Low dimensional linear representations of the mapping class group of a nonorientable surface

TL;DR

This paper classifies low-dimensional linear representations of the mapping class group of nonorientable surfaces, showing they mostly factor through abelianization or are finite, with specific exceptions linked to homological representations.

Contribution

It provides a comprehensive classification of low-dimensional linear representations of the mapping class group of nonorientable surfaces, revealing their structure and limitations.

Findings

Representations with dimension ≤ g-2 factor through abelianization.

Finite or conjugate to homological representations in certain cases.

Homomorphisms between different genus mapping class groups factor through abelianization.

Abstract

Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of $\mathcal{M}(N_{g,n})$, which is $\mathbb{Z}_2\times\mathbb{Z}_2$ for $g\in\{5,6\}$ and $\mathbb{Z}_2$ for $g\ge 7$. If $g\ge 7$, $n=0$ and $m=g-1$, then either $f$ has finite image (of order at most two if $g\ne 8$), or it is conjugate to one of four "homological representations". As an application we prove that for $g\ge 5$ and $h<g$, every homomorphism $\mathcal{M}(N_{g,0})\to\mathcal{M}(N_{h,0})$ factors through the abelianization of $\mathcal{M}(N_{g,0})$.