Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus

TL;DR

This paper develops a novel enumeration technique for unlabelled orientable maps on surfaces of a given genus, considering all homeomorphisms, including orientation-reversing ones, and introduces quotient maps as an intermediate step.

Contribution

It provides the first enumeration of unlabelled maps on orientable surfaces considering all homeomorphisms and develops a new method involving quotient maps for this purpose.

Findings

Enumeration of unlabelled maps on orientable surfaces with all homeomorphisms.

Introduction of quotient maps on surfaces with boundaries and branch points.

Development of a specialized enumeration technique for these maps.

Abstract

In this work for the first time we enumerate unlabelled maps on orientable genus $g$ surfaces with respect to all homeomorphisms, including both orientation-preserving and orientation-reversing. We show that in the latter case as an intermediate step one has to enumerate rooted maps of a special kind (quotient maps) on orientable and non-orientable surfaces possibly having a boundary and a certain number of branch points. In this work we develop a special technique for enumerating such maps.