Skip to main content
SpringerLink
Log in

Morava K-theories: A survey

  • Survey Articles
  • Conference paper
  • First Online:
Algebraic Topology Poznań 1989

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, J.F.: Stable homotopy and generalised homology, Univ. of Chicago press, Chicago, Illinois and London (1974).

    MATH  Google Scholar 

  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. Baas, N.A.: On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302.

    MathSciNet  MATH  Google Scholar 

  4. Baas, N.A. and Madsen, I.: On the realization of certain modules over the Steenrod algebra, Math. Scand 31 (1972), 220–224.

    MathSciNet  MATH  Google Scholar 

  5. Baker, A.: Some families of operations in Morava K-theory, Amer. J. Math. 111(1989),95–109.

    Article  MathSciNet  MATH  Google Scholar 

  6. —A ∞-structures on some spectra related to Morava K-theories, preprint Manchester Univ., (1988).

    Google Scholar 

  7. Baker, A. and Würgler, U.: Liftings of formal groups and the Artinian completion of v −1 n BP, Math. Proc. Camb. Phil. Soc. 106 (1989),511–530.

    Article  MATH  Google Scholar 

  8. —:Bockstein operations in Morava K-theory, preprint 1989.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cartier, P.: Modules associés à un groupe formel commutatif, courbes typiques, C. R. Acad. Sci. Paris Série A 265(1965), 129–132.

    MathSciNet  MATH  Google Scholar 

  11. Devinatz, E.S., Hopkins, M.J. and Smith, J.H.: Nilpotence and stable homotopy I, Ann. Math. 128(1988), 207–241.

    Article  MathSciNet  MATH  Google Scholar 

  12. Dold, A.: Chern classes in general cohomology. Symp. Math. V(1970),385–410.

    MathSciNet  Google Scholar 

  13. Fröhlich,A.: Formal groups, Lecture Notes in Math. 74(1968).

    Google Scholar 

  14. Hazewinkel M.: Formal groups and applications. Academic press, 1978.

    Google Scholar 

  15. Hopkins, J.R.: Global methods in homotopy theory, Proc. Durham Symp. 1985, Cambridge Univ. Press (1987), 73–96.

    Google Scholar 

  16. Hunton, J.: The Morava K-theories of wreath products, Preprint Cambridge Univ. (1989).

    Google Scholar 

  17. —: Ph.D. Thesis, Cambridge Univ. (1989).

    Google Scholar 

  18. Johnson, D.C. and Wilson, W.S.: BP-operations and Morava's extraordinary K-theories, Math.Z. 144(1975),55–75.

    Article  MathSciNet  MATH  Google Scholar 

  19. —: The Brown-Peterson homology of elementary p-groups, Amer. J. Math. 107(1984), 427–453.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kane, R.M.: Implications in Morava K-theory, Mem. Amer. Math. Soc. 59 (1986), No.340.

    Google Scholar 

  21. Knus M.,and Ojanguren, M.: Théorie de la descente et algébres d'Azumaya. Lecture Notes in Mathematics 389, 1974.

    Google Scholar 

  22. Kuhn, N.J.: Morava K-theories and infinite loop spaces, Springer Lect. Notes in Math. 1370(1989),243–257.

    Article  MathSciNet  MATH  Google Scholar 

  23. —: The Morava K-theories of some classifying spaces, TAMS 304(1987),193–205.

    MathSciNet  MATH  Google Scholar 

  24. —: Character rings in algebraic topology, London Math. Soc. Lecture Notes 139 (1989), 111–126.

    MathSciNet  MATH  Google Scholar 

  25. Kultze,R.: Die Postnikov-Faktoren von k(n), Manuskript, Universität Frankfurt (1989).

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  27. —: The algebra k(n)*(k(n)) for the prime 2, Arch. Math. 51(1988),141–146.

    Article  MathSciNet  MATH  Google Scholar 

  28. Landweber, P.S.: BP *(BP) and typical formal groups, Osaka J. Math. 12(1975),357–363.

    MathSciNet  MATH  Google Scholar 

  29. —: Homological properties of comodules over MU *(MU) and BP *(BP), Amer. J. Math. 98(1976),591–610.

    Article  MathSciNet  MATH  Google Scholar 

  30. Lazard, M.: Sur les groupes de Lie formels á un paramètre, Bull. Soc. Math. France 83, 251–274.

    Google Scholar 

  31. Lellmann, W.: Connected Morava K-theories, Math. Z. 179 (1982), 387–399.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  34. Mischenko: Appendix 1 in Novikov [41].

    Google Scholar 

  35. Mitchell, S.A.: Finite complexes with A(n)-free cohomology, Topology 24(1985), 227–248.

    Article  MathSciNet  MATH  Google Scholar 

  36. Morava, J.:A product for odd-primary bordism of manifolds with singularities, Topology 18(1979), 177–186.

    Article  MathSciNet  MATH  Google Scholar 

  37. —, Completions of complex cobordism, Lecture Notes in Math. 658(1978),349–361.

