Kyoto Journal of Mathematics

KO-theory of exceptional flag manifolds

Daisuke Kishimoto and Akihiro Ohsita

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Abstract

The KO-theory of the flag manifold G/T is determined by calculating the Atiyah–Hirzebruch spectral sequence when G is one of the exceptional Lie groups G2, F4, E6, where T is a maximal torus of G.

Article information

Source
Kyoto J. Math., Volume 53, Number 3 (2013), 673-692.

Dates
First available in Project Euclid: 19 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1376917629

Digital Object Identifier
doi:10.1215/21562261-2265923

Mathematical Reviews number (MathSciNet)
MR3102565

Zentralblatt MATH identifier
1277.55002

Subjects
Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 55T25: Generalized cohomology

Citation

Kishimoto, Daisuke; Ohsita, Akihiro. $KO$ -theory of exceptional flag manifolds. Kyoto J. Math. 53 (2013), no. 3, 673--692. doi:10.1215/21562261-2265923. https://projecteuclid.org/euclid.kjm/1376917629


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References

  • [A] M. F. Atiyah, $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386.
  • [F] M. Fujii, $K_{O}$-groups of projective spaces, Osaka J. Math. 4 (1967), 141–149.
  • [IT] K. Ishitoya and H. Toda, On the cohomology of irreducible symmetric spaces of exceptional type, J. Math. Kyoto Univ. 17 (1977), 225–243.
  • [K] D. Kishimoto, $KO$-theory of complex Stiefel manifolds, J. Math. Kyoto Univ. 44 (2004), 669–674.
  • [KKO] D. Kishimoto, A. Kono, and A. Ohsita, $K\mathrm{O}$-theory of flag manifolds, J. Math. Kyoto Univ. 44 (2004), 217–227.
  • [KH1] A. Kono and S. Hara, $K\mathrm{O}$-theory of complex Grassmannians, J. Math. Kyoto Univ. 31 (1991), 827–833.
  • [KH2] A. Kono and S. Hara, $K\mathrm{O}$-theory of Hermitian symmetric spaces, Hokkaido Math. J. 21 (1992), 103–116.
  • [KI1] A. Kono and K. Ishitoya, Squaring operations in the $4$-connective fibre spaces over the classifying spaces of the exceptional Lie groups, Publ. Res. Inst. Math. Sci. 21 (1985), 1299–1310.
  • [KI2] A. Kono and K. Ishitoya, “Squaring operations in $\operatorname{mod}2$ cohomology of quotients of compact Lie groups by maximal tori” in Algebraic Topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, Berlin, 1987, 192–206.
  • [MT] M. Mimura and H. Toda, Topology of Lie Groups, I, II, Trans. Math. Monogr. 91, Amer. Math. Soc., Providence, 1991.
  • [T] H. Toda, On the cohomology ring of some homogeneous spaces, J. Math. Kyoto Univ. 15 (1975), 185–199.
  • [TW] H. Toda and T. Watanabe, The integral cohomology ring of $\textbf{F}_{4}/\textbf{T}$ and $\textbf{E}_{6}/\textbf{T}$, J. Math. Kyoto Univ. 14 (1974), 257–286.
  • [Y1] N. Yagita, A note on the Witt group and the $KO$-theory of complex Grassmannians, J. $K$-theory 9 (2012), 161–175.
  • [Y2] N. Yagita, Witt groups of algebraic groups, preprint.
  • [Z] M. Zibrowius, Witt groups of complex cellular varieties, Doc. Math. 16 (2011), 465–511.