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Annals of Global Analysis and Geometry

, Volume 34, Issue 1, pp 21–37 | Cite as

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

  • Camille Laurent-Gengoux
  • Friedrich WagemannEmail author
Original Paper

Abstract

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 \(\hat{P}/Z\cong P\) . Neeb (Commun. Algebra 34:991–1041, 2006) 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 article, 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.

Keywords

Crossed modules of Lie algebroids Crossed modules of Lie groupoids Crossed modules of topological Lie algebras Obstruction class Bundle gerbe Deligne cohomology 

Mathematics Subject Classification

55S35 55N35 19JXX 22A22 

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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Université de Poitiers, SP2MI, Boulevard Marie et Pierre CurieFuturoscope-Chasseneuil CedexFrance
  2. 2.Laboratoire de Mathematiques Jean LerayUniversité de NantesNantes Cedex 3France

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