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

Luis Paris, Błażej Szepietowski|arXiv (Cornell University)|Aug 27, 2013

Geometric and Algebraic Topology27 references10 citations

TL;DR

This paper provides the first explicit finite presentation for the mapping class group of a nonorientable surface $\mathcal{M}(N_{g,n})$ when $n \in \{0,1\}$ and $g+n > 3$. Using an algorithm based on Brown's method and the action on the curve complex, the authors construct presentations by combining Dehn twists from an embedded orientable surface and crosscap transpositions from a braid-like structure, with relations categorized into three families: Dehn twist relations, braid relations, and mixed relations.

ABSTRACT

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

Motivation & Objective

• To provide an explicit finite presentation for the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface with genus $g$ and $n \in \{0,1\}$ boundary components.
• To extend the known finite presentations of mapping class groups from orientable to nonorientable surfaces, where such presentations were previously known only for small genus.
• To establish a uniform method for constructing presentations using the action of $\mathcal{M}(N_{g,n})$ on the ordered curve complex.
• To prove the presentation via induction, using base cases $ (g,n) = (3,1) $ and $ (4,0) $, and verifying relations through stabilizer computations and simplicial complex analysis.

Proposed method

• Apply an algorithm based on Brown's method for computing group presentations from group actions on a simply connected complex.
• Use the action of $\mathcal{M}(N_{g,n})$ on the ordered complex of curves to compute stabilizers of vertices and edges.
• Construct $\mathcal{M}(N_{g,1})$ as a quotient of the free product $\mathcal{M}(S_{\rho,r}) \ast \mathcal{M}(S_{0,1}, \mathcal{P}_g) $, where $g = 2\rho + r$, $r \in \{1,2\}$.
• Identify generators: Dehn twists from $\mathcal{M}(S_{\rho,r})$ and $g-1$ crosscap transpositions from the braid group $\mathcal{M}(S_{0,1}, \mathcal{P}_g)$.
• Derive three families of relations: (A) Dehn twist relations in $\mathcal{M}(S_{\rho,r})$, (B) braid relations among crosscap transpositions, and (C) mixed relations between the two types.
• Obtain $\mathcal{M}(N_{g,0})$ from $\mathcal{M}(N_{g,1})$ by adding three specific relations, completing the presentation for the closed case.

Experimental results

Research questions

• RQ1 Can a finite presentation be explicitly constructed for $\mathcal{M}(N_{g,n})$ when $n \in \{0,1\}$ and $g+n > 3$?
• RQ2 How can the mapping class group of a nonorientable surface be decomposed algebraically using embeddings of orientable surfaces and braid-like structures?
• RQ3 What are the defining relations that govern the interaction between Dehn twists and crosscap transpositions in $\mathcal{M}(N_{g,n})$?
• RQ4 How does the action of $\mathcal{M}(N_{g,n})$ on the ordered curve complex facilitate the derivation of a finite presentation?
• RQ5 What is the relationship between the mapping class group of a nonorientable surface and the mapping class group of its associated double cover?

Key findings

• The paper provides an explicit finite presentation for $\mathcal{M}(N_{g,1})$ for all $g > 3$, realized as a quotient of the free product $\mathcal{M}(S_{\rho,r}) \ast \mathcal{M}(S_{0,1}, \mathcal{P}_g)$ with $g = 2\rho + r$, $r \in \{1,2\}$.
• The presentation for $\mathcal{M}(N_{g,1})$ is given by three families of relations: (A) Dehn twist relations from $\mathcal{M}(S_{\rho,r})$, (B) braid relations among $g-1$ crosscap transpositions, and (C) mixed relations between Dehn twists and crosscap transpositions.
• The presentation for $\mathcal{M}(N_{g,0})$ is obtained from $\mathcal{M}(N_{g,1})$ by adding exactly three additional relations.
• The base cases $\mathcal{M}(N_{3,1})$ and $\mathcal{M}(N_{4,0})$ are verified in Section 4, serving as the foundation for the inductive proof.
• The proof of the main theorem relies on computing stabilizers of vertices and edges in the ordered curve complex and verifying relations for simplices of dimension 1 and 2.
• The authors show that all relations (2)–(9) and (H1)–(H6) are consequences of the initial relations (0), (1), and (H1)–(H6), confirming the consistency of the presentation.