MANY FINITE-DIMENSIONAL LIFTING BUNDLE GERBES ARE TORSION

Many bundle gerbes are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson [‘A note on bundle gerbes and infinite-dimensionality’, J. Aust. Math. Soc.90(1) (2011), 81–92] proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$ -class. I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles and finite-dimensional twists of topological K-theory.

Transgression maps for crossed modules of groupoids

Given a crossed module of groupoids [Formula: see text], we construct (1) a natural homomorphism from the product groupoid [Formula: see text] to the crossed product groupoid [Formula: see text] and (2) a transgression map from the singular cohomology [Formula: see text] of the nerve of the groupoid [Formula: see text] to the singular cohomology [Formula: see text] of the nerve of the crossed product groupoid [Formula: see text]. The latter turns out to be identical to the transgression map obtained by Tu–Xu in their study of equivariant [Formula: see text]-theory.