Towers of fibrations

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


In this chapter we generalize two well-known results about towers of fibrations:
  1. (i)
    We will show that, for a (pointed) tower of fibrations {Xn}, the short exact sequence
    $$* \to {\mathop {\lim }\limits_ \leftarrow ^1}{\pi _{i + 1}}{X_n} \to {\pi _i}\mathop {\lim }\limits_ \leftarrow {X_n} \to \mathop {\lim }\limits_ \leftarrow {\pi _i}{X_n} \to *$$
    which is “well known” for i ≥ 1, also exists for i = 0, if one uses a suitable notion of \({\mathop {\lim }\limits_ \leftarrow ^1}\) for not necessarily abelian groups.
  2. (ii)

    We will generalize the usual homotopy spectral sequence of a (pointed) tower of fibrations, to an “extended” homotopy spectral sequence, which in dimension 1 consists of (possibly non-abelian) groups, and in dimension 0 of pointed sets, acted on by the groups is dimension 1.



Abelian Group Spectral Sequence Short Exact Sequence Homotopy Type Homotopy Group 
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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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