#### Volume 14, issue 4 (2014)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Low-dimensional linear representations of the mapping class group of a nonorientable surface

### Błażej Szepietowski

Algebraic & Geometric Topology 14 (2014) 2445–2474
##### Abstract
[an error occurred while processing this directive]

Suppose that $f$ is a homomorphism from the mapping class group $\mathsc{ℳ}\left({N}_{g,n}\right)$ 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 $\mathsc{ℳ}\left({N}_{g,n}\right)$, which is ${ℤ}_{2}×{ℤ}_{2}$ for $g\in \left\{5,6\right\}$ and ${ℤ}_{2}$ 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, every homomorphism $\mathsc{ℳ}\left({N}_{g,0}\right)\to \mathsc{ℳ}\left({N}_{h,0}\right)$ factors through the abelianization of $\mathsc{ℳ}\left({N}_{g,0}\right)$.

##### Keywords
mapping class group, nonorientable surface, linear representation
Primary: 20F38
Secondary: 57N05