  A detailed analysis of Chern-Simons (CS) theories in which a compact abelian
direct product gauge group U(1)^k is spontaneously broken down to a direct
product H of (finite) cyclic groups is presented. The spectrum features global
H charges, vortices carrying flux labeled by the elements of H and dyonic
combinations. Due to the Aharonov-Bohm effect these particles exhibit
toplogical interactions. The remnant of the U(1)^k CS term in the discrete H
gauge theory describing the effective long distance physics of such a model is
shown to be a 3-cocycle for H summarizing the nontrivial topological
interactions cast upon the magnetic vortices by the U(1)^k CS term. It is noted
that there are in general three types of 3-cocycles for a finite abelian gauge
group H: one type describes topological interactions among vortices carrying
flux w.r.t. the same cyclic group in the direct product H, another type gives
rise to topological interactions between vortices carrying flux w.r.t. two
different cyclic factors of H and a third type leading to topological
interactions between vortices carrying flux w.r.t. three different cyclic
factors. Among other things, it is demonstrated that only the first two types
can be obtained from a spontaneously broken U(1)^k CS theory. The 3-cocycles
that can not be reached in this way turn out to be the most interesting. They
render the theory nonabelian and in general lead to dualities with planar
theories with a nonabelian finite gauge group. In particular, the CS theory
with finite gauge group H = Z_2 x Z_2 x Z_2 defined by such a 3-cocycle is
shown to be dual to the planar discrete D_4 gauge theory with D_4 the dihedral
group of order 8.
