40 - Smooth coverings of regular maps

Summary

Abstract

This paper describes a method, algorithmic in part, for determining thoseregular maps N which are smooth covers of a given regular map M. In the case where the base map M is reflexible, the method is able to distinguish between the chiral and the reflexible covers. The method is illustrated with an important example, M=k₂(O), a 9-fold covering of which is one of the two smallest chiral maps with triangular faces.

Preliminaries

The paper contains a fuller account of the following definitions and preliminary results: a map M is an embedding of a (very general) graph into a surface. We consider the map to be barycentrically subdivided into triangular regions called flags, and choose one flag to be special; this is the root flag, I. Each flag f has three neighbors, denoted fr₀, fr₁, fr₂ as in Figure 1:

Then r₀, r₁, r₂ are permutations on Ω, the collection of flags. These three involutions generate a group C, the connection group. A symmetry of M is a permutation of ω which commutes with C, and G(M) is the group of all symmetries. We want to discuss two kinds of regularity: (1) M is rotary if G(M) has symmetries R and S which send the root flag I to Ir₁r₀ and Ir₁r₂, respectively. G⁺(M) is the subgroup of G(M) generated by R and S. This is a (2,p, q) group, where p and q are the orders of R and S respectively. (2) If M is rotary, then it is reflexible provided that G(M)also contains a symmetry X which sends I to Ir₁, and otherwise it is chiral.