[Paper Review] Generators for the mapping class group of a nonorientable surface

Susumu Hirose|arXiv (Cornell University)|Feb 24, 2016

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper proves that the mapping class group of a nonorientable surface of genus $g \geq 4$ requires at least $g$ Dehn twists and one $Y$-homeomorphism to generate the group, establishing Szepietowski's generating set as minimal. The proof uses the action on homology with $\mathbb{Z}_2$ coefficients and a $\mathbb{Z}_4$-quadratic form invariant under the group action, showing that no smaller set of Dehn twists and $Y$-homeomorphisms can generate the full group.

ABSTRACT

We show that Szepietowski's system of generators for the mapping class group of a non-orientable surface is a minimal generating set by Dehn twists and $Y$-homemorphisms.

Motivation & Objective

• To establish the minimality of Szepietowski's generating set for the mapping class group $\mathcal{M}(N_g)$ of a nonorientable surface of genus $g \geq 4$.
• To determine the minimal number of Dehn twists and $Y$-homeomorphisms required to generate $\mathcal{M}(N_g)$.
• To show that no proper subset of Szepietowski's generators suffices for $g \geq 4$.
• To use the action on $H_1(N_g; \mathbb{Z}_2)$ and $\mathbb{Z}_4$-quadratic forms to rule out smaller generating sets.
• To extend the analogy of Humphries' minimality result for orientable surfaces to the nonorientable case.

Proposed method

• Define $\mathbb{Z}_4$-quadratic forms on $H_1(N_g; \mathbb{Z}_2)$, which satisfy $q(x+y) = q(x) + q(y) + 2\times(x,y)$.
• Use the first Stiefel-Whitney class $w_1$ to distinguish one-sided and two-sided curves in homology.
• Characterize admissible $A$-circles as two-sided curves whose complement is connected and nonorientable.
• Apply the change of coordinates principle to relate curves with isomorphic complements via mapping class group elements.
• Show that Dehn twists act on homology via $x \mapsto x + (x,[c])\cdot[c]$, and $Y$-homeomorphisms act trivially.
• Construct a $\mathbb{Z}_4$-quadratic form preserved by $g-1$ Dehn twists and $Y$-homeomorphisms, but not by the full group, to derive a contradiction if $n < g$.

Experimental results

Research questions

• RQ1 What is the minimal number of Dehn twists and $Y$-homeomorphisms needed to generate $\mathcal{M}(N_g)$ for $g \geq 4$?
• RQ2 Can the generating set of $\mathcal{M}(N_g)$ proposed by Szepietowski be reduced further?
• RQ3 Is there a homological invariant that distinguishes the full mapping class group from subgroups generated by fewer than $g$ Dehn twists and one $Y$-homeomorphism?
• RQ4 Does the action of the mapping class group on $H_1(N_g; \mathbb{Z}_2)$ preserve any $\mathbb{Z}_4$-quadratic form?
• RQ5 Why is $\mathcal{M}(N_g)$ not generated by Dehn twists alone, and how does the $Y$-homeomorphism compensate?

Key findings

• The mapping class group $\mathcal{M}(N_g)$ for $g \geq 4$ requires at least $g$ Dehn twists and one $Y$-homeomorphism to be generated.
• Any generating set of $\mathcal{M}(N_g)$ must include at least one $Y$-homeomorphism, as the group is not generated by Dehn twists alone.
• The system $\{t_{a_i} \mid i=1,\dots,g-1\}, t_{b_2}, Y_{m_{g-1},a_{g-1}}$ is minimal: no proper subset generates $\mathcal{M}(N_g)$.
• For $g=3$, the minimal generating set consists of two Dehn twists and one $Y$-homeomorphism, and no smaller set suffices.
• A $\mathbb{Z}_4$-quadratic form exists that is preserved by $g-1$ Dehn twists and $Y$-homeomorphisms, but no such form is preserved by the full group, proving minimality.
• The action of $\mathcal{M}(N_g)$ on $H_1(N_g; \mathbb{Z}_2)$ cannot preserve any $\mathbb{Z}_4$-quadratic form, which is key to the contradiction in the minimality proof.