Classification of principal bundles and lie groupoids with prescribed gauge group bundle

Abstract

By the gauge group bundle of a principal bundle P( B,G) we mean the Lie group bundle associated to P( B,G) through the conjugacy action of G on itself. Given only B and a Lie group bundle M on B, we ask if there exists P( B,G) with gauge group bundle isomorphic to M and, if so, how they can be described. Using a form of Whitehead's concept of crossed module, in place of the idea of an ‘abstract kernel’, we find an obstruction class in Ȟ 2( B, ZG ) ( G the fibre-type of M) whose vanishing gives a necessary and sufficient condition for the existence of such a P( B,G); and, when this class vanishes, a simple transitive action of Ȟ 1( B, ZG ) on the set of equivalence classes of possible bundles. We work mainly in terms of Lie groupoids, which language seems well-adapted to these questions.