Segal’s Burnside Ring Conjecture for Compact Lie Groups

  • Chun-Nip Lee
  • Norihiko Minami
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 27)


What is Segal’s Burnside ring conjecture for compact Lie groups? In order to explain it, we will have to go back thirty years when Michael Atiyah proved the following remarkable result in topological K-theory.


Finite Group Closed Subgroup Finite Subgroup Stable Homotopy London Mathematical Society Lecture Note 
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  1. [A1]
    J.F. Adams, On the groups, J(X) − I, Topology 2 (1963), 181–195.CrossRefMathSciNetGoogle Scholar
  2. [A2]
    J.F. Adams, Stable homotopy and generalized homology, The University of Chicago Press, Chicago 1974.Google Scholar
  3. [A3]
    J.F. Adams, Maps between classifying spaces II, Inv. Math. 49 (1978), 1–65.CrossRefMATHGoogle Scholar
  4. [AGM]
    J.F. Adams, J.H. Gunawardena, and H.R. Miller, The Segal conjecture for elementary p-groups, Topology 24 (1985), 435–460.CrossRefMATHMathSciNetGoogle Scholar
  5. [AHJM]
    J.F. Adams, J.-P. Haeberly, S. Jackowski, and J.P. May, A generalization of the Segal conjecture, Topology 27 (1988), 7–21.CrossRefMATHMathSciNetGoogle Scholar
  6. [At]
    M.F. Atiyah, Characters and cohomology of finite groups, Publ. Math. I. H. E. S. 9 (1961), 23–64.MathSciNetGoogle Scholar
  7. [AS]
    M.F. Atiyah and G.B. Segal, Equivariant K-theory and completion, J. Diff. Geom. 3 (1969), 1–18.MATHMathSciNetGoogle Scholar
  8. [B]
    S. Bauer, On the Segal conjecture for compact Lie groups, J. reine angew. Math. 400 (1989), 134–145.MATHMathSciNetGoogle Scholar
  9. [BG]
    J. Becker and D. Gottlieb, The transfer and fiber bundles, Topology 14 (1975), 1–13.CrossRefMATHMathSciNetGoogle Scholar
  10. [BF]
    D. Benson and M. Feshbach, Stable splittings of classifying spaces of finite groups,Topology (to appear).Google Scholar
  11. [BHM]
    M. Bökstedt, W.C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces,Aarhus Universitet preprint series 14 (1989/90).Google Scholar
  12. [BK]
    A. Bousfield and D. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Math. # 304, Springer-Verlag, New York 1972.CrossRefGoogle Scholar
  13. [Br]
    G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972.MATHGoogle Scholar
  14. [BM]
    G. Brumfiel and I. Madsen, Evaluation of the transfer and the universal surgery class, Inventiones Math. 32 (1976), 133–169.CrossRefMATHMathSciNetGoogle Scholar
  15. [C1]
    Carlsson, G., Equivariant stable homotopy and Segal’s Burnside ring conjecture, Annals of Mathematics 120 (1984), 189–224.CrossRefMATHMathSciNetGoogle Scholar
  16. [C2]
    G. Carlsson, Homotopy Theory, edited by E. Rees and J.D.S. Jones, London Mathematical Society Lecture Note Series 117, Cambridge University Press, 1987.Google Scholar
  17. [C3]
    G. Carlsson, Advances in Homotopy Theory, edited by S. M. Salamon, B. Steer and W.A. Sutherland, London Mathematical Society Lecture Note Series 139, Cambridge University Press, 1989.Google Scholar
  18. [C4]
    G. Carlsson, On the homotopy fixed point problem for free loop space and other function complexes,to appear.Google Scholar
  19. [Co]
    R.L. Cohen, The immersion conjecture for di f ferentiable manifolds, Annals of Math. 122 (1985), 237–328.CrossRefMATHGoogle Scholar
  20. [CR]
    C. Curtis and I. Reiner, Methods of Representation theory Vol.1, Wiley, New York, 1981.Google Scholar
  21. [tD1]
    T. tom Dieck, Orbittypen und äquivariante Homologie II,Arch. Math. 26 (1975), 650–662.CrossRefMATHGoogle Scholar
  22. [tD2]
    T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. # 766, Springer-Verlag, New York 1979.Google Scholar
  23. [tDP]
    T. tom Dieck and T. Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 (1978), 273–287.CrossRefMATHMathSciNetGoogle Scholar
  24. [F1]
    M. Feshbach, The transfer and compact Lie groups, Trans. Amer Math. 251 (1979), 139–169.CrossRefMathSciNetGoogle Scholar
  25. [F2]
    M. Feshbach, The Segal conjecture for compact Lie group, Topology 26 (1987), 1–20.CrossRefMATHMathSciNetGoogle Scholar
  26. [F3]
    M. Feshbach, Essential maps exist from BU to CokerJ, Proc. A. M. S. (1986), 539–543.Google Scholar
  27. [Fr]
    E.Friedlander, Fibrations in etale homotopy theory, Publ. Math. I. H. E. S. 42 (1972), 5–46.MATHGoogle Scholar
  28. [F-P]
    Z. Fiedorowicz and S. Priddy, Homology of classical groups over finite fields and their assiciated infinite loop spaces,Lecture Notes in Math. # 674, Springer-Verlag, New York.Google Scholar
  29. [FM 1]
    E.Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helvetici 59 (1984), 347–361.CrossRefMATHMathSciNetGoogle Scholar
  30. [FM 2]
    E. Friedlander and G. Mislin, Locally finite approximation of Lie groups I, Invent. math. 83 (1986), 425–436.CrossRefMATHMathSciNetGoogle Scholar
