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

Camille Laurent-Gengoux (1) , Friedrich Wagemann (2)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]
2 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: crossed modules of Lie algebroids; crossed module of LIe groupoids; obstruction class; gerbe

Domaines: Topologie algébrique [math.AT]

Camille Laurent-Gengoux, Friedrich Wagemann. Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles. Annals of Global Analysis and Geometry, 2007, 34, pp.21--37. ⟨10.1007/s10455-007-9098-0⟩. ⟨hal-00112323v3⟩