Volume 14, issue 4 (2014)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Low-dimensional linear representations of the mapping class group of a nonorientable surface

Błażej Szepietowski

Algebraic & Geometric Topology 14 (2014) 2445–2474
Abstract
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Suppose that f is a homomorphism from the mapping class group (Ng,n) of a nonorientable surface of genus g with n boundary components to GL(m, ). We prove that if g 5, n 1 and m g 2, then f factors through the abelianization of (Ng,n), which is 2 × 2 for g {5,6} and 2 for g 7. If g 7, n = 0 and m = g 1, then either f has finite image (of order at most two if g8), or it is conjugate to one of four “homological representations”. As an application we prove that for g 5 and h < g, every homomorphism (Ng,0) (Nh,0) factors through the abelianization of (Ng,0).

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