  31. [H]
    H. Hauchild, Zerspaltung aquivarianter Homotopiemengen, Math. Annalen 230 (1977), 279–292.CrossRefGoogle Scholar
  32. [La]
    Laitinen, On the Burnside ring and stable cohomotopy of a finite group,Math. Scand. 44 (1979), 37–72.CrossRefGoogle Scholar
  33. [L1]
    C-N. Lee, Stable splittings of the dual spectrum of the classifying space of a compact Lie group,to appear in Transaction A.M.S.Google Scholar
  34. [L2]
    C-N. Lee, On the homotopy inverse limit of transfer maps, Math. Zeit. 205 (1990), 97–103.CrossRefMATHGoogle Scholar
  35. [LMM]
    L. Lewis, J. May, and J. McClure, Classifying G-spaces and the Segal conjecture, Canadian Mathematical Society Conference Proceedings Volume 2, Part 2, American Math. Soc., Rhode Island, 1982, pp. 165–179.Google Scholar
  36. [LMS]
    L. Lewis, J. May, and M. Steinberger, Equivariant Stable Homotopy Theory, Lecture Notes in Math. # 1213, Springer-Verlag, New York 1986.Google Scholar
  37. [MMM]
    B. Mann, E.Miller, and H. Miller, S 1 -Equivariant function spaces and characteristic classes, Trans. of the Amer. Math. Soc. 295 (1986), 233–256.MATHMathSciNetGoogle Scholar
  38. [MaP]
    J. Martino and S. Priddy, The complete stable splitting for the classifying space of a finite group,Topology (to appear).Google Scholar
  39. [Ma1]
    J.P. May, The completion conjecture in equivariant cohomology, Springer Lecture Notes in Mathematics 1051 (1983), 620–637.CrossRefGoogle Scholar
  40. [Ma2]
    J.P. May, Contemp. Math., vol. 37, Amer. Math. Soc., 1983.Google Scholar
  41. [MM]
    J.P. May and J.E. McClure, A reduction of the Segal conjecture, Canadian Mathematical Society Conference Proceedings Volume 2, Part 2, American Math. Soc., Rhode Island, 1982, pp. 209–222.Google Scholar
  42. [MSZ]
    J.P. May, V.P. Snaith, and P. Zelewski, A further generalization of the Segal conjecture, Q. J. Math. 40 (1989), 457–473.CrossRefMATHMathSciNetGoogle Scholar
  43. [Mc]
    J.E. McClure, Restriction maps in equivariant K-theory, Topology 25 (1986), 399–409.CrossRefMATHMathSciNetGoogle Scholar
  44. [MW]
    H. Miller and C. Wilkerson, On the Segal conjecture for Periodic Groups, Contempory Mathematics, vol. 19, American Math. Soc., Rhode Island, 1983, pp. 233–246.Google Scholar
  45. [M1]
    N. Minami, On the I(G)-adic topology of the Burnside rings of compact Lie groups, Publ. Res. Inst. Math. Sci. 20 (1984), 447–460.CrossRefMATHMathSciNetGoogle Scholar
  46. [M2]
    N. Minami, Group homomorphisms inducing an isomorphism of a functor, Math. Proc. Cambridge Phil. Soc. 104 (1988), 81–93.CrossRefMATHMathSciNetGoogle Scholar
  47. [M3]
    N. Minami, On the classifying spaces of SL3(7G), St3(Z) and finite Chevalley groups, MSRI preprint series 07025–90 (1990).Google Scholar
  48. [M4]
    N. Minami, The relative Burnside ring and stable maps between classifying spaces of compact Lie groups.Google Scholar
  49. [Mi]
    S. Mitchell, Advances in Homotopy Theory, edited by S. M. Salamon, B. Steer and W. A. Sutherland, London Mathematical Society Lecture Note Series 139,Cambridge University Press, 1989.Google Scholar
  50. [MP1]
    S. Mitchell and S. Priddy, Symmetric product spectra and splittings of classifying spaces, American J. Math. 106 (1984), 219–232.CrossRefMATHMathSciNetGoogle Scholar
  51. [MP2]
    S. Mitchell and S. Priddy, A double coset formula for Levi subgroups and splitting BGL n,Springer Lecture Notes in Mathematics 1370 (1989), 325–334.CrossRefMathSciNetGoogle Scholar
  52. [N1]
    G. Nishida, On the S l -Segal conjecture, Publ. Res. Inst. Math. Sci. 19 (1983), 1153–1162.CrossRefMATHMathSciNetGoogle Scholar
  53. [N2]
    G. Nishida, On stable p-equivalence of classifying spaces of compact Lie groups (1983).Google Scholar
  54. [Q]
    D. Quillen, The Adams conjecture, Topology 10 (1970), 67–80.CrossRefMathSciNetGoogle Scholar
  55. [R]
    D. Ravenel, The Segal conjecture for cyclic groups and its consequences, American J. Math. 106 (1984), 415–446.CrossRefGoogle Scholar
  56. [S]
    G. Segal, in International Congress of Mathematics at Nice.Google Scholar
  57. [S1]
    V.P. Snaith, Algebraic cobordism and K-theory, Memoir of Amer. Math. Soc. 221 (1979).Google Scholar
  58. [S2]
    V.P. Snaith, Topological Methods in Galois Representation Theory, Wiley Interscience, New-York, 1989.MATHGoogle Scholar
  59. [Su]
    D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math. 100 (1974), 1–79.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Chun-Nip Lee
    • 1
    • 2
  • Norihiko Minami
    • 1
    • 2
  1. 1.Dept. of Math.Northwestern UniversityWanstonUSA
  2. 2.Dept. of Math.University of AlabamaTuscaloosaUSA

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