[Paper Review] A finite presentation for the hyperelliptic mapping class group of a nonorientable surface

Michał Stukow|arXiv (Cornell University)|Feb 17, 2014

Geometric and Algebraic Topology11 references4 citations

TL;DR

This paper provides a finite presentation for the hyperelliptic mapping class group of a nonorientable surface, extending the classical theory from orientable to nonorientable surfaces. Using a centralizer construction via a double cover, it derives explicit generators and relations, and applies this to compute the first homology group with coefficients in the first homology of the surface, yielding $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ for the orientation-preserving subgroup and $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ for the full hyperelliptic group when genus $g \geq 3$. The results generalize known results from the orientable case to the nonorientable setting.

ABSTRACT

We obtain a simple presentation of the hyperelliptic mapping class group $M^h(N)$ of a nonorientable surface N. As an application we compute the first homology group of $M^h(N)$ with coefficients in $H_1(N;Z)$.

Motivation & Objective

• To extend the concept of the hyperelliptic mapping class group from orientable to nonorientable surfaces.
• To provide a finite presentation for the hyperelliptic mapping class group $\mathcal{M}^h(N_g)$ of a closed nonorientable surface of genus $g$.
• To compute the first homology group of $\mathcal{M}^h(N_g)$ with coefficients in $H_1(N_g;\mathbb{Z})$.
• To establish a structural analogy between the nonorientable and orientable cases, particularly in terms of group presentations and homological invariants.

Proposed method

• Define the hyperelliptic mapping class group $\mathcal{M}^h(N_g)$ as the centralizer of the image of the hyperelliptic involution $\varrho$ in the mapping class group $\mathcal{M}(N_g)$, derived from a double cover of an orientable surface.
• Use the centralizer construction via the involution $j$ on $S_{g-1}$ to lift the structure from the orientable case to the nonorientable setting.
• Derive a finite presentation for $\mathcal{M}^h(N_g)$ and its orientation-preserving subgroup $\mathcal{M}^{h+}(N_g)$ using generators corresponding to Dehn twists and the hyperelliptic involution.
• Apply the presentation to compute homology by analyzing relations in the group ring $\mathbb{Z}[\mathcal{M}^h(N_g)]$ acting on $H_1(N_g;\mathbb{Z})$, focusing on torsion in the first homology group.
• Use algebraic relations, including those involving the hyperelliptic involution $\varrho$, to determine the order of generators in homology and eliminate redundant ones.
• Leverage known results from the orientable case (e.g., Wajnryb’s presentation) and adapt them to the nonorientable context through the quotient map $\pi_j$.

Experimental results

Research questions

• RQ1 What is a finite presentation for the hyperelliptic mapping class group of a nonorientable surface?
• RQ2 How does the structure of the hyperelliptic mapping class group on nonorientable surfaces compare to that on orientable surfaces?
• RQ3 What is the first homology group of $\mathcal{M}^h(N_g)$ with coefficients in $H_1(N_g;\mathbb{Z})$?
• RQ4 Are the generators of the homology group torsion, and if so, what is their order?
• RQ5 Can the presentation of the hyperelliptic group be used to recover or generalize known results from the orientable case?

Key findings

• The hyperelliptic mapping class group $\mathcal{M}^h(N_g)$ admits a finite presentation with explicit generators and relations, extending the known presentation for the orientable case.
• The orientation-preserving subgroup $\mathcal{M}^{h+}(N_g)$ has a finite presentation with generators corresponding to Dehn twists and the hyperelliptic involution.
• For $g \geq 3$, the first homology group $H_1(\mathcal{M}^{h+}(N_g); H_1(N_g;\mathbb{Z}))$ is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$, indicating nontrivial torsion in the homology.
• For the full hyperelliptic group $\mathcal{M}^h(N_g)$, the first homology group with coefficients in $H_1(N_g;\mathbb{Z})$ is $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$, showing an additional torsion component.
• The hyperelliptic involution $\varrho$ plays a central role in the relations, and its action helps eliminate redundant generators in the homology computation.
• The results generalize known results from the orientable case: while $H_1(\mathcal{M}^h(S_g); H_1(S_g;\mathbb{Z})) \cong \mathbb{Z}_2$ for orientable surfaces, the nonorientable case yields a larger torsion group.