2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168328Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension 1\to Z\to \hat{K}\to K\to 1 of K. It is a classical question whether there exists a \hat{K}-principal bundle \hat{P} on M such that \hat{P}/Z is isomorphic to P. Neeb defines in this context a crossed module of topological Lie algebras whose cohomology class [ω_{\rm top alg}] is an obstruction to the existence of \hat{P}. In the present paper, we show that [ω_{\rm top alg}] is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.19 pagesAlgebraic Topology22A22, 17B56, 18F20, 18G40text