Unstable maps

TL;DR

This paper investigates the stability of maps, focusing on their automorphisms and how they relate to their orientable double covers, revealing new families of unstable maps and connections to other mathematical structures.

Contribution

It introduces several infinite families of unstable maps and explores their relationships with graphs, hypermaps, and Klein surfaces.

Findings

Identification of infinite families of unstable maps

Relationship between unstable maps and automorphisms of their double covers

Connections to similar phenomena in graphs, hypermaps, and Klein surfaces

Abstract

A map which is non-orientable or has non-empty boundary has a canonical double cover which is orientable and has empty boundary. The map is called stable if every automorphism of this cover is a lift of an automorphism of the map. This note describes several infinite families of unstable maps, and relates them to similar phenomena for graphs, hypermaps and Klein surfaces.