Approximating groups of bundle automorphisms by loop spaces

Bencivenga, Roberto

doi:10.1090/S0002-9947-1984-0752499-4

Approximating groups of bundle automorphisms by loop spaces
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Trans. Amer. Math. Soc. 285 (1984), 703-715 Request permission

Abstract:

D. H. Gottlieb proved in 1972 that the group of automorphisms of a numerable $G$-bundle $p:X \to B$ is weakly homotopy equivalent to $\Omega \;\operatorname {Map}(B,{B_G};k)$, where $k:B \to {B_G}$ is a classifying map for $p$. We refine this classical result by constructing a genuine homotopy equivalence between these two spaces which is natural with respect to numerable bundle morphisms, can be generalized to fibre bundles, and can be interpreted as a natural isomorphism between two suitably defined functors.

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