. Torsion cohomology classes and algebraic cycles on complex projective manifolds. Adv. Math. 198 (2005), no. 1, 107–127.CrossRefGoogle Scholar
Totaro, B.. Torsion algebraic cycles and complex cobordism. J. Amer. Math. Soc. 10 (1997), no. 2, 467–493.10.1090/S0894-0347-97-00232-4CrossRefGoogle Scholar
Yagita, N.. Applications of Atiyah–Hirzebruch spectral sequences for motivic cobordism. Proc. London Math. Soc. (3) 90 (2005), no. 3, 783–816.10.1112/S0024611504015084CrossRefGoogle Scholar