Hatcher-Algebraic-Topology · Problem 3H.3

Homotopy Theory

Mathematics — Graph Theory · bundles-of-groups-classifying-spaces

Problem

Let B(X;G)\mathcal{B}(X; G) be the set of isomorphism classes of bundles of groups E→XE \to X with fiber GG, and let E0→BAut(G)E_0 \to B\mathrm{Aut}(G) be the bundle corresponding to the 'identity' action ρ:Aut(G)→Aut(G)\rho : \mathrm{Aut}(G) \to \mathrm{Aut}(G). Show that the map [X,BAut(G)]→B(X,G)[X, B\mathrm{Aut}(G)] \to \mathcal{B}(X, G), [f]↦f∗(E0)[f] \mapsto f^*(E_0), is a bijection if XX is a CW complex, where [X,Y][X, Y] denotes the set of homotopy classes of maps X→YX \to Y.

Related theory

Browse the problem library