Proceedings of the Japan Academy, Series A, Mathematical Sciences

Chern classes and generators

Masaki Kameko

Full-text: Open access

Abstract

We give a simple proof for the fact that certain algebra generators of the $\mathrm{mod}\,2$ cohomology of classifying spaces of exceptional Lie groups are given by Chern classes and Stiefel-Whitney classes of certain representations.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 1 (2012), 21-23.

Dates
First available in Project Euclid: 30 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1325264392

Digital Object Identifier
doi:10.3792/pjaa.88.21

Mathematical Reviews number (MathSciNet)
MR2872211

Zentralblatt MATH identifier
1238.55007

Subjects
Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]

Keywords
Classifying space exceptional Lie group characteristic class

Citation

Kameko, Masaki. Chern classes and generators. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 1, 21--23. doi:10.3792/pjaa.88.21. https://projecteuclid.org/euclid.pja/1325264392


Export citation

References

  • J. F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1996.
  • M. Kameko and N. Yagita, Chern subrings, Proc. Amer. Math. Soc. 138 (2010), no. 1, 367–373.
  • A. Kono and M. Mimura, Cohomology $\mathrm{mod}\,2$ of the classifying space of the compact connected Lie group of type $E_{6}$, J. Pure Appl. Algebra 6 (1975), 61–81.
  • A. Kono, M. Mimura and N. Shimada, On the cohomology $\mathrm{mod}\,2$ of the classifying space of the 1 connected exceptional Lie group $E_{7}$, J. Pure Appl. Algebra 8 (1976), no. 3, 267–283.
  • A. Kono, A note on the Stiefel-Whitney classes of representations of exceptional Lie groups, J. Math. Kyoto Univ. 45 (2005), no. 1, 217–219.
  • M. Mimura and T. Nishimoto, On the Stiefel-Whitney classes of the representations associated with $\mathrm{Spin}(15)$, in Proceedings of the School and Conference in Algebraic Topology, Geom. Topol. Monogr., 11, Geom. Topol. Publ., Coventry, 2007, pp. 141–176.
  • D. Quillen, The $\mathrm{mod}\,2$ cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197–212.
  • B. Schuster and N. Yagita, Transfers of Chern classes in BP-cohomology and Chow rings, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1039–1054 (electronic).
  • H. Toda, Cohomology of the classifying space of exceptional Lie groups, in Manifolds–-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), 265–271, Univ. Tokyo Press, Tokyo, 1975.