Equivariant Bundle Gerbes and Simplicial Extensions

Abstract

Bundle gerbes, introduced in Michael Murray’s 1996 paper “Bundles Gerbes” are a way of geometrically representing degree three integral cohomology for a manifold in the same way that line bundles represent degree two integral cohomology. We want to explore the notion of a bundle gerbe on a simplicial manifold and relate this to simplicial cohomology. We define the simplicial extension of a bundle gerbe and show that appropriate equivalence classes of simplicial extensions are classified by degree two U(1) cohomology and in some cases degree three integral cohomology. One important example we come across is the simplicial space generated by the action of a Lie group on a manifold. These simplicial techniques give us a method of classifying degree three integral equivariant cohomology using the idea of weak actions and strong actions of a Lie group on a bundle gerbe. We introduce the universal strongly equivariant bundle gerbe which is universal in the sense that every strongly equivariant bundle gerbe is a pullback of this bundle gerbe.