    Article  MathSciNet  MATH  Google Scholar 

  38. —,Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121(1985), 1–39.

    Article  MathSciNet  MATH  Google Scholar 

  39. —,Forms of K-theory, Math. Z. 201(1989),401–428.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. Pazhitmov, A.V.: Uniqueness theorems for generalized cohomology theories, Math. USSR Izvestiyah 22(1984),483–506.

    Article  Google Scholar 

  43. Quillen, D.G.: On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75(1969),1293–1298.

    Article  MathSciNet  MATH  Google Scholar 

  44. Ravenel,D.C.: Complex cobordism and stable homotopy groups of spheres,Academic Press (1986).

    Google Scholar 

  45. —, The structure of BP *(BP) modulo an invariant prime ideal, Topology 15(1976),149–153.

    Article  MathSciNet  MATH  Google Scholar 

  46. —, The structure of Morava stabilizer algebras, Invent. Math. 37(1976),109–120.

    Article  MathSciNet  MATH  Google Scholar 

  47. —,Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984),351–414.

    Article  MathSciNet  MATH  Google Scholar 

  48. —, Morava K-theories and finite groups, Contemp. Math. AMS 12 (1982), 289–292.

    Article  MathSciNet  MATH  Google Scholar 

  49. —, The homology and Morava K-theory of Ω 2 SU(n), preprint Univ. of Rochester (1989).

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  51. —, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9(1977),241–280.

    Article  MathSciNet  MATH  Google Scholar 

  52. Robinson, A.: Obstruction theory and the strict associativity of Morava K-theories, London Math. Soc. Lecture Notes 139 (1989), 143–152.

    MathSciNet  MATH  Google Scholar 

  53. —: Derived tensor products in stable homotopy theory, Topology 22(1983),1–18.

    Article  MathSciNet  MATH  Google Scholar 

  54. —: Spectra of derived module homomorphisms, Math. Proc. Camb. Philos. Soc. 101(1987), 249–257.

    Article  MathSciNet  MATH  Google Scholar 

  55. —:Composition products in RHom and ring spectra of derived homomorphisms, Springer Lecture Notes in Math. 1370(1989), 374–386.

    Article  MATH  Google Scholar 

  56. Sanders, J.P.: The category of H-modules over a spectrum, Mem. Am. Math. Soc. 141(1974).

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  58. Wilson, S.W.:Brown-Peterson homology, an introduction and sampler, Regional Conference series in Math. No. 48, AMS, Providence, Rhode Island (1980).

    MATH  Google Scholar 

  59. —:The Hopf ring for Morava K-theory, Pub. RIMS Kyoto Univ. 20(1984), 1025–1036.

    Article  MathSciNet  MATH  Google Scholar 

  60. —:The complex cobordism of BO n , J. London Math. Soc. 29(1984), 352–366.

    Article  MathSciNet  MATH  Google Scholar 

  61. Würgler, U.: Cobordism theories of unitary manifolds with singularities and formal group laws, Math. Z. 150(1976),239–260.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. —:A splitting theorem for certain cohomology theories associated to BP *(-), Manuscripta Math. 29(1979), 93–111.

    Article  MathSciNet  MATH  Google Scholar 

  65. —:On a class of 2-periodic cohomology theories, Math. Ann. 267(1984), 251–269.

    Article  MathSciNet  MATH  Google Scholar 

  66. —:Commutative ring-spectra of characteristic 2, Comment. Math. Helv. 61(1986), 33–45.

    Article  MathSciNet  MATH  Google Scholar 

  67. Yagita, N.:On the Steenrod algebra of Morava K-theory, J. London Math. Soc. 22(1980), 423–438.

    Article  MathSciNet  MATH  Google Scholar 

  68. —:The exact functor theorem for BP */I n -theory, Proc. Japan Acad. 52(1976),1–3.

    Article  MathSciNet  MATH  Google Scholar 

  69. —,On the algebraic structure of cobordism operations with singularities, J. London Math. Soc. 16(1977),131–141.

    Article  MathSciNet  MATH  Google Scholar 

  70. —,A topological note on the Adams spectral sequence based on Morava's K-theory, Proc. Am. Math. Soc. 72(1978),613–617.

    MathSciNet  MATH  Google Scholar 

  71. Yamaguchi, A.: Morava K-theory of double loop spaces of spheres, Math. Z. 199 (1988),511–523.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität Bern, CH 3012, Bern

    Urs Würgler

Authors
  1. Urs Würgler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stefan Jackowski Bob Oliver Krzystof Pawałowski

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Würgler, U. (1991). Morava K-theories: A survey. In: Jackowski, S., Oliver, B., Pawałowski, K. (eds) Algebraic Topology Poznań 1989. Lecture Notes in Mathematics, vol 1474. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084741

Download citation

  • DOI: https://doi.org/10.1007/BFb0084741

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54098-4

  • Online ISBN: 978-3-540-47403-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation