Morava K-theories: A survey

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


Hopf Algebra Spectral Sequence Cohomology Theory Stable Homotopy Finite Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adams, J.F.: Stable homotopy and generalised homology, Univ. of Chicago press, Chicago, Illinois and London (1974).zbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Baker, A.: Some families of operations in Morava K-theory, Amer. J. Math. 111(1989),95–109.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Devinatz, E.S., Hopkins, M.J. and Smith, J.H.: Nilpotence and stable homotopy I, Ann. Math. 128(1988), 207–241.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    —: The Brown-Peterson homology of elementary p-groups, Amer. J. Math. 107(1984), 427–453.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    —: The Morava K-theories of some classifying spaces, TAMS 304(1987),193–205.MathSciNetzbMATHGoogle Scholar
  24. [24]
    —: Character rings in algebraic topology, London Math. Soc. Lecture Notes 139 (1989), 111–126.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    —: The algebra k(n)*(k(n)) for the prime 2, Arch. Math. 51(1988),141–146.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Landweber, P.S.: BP *(BP) and typical formal groups, Osaka J. Math. 12(1975),357–363.MathSciNetzbMATHGoogle Scholar
  29. [29]
    —: Homological properties of comodules over MU *(MU) and BP *(BP), Amer. J. Math. 98(1976),591–610.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Morava, J.:A product for odd-primary bordism of manifolds with singularities, Topology 18(1979), 177–186.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    —, Completions of complex cobordism, Lecture Notes in Math. 658(1978),349–361.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    —,Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121(1985), 1–39.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    —,Forms of K-theory, Math. Z. 201(1989),401–428.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    —, The structure of Morava stabilizer algebras, Invent. Math. 37(1976),109–120.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    —,Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984),351–414.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    —, Morava K-theories and finite groups, Contemp. Math. AMS 12 (1982), 289–292.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    —, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9(1977),241–280.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    Robinson, A.: Obstruction theory and the strict associativity of Morava K-theories, London Math. Soc. Lecture Notes 139 (1989), 143–152.MathSciNetzbMATHGoogle Scholar
  53. [53]
    —: Derived tensor products in stable homotopy theory, Topology 22(1983),1–18.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    —: Spectra of derived module homomorphisms, Math. Proc. Camb. Philos. Soc. 101(1987), 249–257.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    —:Composition products in RHom and ring spectra of derived homomorphisms, Springer Lecture Notes in Math. 1370(1989), 374–386.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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).zbMATHGoogle Scholar
  59. [59]
    —:The Hopf ring for Morava K-theory, Pub. RIMS Kyoto Univ. 20(1984), 1025–1036.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    —:The complex cobordism of BO n, J. London Math. Soc. 29(1984), 352–366.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    Würgler, U.: Cobordism theories of unitary manifolds with singularities and formal group laws, Math. Z. 150(1976),239–260.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    —:On a class of 2-periodic cohomology theories, Math. Ann. 267(1984), 251–269.MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    —:Commutative ring-spectra of characteristic 2, Comment. Math. Helv. 61(1986), 33–45.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    Yagita, N.:On the Steenrod algebra of Morava K-theory, J. London Math. Soc. 22(1980), 423–438.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    —:The exact functor theorem for BP */I n-theory, Proc. Japan Acad. 52(1976),1–3.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    —,On the algebraic structure of cobordism operations with singularities, J. London Math. Soc. 16(1977),131–141.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  71. [71]
    Yamaguchi, A.: Morava K-theory of double loop spaces of spheres, Math. Z. 199 (1988),511–523.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität BernBern

Personalised recommendations