Cohomological Constructions of Regular Cyclic Coverings of the Platonic Maps

A Platonic map is a regular map M on the sphere S. Following [CMo] we say that M has type {n,m} if it has n-gonal faces and the vertices have valency m; since these parameters determine a Platonic map uniquely, one can unambiguously write M = {n,m}. As one must have 0 ≤ (m − 2)(n − 2) < 4, there are precisely the possibilities M = {n, 2} (dihedron), M = {2,m} (hosohedron), M = {3, 3} (tetrahedron), M = {4, 3} (cube), M = {3, 4} (octahedron), M = {5, 3} (dodecahedron) and M = {3, 5} (icosahedron). In [JS] we determined the regular maps which occur as d-sheeted coverings ofM, branched over the vertices, edge-midpoints or face-centers, with a cyclic group of covering transformations. In case the ramification is over the face-centers of M, the parametrization is as follows: