2006/11/30 by Camille Laurent-Gengoux, Friedrich Wagemann
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Characteristic class #Cohomology #Differential geometry #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Lie group #Mathematics #Pure mathematics #Torsion (gastropod) #math.AT #msc:17B56 #msc:18F20 #msc:18G40 #msc:22A22
paper · pdf · doi:10.1007/s10455-007-9098-0
published as Annals of Global Analysis and Geometry 34 (2007) 21--37 · 19 pages
arxiv created 2007/03/19 · openalex publication_date 2007/12/18 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension 1→ Z→ K→ K→ 1 of K. It is a classical question whether there exists a K-principal bundle P on M such that 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 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.