Many Finite-Dimensional Lifting Bundle Gerbes Are Torsion. 2
Many finite-dimensional lifting bundle gerbes are 1 torsion. 1 This document is released under a Creative Commons Zero license 2 David Michael Roberts creativecommons.org/publicdomain/zero/1.0/ 2 orcid.org/0000-0002-3478-0522 April 26, 2021 Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from be- ing 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. 1 Introduction A bundle gerbe [Mur96] is a geometric object that sits over a given space or manifold X classified by elements of H3(X, Z), in the same way that (complex) line bundles on X are classified by elements of H2(X, Z). And, just as line bundles on manifolds have connections 3 giving rise to curvature, a 2-form, giving a class in HdR(X), bundle gerbes have a notion of geometric ‘connection’ data, with curvature a 3 3-form and hence a class in HdR(X). Since de Rham cohomology sees only the non-torsion part of integral cohomology, bundle gerbes that are classified by torsion classes in H3 are thus trickier in one sense to ‘see’ geometrically.
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