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

Abstract

Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{ℳ}\left(N_{g,n}\right)$ of a nonorientable surface of genus $g$ with $n$ boundary components to $GL\left(m,ℂ\right)$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of $\mathcal{ℳ}\left(N_{g,n}\right)$, which is ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for $g\in \left\{5,6\right\}$ 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 $\mathcal{ℳ}\left(N_{g,0}\right)\to \mathcal{ℳ}\left(N_{h,0}\right)$ factors through the abelianization of $\mathcal{ℳ}\left(N_{g,0}\right)$.