## Algebraic & Geometric Topology

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

Błażej Szepietowski

#### Abstract

Suppose that $f$ is a homomorphism from the mapping class group $ℳ(Ng,n)$ of a nonorientable surface of genus $g$ with $n$ boundary components to $GL(m,ℂ)$. We prove that if $g≥5$, $n≤1$ and $m≤g−2$, then $f$ factors through the abelianization of $ℳ(Ng,n)$, which is $ℤ2×ℤ2$ for $g∈{5,6}$ and $ℤ2$ for $g≥7$. If $g≥7$, $n=0$ and $m=g−1$, then either $f$ has finite image (of order at most two if $g≠8$), or it is conjugate to one of four “homological representations”. As an application we prove that for $g≥5$ and $h, every homomorphism $ℳ(Ng,0)→ℳ(Nh,0)$ factors through the abelianization of $ℳ(Ng,0)$.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2445-2474.

Dates
Revised: 17 January 2014
Accepted: 31 January 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715969

Digital Object Identifier
doi:10.2140/agt.2014.14.2445

Mathematical Reviews number (MathSciNet)
MR3331618

Zentralblatt MATH identifier
1304.57031

Subjects
Primary: 20F38: Other groups related to topology or analysis
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

#### Citation

Szepietowski, Błażej. Low-dimensional linear representations of the mapping class group of a nonorientable surface. Algebr. Geom. Topol. 14 (2014), no. 4, 2445--2474. doi:10.2140/agt.2014.14.2445. https://projecteuclid.org/euclid.agt/1513715969

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