Fibre lemmas

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


For a general fibration of connected spaces F → E → B, the map RE → RB is always a fibration (Ch.I, 4.2), but RF need not have the same homotopy type as the fibre of RE → RB. For example, if R = Q, then
$${S^2} \to {P^2} \to K\left( {{Z_2},1} \right)$$
is, up to homotopy, a fibration, but RS2 → RP2 → RK(Z2,1) is not, because (Ch.I, 5.5) RP2 and RK(Z2,1) are contractible, while RS2 is not.


Exact Sequence Spectral Sequence Short Exact Sequence Homotopy Type Connected Space 
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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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