Geometric representations of the braid group on a nonorientable surface

TL;DR

This paper classifies homomorphisms from the braid group to the pure mapping class group of nonorientable surfaces, revealing that for large n and small genus, such homomorphisms are either cyclic or map generators to specific geometric transformations.

Contribution

It provides a complete classification of these homomorphisms under certain conditions, identifying their structure in terms of Dehn twists and crosscap transpositions.

Findings

Homomorphisms are either cyclic or map generators to Dehn twists or crosscap transpositions.

Classification holds for n ≥ 14 and g ≤ 2⌊n/2⌋+1.

Homomorphisms may be multiplied by elements in the centralizer of the image.

Abstract

We classify homomorphisms from the braid group on $n$ strands to the pure mapping class group of a nonoriantable surface of genus $g$. For $n\ge 14$ and $g\le 2\lfloor{n/2}\rfloor+1$ every such homomorphism is either cyclic, or it maps standard generators of the braid group to either distinct Dehn twists, or distinct crosscap transpositions, possibly multiplied by the same element of the centralizer of the image.