Acta Mathematica Sinica, English Series

, Volume 35, Issue 4, pp 445–462 | Cite as

Suspension Splittings and Self-maps of Flag Manifolds

  • Shizuo KajiEmail author
  • Stephen TheriaultEmail author


If G is a compact connected Lie group and T is a maximal torus, we give a wedge decomposition of ΣG/T by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of G/T.


Flag manifold self-map stable splitting 

MR(2010) Subject Classification

55P40 55S37 57T15 


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  1. [1]
    Bergeron, F., Bergeron, N., Howlett, R. B., et al.: A decomposition of the descent algebra of a finite Coxeter group. J. Algebraic Combin., 1, 23–44 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bernstein, I. N., Gelfand, I. M., Gelfand, S. I.: Schubert cells and the cohomology of the spaces G/P, LMS. Cambridge Univ. Press, 69, 115–140 (1982)Google Scholar
  3. [3]
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math., 57, 115–207 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Borel, A.: Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes. Tôhoku Math. J. (2). 13, 216–240 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Bott, R., Samelson, H.: The integral cohomology ring of G/T. Proc. Nat. Acad. Sci. USA, 41, 490–493 (1955)CrossRefzbMATHGoogle Scholar
  6. [6]
    Chevalley, C.: Sur les décomposition cellulaires des espaces G/B, Algebraic Groups and their Generalizations: Classical Methods (W. Haboush, ed.), Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., 1–23, 1994Google Scholar
  7. [7]
    Duan, H., Zhao, X. A.: The classification of cohomology endomorphisms of certain flag manifolds. Pacific J. Math., 192, 93–102 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Duan, H., Zhao, X. A.: A unified formula for Steenrod operations in flag manifolds. Compositio Mathematica, 143, 257–270 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Glover, H., Homer, W.: Self-maps of flag manifolds. Trans. Amer. Math. Soc., 267, 423–434 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Hilton, P., Mislin, G., Roitberg, J.: Localization of Nilpotent Groups and Spaces, Math. Studies No. 15, North-Holland, Amsterdam, 1975zbMATHGoogle Scholar
  11. [11]
    Kaji, S.: Three presentations of torus equivariant cohomology of flag manifolds, in Proceedings of International Mathematics Conference in honour of the 70th Birthday of Professor S. A. Ilori, Scholar
  12. [12]
    Kitchloo, N.: Cohomology splittings of Stiefel manifolds. J. London Math. Soc. (2), 64(2), 457–471 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Lascoux, A., Schützenberger, M.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math., 294(13), 447–450 (1982)MathSciNetzbMATHGoogle Scholar
  14. [14]
    May, J. P., Ponto, K.: More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012Google Scholar
  15. [15]
    Miller, H.: Stable splitting of Stiefel manifolds. Topology, 24(4), 411–419 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Mimura, M.: The homotopy groups of Lie groups of low rank. J. Math. Kyoto Univ., 6, 131–176 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Mimura, M., Toda, H.: Homotopy groups of SU(3), SU(4) and Sp(2). J. Math. Kyoto Univ., 3, 217–250 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Nishida, G., Yang, Y.: On a p-local stable splitting of U(n). J. Math. Kyoto Univ., 41(2), 387–401 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Papadima, S.: Rigidity properties of compact Lie groups modulo maximal tori. Math. Ann., 275, 637–652 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Priddy, S.: Recent progress on stable splittings. Proc. Durham Symp. Homotopy Theory, 1985, LMS, 117, 149–174, (1987)Google Scholar
  21. [21]
    Stembridge, J. R.: Orthogonal sets of Young symmetrizers. Adv. Appl. Math., 46, 576–582 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Toda, H.: Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton University Press, Princeton 1962zbMATHGoogle Scholar
  23. [23]
    Toda, H., Watanabe, T.: The integral cohomology ring of F4/T and E6/T. J. Math. Kyoto Univ., 14, 257–286 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Ullman, H. E.: An equivariant generalization of the Miller splitting theorem. Algebr. Geom. Topol., 12(2), 643–684 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Yang, Y.: On a p-local stable splitting of Stiefel manifolds. J. Math. Soc. Japan, 54, 911–921 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Zhao, X. A.: Cohomology endomorphisms of flag manifolds. Acta Math. Sinica, 44, 1099–1106 (2001)MathSciNetzbMATHGoogle Scholar
  27. [27]
    Zhao, X. A.: Maps from a simply connected space to a flag manifold G/T. Acta. Math. Sinica, 20, 1131–1134 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  2. 2.Mathematical SciencesUniversity of SouthamptonSouthamptonUnited Kingdom

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