Generating mapping class groups of nonorientable surfaces with boundary

Abstract

We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups

Crosscap slides and the level 2 mapping class group of a nonorientable surface

“…Later, Chillingworth [3] found finite generating sets for M(N ) and T (N ). Korkmaz [8,9] and Stukow [19,21] In this paper we consider the subgroup Y(N ) of M(N ) generated by all crosscap slides. Our main result is Theorem 5.5, which asserts that for a closed nonorientable surface N of genus g ≥ 2, Y(N ) is equal to the level 2 subgroup 2 (N ) of M(N ) consisting of those mapping classes which act trivially on H 1 (N ; Z 2 ).…”

On finite index subgroups of the mapping class group of a nonorientable surface

“…The reason is that for n ≤ 1, M(N h,n ) and T (N h,n ) have particularly simple generators. We emphasise, however, that finite generating sets for these groups are known for arbitrary n [15,16]. It is worth mentioning at this point, that the first finite generating set for M(N h,0 ), h ≥ 3, was obtained by Chillingworth [3] using Lickorish's results [8,9].…”

A presentation for the mapping class group of the closed non-orientable surface of genus 4

“…Notice that a 4 , u 1 , u 2 and t are expressed in terms of the remaining generators by the relations (9,15,16,17). Thus the presentation in Theorem 2.1 can be reduced by Tietze transformations to a presentation with generators a 1 , a 2 , a 3 , b and u 3 .…”

“…If F is not closed, then a E-mail address: blaszep@math.univ.gda.pl. finite set of generators for M(F ) was found by Korkmaz [10] if F has punctures, and by Stukow [15] if F has punctures and boundary and g ≥ 3.…”