The simplicial bundle of a CW fibration

Author:
Donald W. Barnes

Journal:
Proc. Amer. Math. Soc. **101** (1987), 559-562

MSC:
Primary 55R05; Secondary 55R10, 55R20, 55T10

MathSciNet review:
908669

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Abstract: Suppose we are given a fibration over a connected base, with both base and fibre having the homotopy type of CW complexes. We construct a fibre bundle over fibre homotopy equivalent to the given fibration and with fibre a simplicial complex. Further, the transformations of the fibre arising from the transition functions of this bundle are simplicial maps. From this, we deduce that the weak spectral sequence constructor axioms are sufficient to determine the Serre spectral sequence of a CW fibration.

**[1]**D. W. Barnes,*Spectral sequence constructors in algebra and topology*, Mem. Amer. Math. Soc.**53**(1985), no. 317, viii+174. MR**776177**, 10.1090/memo/0317**[2]**Edward Fadell,*On fiber homotopy equivalence*, Duke Math. J**26**(1959), 699–706. MR**0109347****[3]**Edward Fadell,*The equivalence of fiber spaces and bundles*, Bull. Amer. Math. Soc.**66**(1960), 50–53. MR**0112137**, 10.1090/S0002-9904-1960-10389-8**[4]**John Milnor,*Construction of universal bundles. I*, Ann. of Math. (2)**63**(1956), 272–284. MR**0077122****[5]**N. E. Steenrod,*A convenient category of topological spaces*, Michigan Math. J.**14**(1967), 133–152. MR**0210075**

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0908669-3

Article copyright:
© Copyright 1987
American Mathematical Society