# Generating the mapping class group of a nonorientable surface by crosscap transpositions

## TL;DR

This paper proves that the mapping class group of a nonorientable surface of genus at least 7 can be generated by conjugates of a single crosscap transposition, providing explicit generating sets for certain cases.

## Contribution

It establishes that a small set of crosscap transpositions generates the entire mapping class group of nonorientable surfaces, advancing understanding of their algebraic structure.

## Key findings

- Mapping class group of genus ≥7 generated by conjugates of one crosscap transposition
- Explicit generating sets of size g+2 for closed or one-boundary surfaces
- Simplifies the understanding of the algebraic structure of these groups

## Abstract

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus $g\ge 7$ is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of $g+2$ crosscap transpositions generating the mapping class group.