2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66417In this paper we construct abelian extensions of the group of diffeomorphisms of a torus. We consider the jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of a central 2-cocycle on a gauge group turns out to be an abelian cocycle on the group of diffeomorphisms. We show that in the case of a circle, the Virasoro-Bott cocycle is a pull-back of the Heisenberg cocycle. We also give an abelian generalization of the Virasoro-Bott cocycle to the case of a manifold with a volume form.14 pagesGroup Theory22E65; 58B25; 81R10text