OBSERVATIONS ON THE CONSTRUCTION OF COVERS USING PERMUTATION VOLTAGE ASSIGNMENTS

In the usual treatment of covering spaces, the algebraic object associated with a cover is a conjugacy class of subgroups of the fundamental group of the target space. For finite degree covers, one could as easily choose the algebraic object to be a class of homomorphisms from the fundamental group of the target space to the symmetric group on n letters, where n is the degree of the cover. A description of the representation of covering spaces is given in this paper. Then the permutation voltage assignment used in the permutation voltage construction of graph and surface covers described in [6] is viewed as a means of defining one such homomorphism. Using this view, methods are given for counting the number of connected components in the constructed cover and for determining the orientability of the surface cover. In addition, it is observed that every branched surface cover can be constructed by permutation voltage assignments. © 1979.