Third group cohomology and gerbes over Lie groups

Abstract

The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space H is given by the third cohomology H 3 ( H , Z ) . When H is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of H . We shall study in more detail this relation in the case of a group extension 1 → N → G → H → 1 when the gerbe is defined by an abelian extension 1 → A → N ˆ → N → 1 of N . In particular, when H s 1 ( N , A ) vanishes we shall construct a transgression map H s 2 ( N , A ) → H s 3 ( H , A N ) , where A N is the subgroup of N -invariants in A and the subscript s denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.