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Morse Functions and Cohomology of Homogeneous Spaces

  • Haibao Duan
Part of the Trends in Mathematics book series (TM)

Keywords

Homogeneous Space Morse Function Cell Decomposition Schubert Variety Multiplicative Rule 
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References

  1. [AB]
    M. Atiyah and R. Bott, On the periodicity theorem for complex vector bundles, Acta Mathematica, 112(1964), 229–247.Google Scholar
  2. [B]
    A. Borel, Sur la cohomologie des spaces fibré principaux et des spaces homogènes de groups de Lie compacts, Ann. of Math. (2) 57, (1953). 115–207.Google Scholar
  3. [BGG]
    I.N. Bernstein, I.M. Gel’fand and S.I. Gel’fand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys 28 (1973), 1–26.Google Scholar
  4. [Br]
    M. Brion, Lectures on the geometry of flag varieties, this volume.Google Scholar
  5. [BS1]
    R. Bott and H. Samelson, The cohomology ring of G/T, Nat. Acad. Sci. 41(7) (1955), 490–492.Google Scholar
  6. [BS2]
    R. Bott and H. Samelson, Application of the theory of Morse to symmetric spaces, Amer. J. Math., Vol. LXXX, no. 4 (1958), 964–1029.Google Scholar
  7. [Ch]
    C. Chevalley, Sur les Décompositions Cellulaires des Espaces G/B, in Algebraic groups and their generalizations: Classical methods, W. Haboush ed. Proc. Symp. in Pure Math. 56 (part 1) (1994), 1–26.Google Scholar
  8. [D]
    J. Dieudonné, A history of Algebraic and Differential Topology, 1900–1960, Boston, Basel, 1989.Google Scholar
  9. [D1]
    H. Duan, Morse functions on Grassmannian and Blow-ups of Schubert varieties, Research report 39, Institute of Mathematics and Department of Mathematics, Peking Univ., 1995.Google Scholar
  10. [D2]
    H. Duan, Morse functions on Stiefel manifolds Via Euclidean geometry, Research report 20, Institute of Mathematics and Department of Mathematics, Peking Univ., 1996.Google Scholar
  11. [D3]
    H. Duan, Some enumerative formulas on flag varieties, Communication in Algebra, 29(10) (2001), 4395–4419.CrossRefGoogle Scholar
  12. [D4]
    H. Duan, Multiplicative rule of Schubert classes, to appear in Invent. Math. (cf. arXiv: math. AG/ 0306227).Google Scholar
  13. [DZ1]
    H. Duan and Xuezhi Zhao, A program for multiplying Schubert classes, arXiv: math.AG/0309158.Google Scholar
  14. [DZ2]
    H. Duan and Xuezhi Zhao, Steenrod operations on Schubert classes, arXiv: math.AT/0306250.Google Scholar
  15. [Eh]
    C. Ehresmann, Sur la topologie de certain spaces homogènes, Ann. of Math. 35(1934), 396–443.MathSciNetGoogle Scholar
  16. [G1]
    G.Z. Giambelli, Risoluzione del problema degli spazi secanti, Mem. R. Accad. Sci. Torino (2)52(1902), 171–211.Google Scholar
  17. [G2]
    G.Z. Giambelli, Alcune proprietà delle funzioni simmetriche caratteristiche, Atti Torino 38(1903), 823–844.Google Scholar
  18. [Han]
    H.C. Hansen, On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274.Google Scholar
  19. [H]
    M. Hirsch, Differential Topology, GTM. No.33, Springer-Verlag, New York-Heidelberg, 1976.Google Scholar
  20. [HM]
    C.S. Hoo and M. Mahowald, Some homotopy groups of Stiefel manifolds, Bull.Amer. Math. Soc., 71(1965), 661–667.Google Scholar
  21. [HPT]
    W.Y. Hsiang, R. Palais and C.L. Terng, The topology of isoparametric submanifolds, J. Diff. Geom., Vol. 27 (1988), 423–460.Google Scholar
  22. [K]
    S. Kleiman, Problem 15. Rigorous foundation of the Schubert’s enumerative calculus, Proceedings of Symposia in Pure Math., 28 (1976), 445–482.Google Scholar
  23. [Ke]
    M.A. Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4(1960), 161–169.Google Scholar
  24. [La]
    A. Lascoux, Polynômes symétriques et coefficients d’intersection de cycles de Schubert. C. R. Acad. Sci. Paris Sér. A 279 (1974), 201–204.Google Scholar
  25. [L]
    L. Lesieur, Les problèmes d’intersections sur une variété de Grassmann, C. R. Acad. Sci. Paris, 225 (1947), 916–917.Google Scholar
  26. [Le]
    C. Lenart, The combinatorics of Steenrod operations on the cohomology of Grassmannians, Advances in Math. 136(1998), 251–283.CrossRefGoogle Scholar
  27. [Ma]
    I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford University Press, Oxford, second ed., 1995.Google Scholar
  28. [M]
    C. Miller, The topology of rotation groups, Ann. of Math., 57(1953), 95–110.Google Scholar
  29. [M1]
    J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965.Google Scholar
  30. [M2]
    J. Milnor, Morse Theory, Princeton University Press, 1963.Google Scholar
  31. [M3]
    J. Milnor, Differentiable structures on spheres, Amer. J. Math., 81(1959), 962–972.Google Scholar
  32. [MS]
    J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies 76, Princeton Univ. Press, 1975.Google Scholar
  33. [P]
    P. Pragacz, Algebro-geometric applications of Schur S-and Q-polynomials, Topics in invariant Theory (M.-P. Malliavin, ed.), Lecture Notes in Math., Vol. 1478, Springer-Verlag, Berlin and New York, 1991, 130–191.Google Scholar
  34. [Sch]
    H. Schubert, Kalkül der abzählenden Geometrie, Teubner, Leipzig, 1879.Google Scholar
  35. [S]
    R.P. Stanley, Some combinatorial aspects of Schubert calculus, Springer Lecture Notes in Math. 1353 (1977), 217–251.Google Scholar
  36. [St]
    N.E. Steenrod and D.B.A. Epstein, Cohomology Operations, Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 1962.Google Scholar
  37. [VD]
    A.P. Veselov and I.A. Dynnikov, Integrable Gradient flows and Morse Theory, Algebra i Analiz, Vol. 8, no 3.(1996), 78–103; Translated in St. Petersburgh Math. J., Vol. 8, no 3.(1997), 429–446.Google Scholar
  38. [Wh]
    J.H.C. Whitehead, On the groups πr(Vn,m), Proc. London Math. Soc., 48(1944), 243–291.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Haibao Duan
    • 1
  1. 1.Institute of MathematicsChinese Academy of SciencesBeijing

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