[Paper Review] An infinite presentation for the mapping class group of a non-orientable surface with boundary

Ryoma Kobayashi, Genki Omori|arXiv (Cornell University)|Oct 17, 2016

Geometric and Algebraic Topology17 references3 citations

TL;DR

This paper presents an infinite presentation for the mapping class group of a non-orientable surface with boundary, generalizing a previous result for surfaces with genus $ g \geq 1 $ and $ n \in \{0,1\} $ to arbitrary $ g \geq 0 $ and $ n \geq 0 $. The presentation uses Dehn twists and crosscap pushing maps as generators, extending Gervais' method to non-orientable surfaces via a finite presentation of the mapping class group.

ABSTRACT

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

Motivation & Objective

• To extend the infinite presentation of the mapping class group of non-orientable surfaces from the case $ n \in \{0,1\} $ to arbitrary $ n \geq 0 $.
• To generalize the generating set of Dehn twists and crosscap pushing maps to all non-orientable surfaces with boundary.
• To provide a uniform infinite presentation for $ \mathcal{M}(N_{g,n}) $ that applies regardless of genus or number of boundary components.
• To establish the validity of the presentation using Gervais' method applied to a newly constructed finite presentation of $ \mathcal{M}(N_{g,n}) $.

Proposed method

• The authors use Gervais' technique of deriving infinite presentations from finite presentations of mapping class groups.
• They construct a finite presentation for $ \mathcal{M}(N_{g,n}) $ when $ n \geq 2 $, building on Stukow's finite presentation for $ n \in \{0,1\} $.
• The generating set consists of Dehn twists along two-sided simple closed curves and crosscap pushing maps along simple loops.
• Relations are derived using braid relations, chain relations, and conjugation identities in the mapping class group.
• The proof relies on analyzing the action of mapping classes on certain curves and their neighborhoods, using $ \Delta $-maps and conjugation rules.
• The authors verify that all defining relations of the finite presentation are satisfied in the infinite presentation, ensuring consistency.

Experimental results

Research questions

• RQ1 Can an infinite presentation for $ \mathcal{M}(N_{g,n}) $ be constructed that applies uniformly for all $ g \geq 0 $ and $ n \geq 0 $?
• RQ2 How do Dehn twists and crosscap pushing maps generate the mapping class group of a non-orientable surface with boundary?
• RQ3 What relations govern the interactions between Dehn twists and crosscap pushing maps in $ \mathcal{M}(N_{g,n}) $?
• RQ4 Is Gervais' method for deriving infinite presentations applicable to non-orientable surfaces?
• RQ5 Can a finite presentation for $ \mathcal{M}(N_{g,n}) $ be constructed for $ n \geq 2 $, enabling the extension of infinite presentations?

Key findings

• The paper provides a complete infinite presentation for $ \mathcal{M}(N_{g,n}) $ for all $ g \geq 0 $ and $ n \geq 0 $, generalizing the earlier result for $ n \in \{0,1\} $.
• The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops, forming a natural extension of the orientable case.
• The presentation is shown to be consistent with the finite presentation of $ \mathcal{M}(N_{g,n}) $ for $ n \geq 2 $, as constructed in Proposition 3.2.
• All defining relations of the finite presentation are verified to hold in the infinite presentation using conjugation, braid, and chain relations.
• The proof confirms that the infinite presentation captures the full structure of $ \mathcal{M}(N_{g,n}) $, including cases where the group is trivial or finite (e.g., $ \mathcal{M}(N_1) $, $ \mathcal{M}(N_{1,1}) $, $ \mathcal{M}(N_2) $).
• The result establishes a uniform framework for studying mapping class groups of non-orientable surfaces with boundary using infinite presentations.