  The geometry of antisymmetric fields with nontrivial transitions over a base
manifold is described in terms of exact sequences of cohomology groups. This
formulation leads naturally to the appearance of nontrivial topological charges
associated to the periods of the curvature of the antisymmetric fields. The
relation between the partition functions of dual theories is carefully studied
under the most general assumptions, and new topological factors related to zero
modes and the Ray Singer torsion are found.
