Israel Journal of Mathematics

, Volume 58, Issue 2, pp 129–143 | Cite as

On phantom maps and a theorem of H. Miller

  • A. Zabrodsky
Article

Abstract

A mapf:XY is a phantom map if any composition off with a map from a finite complex intoX is null homotopic. The proof of the Sullivan conjecture by H. Miller enables us to understand more deeply this phenomena. We prove, among other things, that any map from a space with finitely many non-vanishing homotopy groups into a finite complex is phantom and that any fibration over a 2-connected space with finitely many non-vanishing homotopy groups and with fiber a finite complex is trivial over each skeleton of the base.

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References

  1. 1.
    J. F. Adams and J. Walker,An example in homotopy theory, Proc. Camb. Phil. Soc.60 (1964), 699–700.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    A. K. Bousfield and D. M. Kan,Homotopy Limits, Completions and Localization, Lecture Notes in Math.304, Springer-Verlag, 1972.Google Scholar
  3. 3.
    G. Carlsson,G. B. Segal’s Burnside ring conjecture for (Z/2) k, Topology22 (1983), 83–103.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    E. M. Friedlander and G. Mislin,Locally finite approximation of Lie groups I, mimeographed.Google Scholar
  5. 5.
    B. I. Gray,Spaces of the same n-type for all n, Topology5 (1966), 241–243.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. A. McGibbon and J. A. Neisendorfer,On the homotopy groups of a finite dimensional space, Comm. Math. Helv.59 (1984), 253–257.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W. Meier,Localisation, completion et application fantomes, C.R. Acad. Sci. Paris281 (1975), 787–789.MATHMathSciNetGoogle Scholar
  8. 8.
    W. Meier,Determination de certains groupes d’applications fantomes, C.R. Acad. Sci. Paris282 (1976), 971–974.MathSciNetGoogle Scholar
  9. 9.
    W. Meier,Pullback theorems and phantom maps, Quart. J. Math.29 (1978), 469–481.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. Meier, Personal communication, Feb. 10, 1983.Google Scholar
  11. 11.
    H. Miller,The Sullivan fixed point conjecture on maps from classifying spaces, Ann. of Math.120 (1984), 39–87.CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Zabrodsky,Maps between classifying spaces, to appear.Google Scholar

Copyright information

© Hebrew Univeristy 1987

Authors and Affiliations

  • A. Zabrodsky
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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