Algebraic & Geometric Topology

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

Błażej Szepietowski

Full-text: Open access

Abstract

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 g5, n1 and mg2, then f factors through the abelianization of (Ng,n), which is 2×2 for g{5,6} and 2 for g7. If g7, n=0 and m=g1, 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 g5 and h<g, every homomorphism (Ng,0)(Nh,0) factors through the abelianization of (Ng,0).

Article information

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

Dates
Received: 6 May 2013
Revised: 17 January 2014
Accepted: 31 January 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
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

Keywords
mapping class group nonorientable surface linear representation

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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