Branched coverings of maps and lifts of map homomorphisms

In this article we generalize both ordinary and permutation voltage constructions to obtain all branched coverings of maps. We approach a map as a set of flags together with three fixed-point-free involutions and relate this approach with other standard representations. We define a lift as a function from these flags into a group. Ordinary voltage and ordinary current assignments are special cases of our lifts. We interpret our construction as an assignment of voltages to the corners of an embedded graph. We describe a simple necessary and sufficient condition for a map homomorphism of base graphs to lift to a homomorphism of covering graphs. As an application we construct centrally-symmetric self-dual spherical polyhedra.