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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Automorphisms of fibrations

Authors: E. Dror, W. G. Dwyer and D. M. Kan
Journal: Proc. Amer. Math. Soc. 80 (1980), 491-494
MSC: Primary 55R15; Secondary 55U10
MathSciNet review: 581012
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Abstract: Let X be a simplicial set, G a simplicial group and $ \bar WG$ the classifying complex of G. Then it is well known [1], [3] that the principal fibrations with base X and group G are classified by the components of the function complex $ {(\bar WG)^X}$. The aim of the present note is to prove the following complement to this result (1.2): Let p be a principal fibration with base X and group G, and let aut p be its simplicial group of automorphisms (which keep the base fixed). Then $ \bar W({\operatorname{aut}}\,p)$ has the homotopy type of the component of $ {(\bar WG)^X}$ which (see above) corresponds to p. A similar result holds for ordinary fibrations.

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  • [1] M. G. Barratt, V. K. A. M. Gugenheim, and J. C. Moore, On semisimplicial fibre-bundles, Amer. J. Math. 81 (1959), 639–657. MR 0111028
  • [2] E. Dror, W. G. Dwyer and D. M. Kan, Equivariant self homotopy equivalences, Proc. Amer. Math. Soc. (to appear).
  • [3] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892

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Article copyright: © Copyright 1980 American Mathematical Society