Third group cohomology and gerbes over Lie groups

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 HH is given by the third cohomology H3(H,Z). When HH 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 HH. We shall study in more detail this relation in the case of a group extension 1→N→G→H→11→N→G→H→1 when the gerbe is defined by an abelian extension 1→A→Nˆ→N→1 of NN. In particular, when Hs1(N,A) vanishes we shall construct a transgression map Hs2(N,A)→Hs3(H,AN), where ANAN is the subgroup of NN-invariants in AA and the subscript ss denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.