Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Details Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

2006-11-08

arxiv.org/abs/math/0611226v3

Annals of Global Analysis and Geometry 34 (2007) 21--37

Mathematics

Algebraic Topology

19 pages

Scientific paper

10.1007/s10455-007-9098-0

Let 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 [\omega_{\rm top alg}] is an obstruction to the existence of \hat{P}. In the present paper, we show that [\omega_{\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.

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