  We use Segal-Mitchison's cohomology of topological groups to define a
convenient model for topological gerbes. We introduce multiplicative gerbes
over topological groups in this setup and we define its representations. For a
specific choice of representation, we construct its category of endomorphisms
and we show that it induces a new multiplicative gerbe over another topological
group. This new induced group is fibrewise Pontrjagin dual to the original one
and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We
show that Pontrjagin dual multipliciative gerbes have equivalent categories of
representations and moreover, we show that their monoidal centers are
equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and
discrete, as well as compact and non-compact Lie groups are provided.
