Advertisement

A Novice's guide to the adams-novikov spectral sequence

  • Douglas C. Ravenel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 658)

Keywords

Hopf Algebra Spectral Sequence Short Exact Sequence Stable Homotopy Adams Spectral Sequence 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. F. Adams, On the groups J(X), IV, Topology 5(1966), 21–71.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. F. Adams, Lectures on generalized cohomology, Lecture Notes in Math., Vol. 99 (Springer-Verlag, 1969).Google Scholar
  3. 3.
    J. F. Adams, Localization and completion with an addendum on the use of Brown-Peterson homology in stable homotopy, University of Chicago Lecture Notes in Mathematics, 1975.Google Scholar
  4. 4.
    J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math. 72(1960), 20–103.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. F. Adams, A periodicity theorem in homological algebra, Proc. Cambridge Phil. Soc. 62(1966), 365–377.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. F. Adams, Stable homotopy and generalized homology, University of Chicago Press, 1974.Google Scholar
  7. 7.
    J. F. Adams, Stable homotopy theory, Lecture Notes in Math., Vol. 3 (Springer-Verlag, 1966).Google Scholar
  8. 8.
    J. F. Adams, On the structure and applications of the Steenrod algebra, Comm. Math. Helv. 32(1958), 180–214.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Araki, Typical formal groups in complex cobordism and K-theory, Kinokumiya Book-Store, Kyoto, 1974.zbMATHGoogle Scholar
  10. 10.
    M. G. Barratt, M. E. Mahowald, and M. C. Tangora, Some differentials in the Adams spectral sequence-II, Topology, 9(1970), 309–316.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. K. Bousfield, Types of acyclicity, J. Pure Appl. Algebra 4(1974), 293–298.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., Vol.304(Springer-Verlag, 1972).Google Scholar
  13. 13.
    Browder, The Kervaire invariant of framed manifolds and its generalizations, Ann. of Math. 90(1969), 157–186.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. M. Buhstaber and S. P. Novikov, Formal groups, power systems and Adams operators, Math. USSR Sbornik 13(1971), 70–116.MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1956.Google Scholar
  16. 16.
    P. Cartier, Modules associés à un groupe formel commutatif. Courbes typiques, C. R. Acad. Sci. Paris, 265(1967), A129–132.MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. Fröhlich, Formal groups, Lecture Notes in Math., Vol. 74 (Springer-Verlag, 1968).Google Scholar
  18. 18.
    M. Hazewinkel, Formal groups and applications, Academic Press (to appear).Google Scholar
  19. 19.
    M. Hazewinkel, A universal formal group and complex cobordism, Bull. A.M.S. 81(1975), 930–933.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D. C. Johnson, H. R. Miller, W. S. Wilson, and R. S. Zahler, Boundary homomorphisms in the generalized Adams spectral sequence and the non-triviality of infinitely many γt in stable homotopy Reunion sobre teoria de homotopia, Northwestern Univ. 1974, Soc. Mat. Mexicana, 1975, 47–59.Google Scholar
  21. 21.
    P. S. Landweber, Annihilator ideals and primitive elements in complex bordism, Ill. J. Math 17(1973); 273–283.MathSciNetzbMATHGoogle Scholar
  22. 22.
    P. S. Landweber, BP*(BP) and typical formal groups, Osaka J. Math. 12(1975), 357–369.MathSciNetzbMATHGoogle Scholar
  23. 23.
    M. P. Lazard, Commutative formal groups, Lecture Notes in Math., Vol. 443 (Springer-Verlag, 1975).Google Scholar
  24. 24.
    M. P. Lazard, Groupes analytiques p-adiques, IHES Pub. Math. No. 26(1965).Google Scholar
  25. 25.
    M. P. Lazard, Sur les groupes formels a un parametre, Bull. Soc. Math. France, 83(1955) 251–274.MathSciNetzbMATHGoogle Scholar
  26. 26.
    A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42(1962).Google Scholar
  27. 27.
    M. E. Mahowald, The metastable homotopy of Sn, Memoirs A.M.S. 72, 1967.Google Scholar
  28. 28.
    M. E. Mahowald, A new infinite family in 2Π* S, Topology 16 (1977), 249–256.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    M. E. Mahowald, Some remarks on the Arf invariant problem from the homotopy point of view, Proc. Symp. Pure Math. A.M.S. Vol. 22.Google Scholar
  30. 30.
    M. E. Mahowald and M. C. Tangora, On secondary operations which detect homotopy classes, Bol. Soc. Math. Mexicana (2) 12(1967), 71–75.MathSciNetzbMATHGoogle Scholar
  31. 31.
    M. E. Mahowald and M. C. Tangora, Some differentials in the Adams spectral sequence, Topology 6(1967), 349–369.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Alg. 3(1966), 123–146.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras; application to the Steenrod algebra, Thesis, Princeton University 1964.Google Scholar
  34. 34.
    J. P. May, Matric Massey products, J. Alg. 12(1969), 533–568.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    R. J. Milgram, The Steenrod algebra and its dual for connective K-theory, Reunion sobre teoria de homotopia, Northwestern Univ. 1974, Soc. Mat. Mexicana, 1975, 127–158.Google Scholar
  36. 36.
    H. R. Miller, Some algebraic aspects of the Adams-Novikov spectral sequence, Thesis, Princeton University, 1974.Google Scholar
  37. 37.
