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

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

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Keywords

mapping class group, nonorientable surface, linear representation

Mathematical Subject Classification 2010

Primary: 20F38

Secondary: 57N05

References

Publication

Received: 6 May 2013

Revised: 17 January 2014

Accepted: 31 January 2014

Published: 28 August 2014

Authors

Błażej Szepietowski
Institute of Mathematics Gdańsk University Wita Stwosza 57 80-952 Gdańsk Poland

Volume 14, issue 4 (2014)

Błażej Szepietowski

Abstract