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Israel Journal of Mathematics

, Volume 66, Issue 1–3, pp 54–104 | Cite as

Homotopy spectral sequences and obstructions

  • A. K. Bousfield
Article

Abstract

For a pointed cosimplicial spaceX , the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX . In this paper,X is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.

Keywords

Spectral Sequence Homotopy Theory Compact Hausdorff Space Obstruction Theory Fundamental Action 
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.

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References

  1. 1.
    D. W. Anderson,A generalization of the Eilenberg-Moore spectral sequence, Bull. Am. Math. Soc.78 (1972), 784–786.zbMATHGoogle Scholar
  2. 2.
    D. W. Anderson,Fibrations and geometric realizations, Bull. Am. Math. Soc.84 (1978), 765–788.zbMATHGoogle Scholar
  3. 3.
    M. André,Homologie des Algebres Commutatives, Springer-Verlag, Berlin 1974.zbMATHGoogle Scholar
  4. 4.
    H. Baues,Obstruction Theory, Springer Lecture Notes in Mathematics, Vol. 628, 1977.Google Scholar
  5. 5.
    A. K. Bousfield,Nice homology coalgebras, Trans. Am. Math. Soc.148 (1970), 473–489.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. K. Bousfield,On the homology spectral sequence of a cosimplicial space, Am. J. Math.109 (1987), 361–394.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. K. Bousfield and D. M. Kan,The homotopy spectral sequence of a space with coefficients in a ring, Topology11 (1972), 79–106.CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. K. Bousfield and D. M. Kan,Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics, Vol. 304, 1972.Google Scholar
  9. 9.
    A. K. Bousfield and D. M. Kan,A second quadrant homotopy spectral sequence, Trans. Am. Math. Soc.177 (1973), 305–318.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Brown,Fibrations of groupoids, J. Algebra15 (1970), 103–132.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Dror-Farjoun and A. Zabrodsky,The homotopy spectral sequence for equivariant function complexes, inAlgebraic Topology, Barcelona 1986, Lecture Notes in Mathematics, Vol. 1298, Springer-Verlag, Berlin, 1987.Google Scholar
  12. 12.
    W. Dwyer, H. Miller and J. Neisendorfer,Fibrewise localization and unstable Adams spectral sequences, to appear.Google Scholar
  13. 13.
    H. Federer,A study of function spaces by spectral sequences, Trans. Am. Math. Soc.82 (1956), 340–361.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    P. Gabriel and M. Zisman,Calculus of Fractions and Homotopy Theory, Springer-Verlag, Berlin, 1967.zbMATHGoogle Scholar
  15. 15.
    M. Huber and W. Meier,Linearly compact groups, J. Pure Appl. Algebra16 (1980), 167–182.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. Lannes,Sur la cohomologie modulo p des p-groupes abeliens elementaires, inHomotopy Theory, Proceedings of the Durham Symposium 1985 (E. Rees and J. D. S. Jones, eds.), Cambridge University Press, 1987.Google Scholar
  17. 17.
    S. MacLane,Categories for the Working Mathematician, Springer-Verlag, Berlin, 1971.Google Scholar
  18. 18.
    J. P. May,Simplicial Objects in Algebraic Topology, Van Nostrand, New York, 1967.Google Scholar
  19. 19.
    H. Miller,The Sullivan conjecture on maps from classifying spaces, Ann. of Math.120 (1984), 39–87.Correction:121 (1985), 605–609.CrossRefMathSciNetGoogle Scholar
  20. 20.
    D. G. Quillen,Homotopical Algebra, Springer Lecture Notes in Mathematics, Vol. 43, 1967.Google Scholar
  21. 21.
    D. G. Quillen,On the (co)-homology of commutative rings, Proc. Symp. Pure Math.17 (1970), 65–87.MathSciNetGoogle Scholar
  22. 22.
    M. E. Sweedler,Hopf Algebras, W. A. Benjamin, New York, 1969.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1989

Authors and Affiliations

  • A. K. Bousfield
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA

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