A presentation for the mapping class group of the closed non-orientable surface of genus 4

TL;DR

This paper derives a finite presentation for the mapping class group of a closed non-orientable surface of genus 4 by analyzing its action on the ordered complex of curves, extending known results beyond genus 3.

Contribution

It provides the first finite presentation for the mapping class group of a genus 4 non-orientable surface, expanding the understanding of these groups.

Findings

Finite presentation obtained for genus 4 non-orientable surface

Method based on action on the ordered complex of curves

Extends known results from genus at most 3

Abstract

Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-orientable surface of genus 4 from its action on the so called ordered complex of curves.