    H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the localization of Novikov's E2-term, Duke Math. Journal 44 (1977) 433–446.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    H. R. Miller, D. C. Ravenel, and W. S. Wilson, Novikov's Ext2 and the nontriviality of the gamma family, Bull. Amer. Math. Soc., 81(1975), 1073–1075.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (to appear).Google Scholar
  40. 40.
    H. R. Miller and W. S. Wilson, On Novikov's Ext1 modulo an invariant prime ideal, Topology, 5(1976), 131–141.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    J. Morava, Extensions of cobordism comodules, (to appear).Google Scholar
  42. 42.
    J. Morava, Structure theorems for cobordism comodules, (to appear somewhere).Google Scholar
  43. 43.
    New York Times, editorial page, June 2, 1976.Google Scholar
  44. 44.
    S. P. Novikov, The methods of algebraic topology from the viewpoint of cobordism theories, Math. U.S.S.R.-Izvestiia 1 (1967), 827–913.CrossRefzbMATHGoogle Scholar
  45. 45.
    S. Oka, A new family in the stable homotopy groups of spheres, Hiroshima J. Math., 5(1975), 87–114.MathSciNetzbMATHGoogle Scholar
  46. 46.
    S. Oka, A new family in the stable homotopy groups of spheres II, Hiroshima J. Math. 6 (1976), 331–342.MathSciNetzbMATHGoogle Scholar
  47. 47.
    S. Oka, Realizing some cyclic BP*-modules and applications to homotopy groups of spheres, Hiroshima Math: J. 7(1977), 427–447.MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. Oka and H. Toda, Nontriviality of an element in the stable homotopy groups of spheres, Hiroshima Math. J. 5(1975), 115–125.MathSciNetzbMATHGoogle Scholar
  49. 49.
    D. G. Quillen, The Adams conjecture, Topology 10(1971), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    D. G. Quillen, On the formal group laws of unoriented and complex cobordism, Bull. A.M.S. 75(1969), 115–125.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    D. C. Ravenel, The cohomology of the Morava stabilizer algebras, Math. Z. 152(1977), 287–297.MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    D. C. Ravenel, Computations with the Adams-Novikov spectral sequence at the prime 3 (to appear).Google Scholar
  53. 53.
    D. C. Ravenel, Localization with respect to certain periodic homology theories, to appear.Google Scholar
  54. 54.
    D. C. Ravenel, A May spectral sequence converging to the Adams-Novikov E2-term, (to appear).Google Scholar
  55. 55.
    D. C. Ravenel, A new method for computing the Adams-Novikov E2-term, (to appear).Google Scholar
  56. 56.
    D. C. Ravenel, The nonexistence of odd primary Arf invariant elements in stable homotopy, Math. Proc. Cambridge Phil. Soc. (to appear).Google Scholar
  57. 57.
    D. C. Ravenel, The structure of BP*BP modulo an invariant prime ideal, Topology 15(1976), 149–153.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    D. C. Ravenel, The structure of Morava stabilizer algebras, Inv. Math. 37(1976), 109–120.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, J. of Pure and Applied Algebra (to appear).Google Scholar
  60. 60.
    Science. June 7, 1976.Google Scholar
  61. 61.
    N. Shimada and T. Yamamoshita, On the triviality of the mod p Hopf invariant, Jap. J. Math. 31(1961), 1–24.MathSciNetzbMATHGoogle Scholar
  62. 62.
    C. L. Siegel, Topics in Complex Function Theory, Vol I. Wiley-Interscience, 1969.Google Scholar
  63. 63.
    L. Smith, On realizing complex bordism modules, Amer. J. Math. 92(1970) 793–856.MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    L. Smith, On realizing complex bordism modules IV, Amer. J. Math. 99(1971), 418–436.CrossRefzbMATHGoogle Scholar
  65. 65.
    V. P. Snaith Cobordism and the stable homotopy of classifying spaces, (to appear).Google Scholar
  66. 66.
    N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Studies, 50.Google Scholar
  67. 67.
    M. C. Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116(1970), 18–64.MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    E. Thomas and R. S. Zahler, Nontriviality of the stable homotopy element γ1, J. Pure Appl. Algebra 4(1974), 189–203.MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies 49.Google Scholar
  70. 70.
    H. Toda, Extended p-th powers of complexes and applications to homotopy theory, Proc. Japan Acad. 44(1968), 198–203.MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    H. Toda, An important relation in homotopy groups of spheres, Proc. Japan Acad. 43(1967), 893–942.MathSciNetzbMATHGoogle Scholar
  72. 72.
    H. Toda, p-primary components of homotopy groups, IV, Mem. Coll. Sci., Kyoto, Series A 32(1959), 297–332.MathSciNetzbMATHGoogle Scholar
  73. 73.
    H. Toda, On spectra realizing exterior parts of the Steenrod algebra, Topology 10(1971), 53–65.MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    J. S. P. Wang, On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Ill. J. Math 11(1967), 480–490.MathSciNetzbMATHGoogle Scholar
  75. 75.
    R. S. Zahler, The Adams-Novikov spectral sequence for the spheres, Ann. of Math 96(1972), 480–504.MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    R. S. Zahler, Fringe families in stable homotopy, Trans. Amer. Math. Soc., 224(1976), 243–253.MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    M. E. Mahowald, The construction of small ring spectra, (to appear).Google Scholar
  78. 78.
    S. Oka, Ring spectra with few cells, (to appear).Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Douglas C. Ravenel
    • 1
  1. 1.University of WashingtonSeattle

Personalised recommendations