Many finite-dimensional lifting bundle gerbes are torsion

TL;DR

This paper extends the understanding of finite-dimensional lifting bundle gerbes, showing they are often torsion even under relaxed conditions, impacting models in higher gauge theory and topological K-theory.

Contribution

It proves that a broad class of finite-dimensional gerbes built from principal bundles are torsion, relaxing previous restrictions on the base's fundamental group and fibre components.

Findings

Gerbes built from principal bundles are often torsion.

Relaxed conditions still guarantee torsion properties.

Implications for higher gauge theory and K-theory models.

Abstract

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$-class. In this note 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.