Many finite-dimensional lifting bundle gerbes are torsion
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(Accepted version)
Date
2022
Authors
Roberts, D.M.
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Journal article
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Bulletin of the Australian Mathematical Society, 2022; 105(2):323-338
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David Michael Roberts
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Abstract
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.
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First published online 17 September 2021
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© 2021 Australian Mathematical Publishing Association Inc.