MANY FINITE-DIMENSIONAL LIFTING BUNDLE GERBES ARE TORSION

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that far from being fibre bundles. Murray and Stevenson proved on simply-connected manifolds, bundles with connected fibres, always have a torsion $DD$-class. In this note I prove an analogous result for wide class of principal bundles, relaxing the requirements fundamental group base components fibre, allowing both to be nontrivial. This has consequences possible models basic gerbes, classification crossed modules Lie groups, coefficient Lie-2-algebras higher gauge theory 2-bundles, twists topological $K$-theory.