Liftings of automorphisms of hypermaps

If H is a hypermap with bit set B, a voltage assignment on H is simply a map z : B → Z, where Z is a group. If Z acts on a set X, and if z is a voltage assignment on H, then one constructs a “ramified covering” Hz(X) → H such that in the category of ramified coverings of H, Hz(X) and Hz′(X) are isomorphic whenever z and z′ are equivalent voltages (but not conversely). A lifting criterion is given for an automorphism of H to lift to one of Hz(X) and applied to give a simple and conceptual proof of the hypermap version of Accola’s theorem for hypermaps. Finally, cohomological aspects of these constructions are given in terms of Machi’s homology theory for hypermaps.