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

Szepietowski, Błażej¹

¹Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Algebraic and Geometric Topology, Tome 14 (2014) no. 4, pp. 2445-2474

Résumé

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 < g, every homomorphism ℳ(Ng,0) →ℳ(Nh,0) factors through the abelianization of ℳ(Ng,0).

Détail

DOI : 10.2140/agt.2014.14.2445

Classification : 20F38, 57N05
Keywords: mapping class group, nonorientable surface, linear representation

Bibliographie

[1] J Aramayona, J Souto, Homomorphisms between mapping class groups, Geom. Topol. 16 (2012) 2285

[2] J A Berrick, V Gebhardt, L Paris, Finite index subgroups of mapping class groups, to appear in Proc. London Math. Soc

[3] J S Birman, D R J Chillingworth, On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 71 (1972) 437

[4] J Bray, S Linton, S Norton, R Parker, S Rogers, I Suleiman, J Tripp, P Walsh, R A Wilson, Atlas of finite group representations, (2001)

[5] F Castel, Geometric representations of the braid groups,

[6] D R J Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 65 (1969) 409

[7] D B A Epstein, Curves on $2$–manifolds and isotopies, Acta Math. 115 (1966) 83

[8] J Franks, M Handel, Triviality of some representations of $\mathrm{MCG}(S_g)$ in $\mathrm{GL}(n,\mathbb C)$, $\mathrm{Diff}(S^2)$ and $\mathrm{Homeo}(\mathbb T^2)$, Proc. AMS 141 (2013) 2951

[9] W Fulton, J Harris, Representation theory: A first course, Graduate Texts in Mathematics 129, Springer (1991)

[10] S Gadgil, D Pancholi, Homeomorphisms and the homology of non-orientable surfaces, Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 251

[11] P A Gastesi, A note of the Torelli spaces of non-orientable compact Klein surfaces, Ann. Acad. Sci. Fenn. Math. 24 (1999) 23

[12] F J González-Acuña, J M Márquez-Bobadilla, On the homeotopy group of the non orientable surface of genus three, Rev. Colombiana Mat. 40 (2006) 75

[13] W J Harvey, M Korkmaz, Homomorphisms from mapping class groups, Bull. London Math. Soc. 37 (2005) 275

[14] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[15] M Korkmaz, Low-dimensional linear representations of mapping class groups,

[16] M Korkmaz, The symplectic representation of the mapping class group is unique,

[17] M Korkmaz, First homology group of mapping class groups of nonorientable surfaces, Math. Proc. Cambridge Philos. Soc. 123 (1998) 487

[18] M Korkmaz, Low-dimensional homology groups of mapping class groups: A survey, Turkish J. Math. 26 (2002) 101

[19] M Korkmaz, Problems on homomorphisms of mapping class groups, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 81

[20] M Korkmaz, J D Mccarthy, Surface mapping class groups are ultrahopfian, Math. Proc. Cambridge Philos. Soc. 129 (2000) 35

[21] W B R Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963) 307

[22] J D Mccarthy, U Pinkall, Representing homology automorphisms of nonorientable surfaces, preprint MPI/SFB 85–11 (2004)

[23] L Paris, B Szepietowski, A presentation for the mapping class group of a nonorientable surface,

[24] J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347

[25] M Stukow, Generating mapping class groups of nonorientable surfaces with boundary, Adv. Geom. 10 (2010) 249

[26] B Szepietowski, Mapping class group of a non-orientable surface and moduli space of Klein surfaces, C. R. Math. Acad. Sci. Paris 335 (2002) 1053

[27] B Szepietowski, A presentation for the mapping class group of the closed non-orientable surface of genus $4$, J. Pure Appl. Algebra 213 (2009) 2001

[28] B Szepietowski, Embedding the braid group in mapping class groups, Publ. Mat. 54 (2010) 359

[29] B Szepietowski, Crosscap slides and the level $2$ mapping class group of a nonorientable surface, Geom. Dedicata 160 (2012) 169