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Tower lemmas

  • Aldridge K. Bousfield
  • Daniel M. Kan
Part of the Lecture Notes in Mathematics book series (LNM, volume 304)

Abstract

In this chapter we establish the following simple and useful sufficient conditions on a tower of fibrations {Ys}, in order that it can be used to obtain the homotopy type of the R-completion of a given space X:
  1. (i)
    If f: X → {Ys is a map which induces, for every R-module M, an isomorphism
    $$\mathop {\lim }\limits_ \to H*\left( {{Y_S};M} \right) \approx H*\left( {X;M} \right)$$
    then f induces a homotopy equivalence \({R_\infty }X \simeq \mathop {\lim }\limits_ \to {R_\infty }{Y_S}\).
     
  2. (ii)

    If, in addition, each Ys is R-complete (Ch.I, 5.1) , then the space \(\mathop {\lim }\limits_ \to {Y_S}\) already has the same homotopy type as RX.

     
  3. (iii)

    If, in addition, each Ys satisfies the even stronger condition of being R-nilpotent (4.2), then, in a certain precise sense, the tower {Ys} has the same homotopy type as the tower {Rs}

     

Keywords

Spectral Sequence Nilpotent Group Full Subcategory Homotopy Type Inverse Limit 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1972

Authors and Affiliations

  • Aldridge K. Bousfield
    • 1
  • Daniel M. Kan
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisChicagoUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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