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Covering groups of non-connected topological groups revisited. (English) Zbl 0831.57001

R. L. Taylor [Trans. Am. Math. Soc. 75, 106-135 (1953; Zbl 0053.013) and Proc. Am. Math. Soc. 5, 753-768 (1954; Zbl 0058.021)]showed that if \(X\) is a topological group then there is an obstruction class \(k_X\) in \(H^3 (\pi_0 C, \pi_1 (X, e))\) such that the universal cover \(\widetilde {X}\) of \(X\) has a unique topological group structure for which the covering projection \(p: \widetilde {X}\to X\) is a group homomorphism if and only if \(k_X\) is trivial. The present note reviews this result in the light of modern developments in cohomology and in particular in the development of new techniques of crossed modules, and crossed complexes, and then proves a generalization of the result as follows. Suppose \(\theta: \Phi\to \pi_0 X\) is a morphism of groups where \(X\) is a topological group, and let \(N\) be a \(\pi_0 X\)-invariant subgroup of \(\pi_1 (X, e)\), then there is a covering morphism \(p: \widetilde {X}\to X\) of topological groups and an isomorphism \(\alpha: \pi_0 \widetilde {X}\to \Phi\) such that \(\theta \alpha= \pi_0 (p)\) and \(p\pi_1 (\widetilde {X}, \widetilde {e})=N\) if and only if an obstruction class \(k(X_1 \theta)\in H^3_\theta (\Phi, \pi_1 (X, e))\) vanishes under change of coefficients along \(\pi (X,e)\to \pi_1 (X, e)/N\).
Reviewer: T.Porter (Bangor)

MSC:

57M10 Covering spaces and low-dimensional topology
55Q05 Homotopy groups, general; sets of homotopy classes
22A05 Structure of general topological groups
22A22 Topological groupoids (including differentiable and Lie groupoids)
18G55 Nonabelian homotopical algebra (MSC2010)
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