Elementary abelian regular coverings of Platonic maps, Case I: ordinary representations

TL;DR

This paper classifies certain symmetric maps on surfaces that are coverings of Platonic maps, focusing on cases where the covering and rotation groups are coprime, using representation theory of homology groups.

Contribution

It provides a complete classification of elementary abelian regular coverings of Platonic maps with coprime covering and rotation group orders, using homology and representation theory.

Findings

Complete classification for coverings over faces or vertices of Platonic maps.

Method extends to other branching patterns.

Uses homology representations to analyze regular coverings.

Abstract

We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the representations of G on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of M, and outline how the method extends to other branching patterns.