Morava K-theories: A survey

  • Urs Würgler
Survey Articles
Part of the Lecture Notes in Mathematics book series (LNM, volume 1474)

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References

  1. [1]
    Adams, J.F.: Stable homotopy and generalised homology, Univ. of Chicago press, Chicago, Illinois and London (1974).MATHGoogle Scholar
  2. [2]
    Araki, S.: Typical formal groups in complex cobordism and K-theory, Lecture Notes in Math., Kyoto Univ. 6, Kinokuniya Book Store, 1973.Google Scholar
  3. [3]
    Baas, N.A.: On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302.MathSciNetMATHGoogle Scholar
  4. [4]
    Baas, N.A. and Madsen, I.: On the realization of certain modules over the Steenrod algebra, Math. Scand 31 (1972), 220–224.MathSciNetMATHGoogle Scholar
  5. [5]
    Baker, A.: Some families of operations in Morava K-theory, Amer. J. Math. 111(1989),95–109.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    A -structures on some spectra related to Morava K-theories, preprint Manchester Univ., (1988).Google Scholar
  7. [7]
    Baker, A. and Würgler, U.: Liftings of formal groups and the Artinian completion of v n−1 BP, Math. Proc. Camb. Phil. Soc. 106 (1989),511–530.CrossRefMATHGoogle Scholar
  8. [8]
    —:Bockstein operations in Morava K-theory, preprint 1989.Google Scholar
  9. [9]
    Brown, E.H. and Peterson, F.P.: A spectrum whose Zp-cohomology is the algebra of reduced p-th powers, Topology 5 (1966), 149–154.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Cartier, P.: Modules associés à un groupe formel commutatif, courbes typiques, C. R. Acad. Sci. Paris Série A 265(1965), 129–132.MathSciNetMATHGoogle Scholar
  11. [11]
    Devinatz, E.S., Hopkins, M.J. and Smith, J.H.: Nilpotence and stable homotopy I, Ann. Math. 128(1988), 207–241.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Dold, A.: Chern classes in general cohomology. Symp. Math. V(1970),385–410.MathSciNetGoogle Scholar
  13. [13]
    Fröhlich,A.: Formal groups, Lecture Notes in Math. 74(1968).Google Scholar
  14. [14]
    Hazewinkel M.: Formal groups and applications. Academic press, 1978.Google Scholar
  15. [15]
    Hopkins, J.R.: Global methods in homotopy theory, Proc. Durham Symp. 1985, Cambridge Univ. Press (1987), 73–96.Google Scholar
  16. [16]
    Hunton, J.: The Morava K-theories of wreath products, Preprint Cambridge Univ. (1989).Google Scholar
  17. [17]
    —: Ph.D. Thesis, Cambridge Univ. (1989).Google Scholar
  18. [18]
    Johnson, D.C. and Wilson, W.S.: BP-operations and Morava's extraordinary K-theories, Math.Z. 144(1975),55–75.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    —: The Brown-Peterson homology of elementary p-groups, Amer. J. Math. 107(1984), 427–453.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Kane, R.M.: Implications in Morava K-theory, Mem. Amer. Math. Soc. 59 (1986), No.340.Google Scholar
  21. [21]
    Knus M.,and Ojanguren, M.: Théorie de la descente et algébres d'Azumaya. Lecture Notes in Mathematics 389, 1974.Google Scholar
  22. [22]
    Kuhn, N.J.: Morava K-theories and infinite loop spaces, Springer Lect. Notes in Math. 1370(1989),243–257.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    —: The Morava K-theories of some classifying spaces, TAMS 304(1987),193–205.MathSciNetMATHGoogle Scholar
  24. [24]
    —: Character rings in algebraic topology, London Math. Soc. Lecture Notes 139 (1989), 111–126.MathSciNetMATHGoogle Scholar
  25. [25]
    Kultze,R.: Die Postnikov-Faktoren von k(n), Manuskript, Universität Frankfurt (1989).Google Scholar
  26. [26]
    Kultze, R. and Würgler, U.: A Note on the algebra P(n)*(P(n)) for the prime 2, Manuscripta Math. 57(1987), 195–203.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    —: The algebra k(n)*(k(n)) for the prime 2, Arch. Math. 51(1988),141–146.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Landweber, P.S.: BP *(BP) and typical formal groups, Osaka J. Math. 12(1975),357–363.MathSciNetMATHGoogle Scholar
  29. [29]
    —: Homological properties of comodules over MU *(MU) and BP *(BP), Amer. J. Math. 98(1976),591–610.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    Lazard, M.: Sur les groupes de Lie formels á un paramètre, Bull. Soc. Math. France 83, 251–274.Google Scholar
  31. [31]
    Lellmann, W.: Connected Morava K-theories, Math. Z. 179 (1982), 387–399.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    Miller, H.R. and Ravenel, D.C.: Morava stabilizer algebras and the localization of Novikov's E 2-term, Duke Math. J. 44(1977), 433–447.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    Miller, H.R., Ravenel, D.C. and Wilson, W.S.: Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2)106 (1977), 459–516.MathSciNetMATHGoogle Scholar
  34. [34]
    Mischenko: Appendix 1 in Novikov [41].Google Scholar
  35. [35]
    Mitchell, S.A.: Finite complexes with A(n)-free cohomology, Topology 24(1985), 227–248.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    Morava, J.:A product for odd-primary bordism of manifolds with singularities, Topology 18(1979), 177–186.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    —, Completions of complex cobordism, Lecture Notes in Math. 658(1978),349–361.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    —,Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121(1985), 1–39.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    —,Forms of K-theory, Math. Z. 201(1989),401–428.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    Mironov, O.K.: Existence of multiplicative structures in the theory of cobordism with singularities, Izv. Akad. Nauk SSSR Ser. Mat. 39(1975),No.5, 1065–1092.MathSciNetGoogle Scholar
  41. [41]
    Novikov,S.P.: The methods of algebraic topology from the viewpoint of complex cobordism theories, Math. USSR Izv. (1967), 827–913.Google Scholar
  42. [42]
    Pazhitmov, A.V.: Uniqueness theorems for generalized cohomology theories, Math. USSR Izvestiyah 22(1984),483–506.CrossRefGoogle Scholar
  43. [43]
    Quillen, D.G.: On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75(1969),1293–1298.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    Ravenel,D.C.: Complex cobordism and stable homotopy groups of spheres,Academic Press (1986).Google Scholar
  45. [45]
    —, The structure of BP *(BP) modulo an invariant prime ideal, Topology 15(1976),149–153.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    —, The structure of Morava stabilizer algebras, Invent. Math. 37(1976),109–120.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    —,Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984),351–414.MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    —, Morava K-theories and finite groups, Contemp. Math. AMS 12 (1982), 289–292.MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    —, The homology and Morava K-theory of Ω 2 SU(n), preprint Univ. of Rochester (1989).Google Scholar
  50. [50]
    Ravenel, D.C. and Wilson, S.W.:The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102(1980),691–748.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    —, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9(1977),241–280.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    Robinson, A.: Obstruction theory and the strict associativity of Morava K-theories, London Math. Soc. Lecture Notes 139 (1989), 143–152.MathSciNetMATHGoogle Scholar
  53. [53]
    —: Derived tensor products in stable homotopy theory, Topology 22(1983),1–18.MathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    —: Spectra of derived module homomorphisms, Math. Proc. Camb. Philos. Soc. 101(1987), 249–257.MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    —:Composition products in RHom and ring spectra of derived homomorphisms, Springer Lecture Notes in Math. 1370(1989), 374–386.CrossRefMATHGoogle Scholar
  56. [56]
    Sanders, J.P.: The category of H-modules over a spectrum, Mem. Am. Math. Soc. 141(1974).Google Scholar
  57. [57]
    Shimada, N and Yagita, N.: Multiplication in the complex bordism theory with singularities, Publ. Res. Inst. Math. Sci. 12 (1976/1977), No.1, 259–293.MathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    Wilson, S.W.:Brown-Peterson homology, an introduction and sampler, Regional Conference series in Math. No. 48, AMS, Providence, Rhode Island (1980).MATHGoogle Scholar
  59. [59]
    —:The Hopf ring for Morava K-theory, Pub. RIMS Kyoto Univ. 20(1984), 1025–1036.MathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    —:The complex cobordism of BO n, J. London Math. Soc. 29(1984), 352–366.MathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    Würgler, U.: Cobordism theories of unitary manifolds with singularities and formal group laws, Math. Z. 150(1976),239–260.MathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    —: On products in a family of cohomology theories associated to the invariant prime ideals of π * (BP), Comment. Math. Helv. 52 (1977),457–481.MathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    —:On the relation of Morava K-theories to Brown-Peterson homology, Monographie no. 26 de L'Enseignement Math.(1978),269–280.Google Scholar
  64. [64]
    —:A splitting theorem for certain cohomology theories associated to BP *(-), Manuscripta Math. 29(1979), 93–111.MathSciNetCrossRefMATHGoogle Scholar
  65. [65]
    —:On a class of 2-periodic cohomology theories, Math. Ann. 267(1984), 251–269.MathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    —:Commutative ring-spectra of characteristic 2, Comment. Math. Helv. 61(1986), 33–45.MathSciNetCrossRefMATHGoogle Scholar
  67. [67]
    Yagita, N.:On the Steenrod algebra of Morava K-theory, J. London Math. Soc. 22(1980), 423–438.MathSciNetCrossRefMATHGoogle Scholar
  68. [68]
    —:The exact functor theorem for BP */I n-theory, Proc. Japan Acad. 52(1976),1–3.MathSciNetCrossRefMATHGoogle Scholar
  69. [69]
    —,On the algebraic structure of cobordism operations with singularities, J. London Math. Soc. 16(1977),131–141.MathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    —,A topological note on the Adams spectral sequence based on Morava's K-theory, Proc. Am. Math. Soc. 72(1978),613–617.MathSciNetMATHGoogle Scholar
  71. [71]
    Yamaguchi, A.: Morava K-theory of double loop spaces of spheres, Math. Z. 199 (1988),511–523.MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag 1991

Authors and Affiliations

  • Urs Würgler
    • 1
  1. 1.Mathematisches Institut der Universität BernBern

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