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Extending local homotopy equivalences

  • Martin Fuchs
Article
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Abstract

We give an elementary proof of one of tom Dieck’s theorems. The theorem says that iff:X → Y is a local homotopy equivalence in a strong enough sense, thenf is a homotopy equivalence globally. Applications, 1. The base space of any numerable principalG-bundle is of the sme homotopy type as the Borel space of the bundle. 2. The nerve of a numerable coveringU ofX for which all finite intersections are contractible is of the same homotopy type asX.

Keywords

Topological Space Base Space Homotopy Type Homotopy Theory Formal Covering 
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References

  1. [1]
    Brown R.,Elements of Modern Topology, Mc Graw-Hill, London (1968).MATHGoogle Scholar
  2. [2]
    Dold A.,Partitions of unity in the theory of fibrations, Ann. of Math.78 (1963), 223–255.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Hu S. T.,Homotopy theory, Academic Press, New York (1959).MATHGoogle Scholar
  4. [4]
    Puppe D.,Homotopiemengen und ihre induzierten Abbildungen I, Math. Z.69 (1958), 299–344.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Segal G. B.,Classifying spaces and spectral sequences, I.H.E.S. Pub. Math.34 (1968), 105–112.MATHGoogle Scholar
  6. [6]
    Tom Dieck T.,Partitions of unity in homotopy theory, Comp. Math.23 (1971), 159–167.MATHMathSciNetGoogle Scholar
  7. [7]
    Vogt R. M.,Homotopy limits and colimits, Math.2 (1973) 134, 11–52.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 1983

Authors and Affiliations

  • Martin Fuchs
    • 1
    • 2
  1. 1.Institut für Mathematik UniversitätKonstanzGermania dell'Ovest
  2. 2.Department of MathematicsMichigan State UniversityLansingU. S. A.

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