[Paper Review] The twist subgroup of the mapping class group of a nonorientable surface

Michał Stukow|arXiv (Cornell University)|Sep 18, 2007

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper provides a simple generating set for the twist subgroup 𝒯(N) of the mapping class group of a nonorientable surface N_g,s^n (with genus g ≥ 3, s boundary components, n punctures), and computes its first integral homology group (abelianization). The key result is a complete classification of H₁(𝒯(N), ℤ) depending on genus and topological type, showing trivial homology for g ≥ 7, and finite abelian groups for lower genus, with explicit torsion structures such as ℤ₁₂ and ℤ₂₄ × ℤ₂^{s−1} for g = 3.

ABSTRACT

Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).

Motivation & Objective

• To determine a simple and explicit generating set for the twist subgroup 𝒯(N) of the mapping class group of a nonorientable surface N_g,s^n.
• To compute the first integral homology group H₁(𝒯(N), ℤ), i.e., the abelianization of the twist subgroup.
• To extend previous results on mapping class groups of nonorientable surfaces by focusing specifically on the algebraic structure of the twist subgroup.
• To clarify the role of boundary components, punctures, and genus in determining the homology of the twist subgroup.

Proposed method

• Leverages the fact that 𝒯(N) is a subgroup of index 2 in the pure, orientation-preserving mapping class group PM⁺(N), enabling reduction to known results on PM⁺(N).
• Uses topological decomposition of the surface into standard generators (e.g., Dehn twists about specific two-sided curves a₁, b_{r+1}, ξ, u_i) to construct a generating set.
• Applies results from prior work (e.g., Korkmaz [9,10], Szepietowski [18,19]) on presentations and homology of mapping class groups.
• Employs homomorphisms from 𝒯(N) to mapping class groups of closed surfaces (e.g., genus 4, 5, 6) to analyze relations in homology.
• Uses known results on homology of mapping class groups of closed nonorientable surfaces (e.g., Theorem 1.1 of Korkmaz [9]) to deduce torsion relations.
• Applies algebraic techniques to show that all relations in H₁(𝒯(N), ℤ) are consequences of a finite set of relations derived from torsion and commutativity.

Experimental results

Research questions

• RQ1 What is a minimal and explicit generating set for the twist subgroup 𝒯(N) of the mapping class group of a nonorientable surface with boundary and punctures?
• RQ2 How does the first homology group H₁(𝒯(N), ℤ) depend on the genus, number of boundary components, and number of punctures of the surface?
• RQ3 What is the structure of the abelianization of the twist subgroup, and which torsion elements arise?
• RQ4 Are there universal relations among Dehn twists in the twist subgroup that hold regardless of the surface’s complexity?
• RQ5 How do boundary components and punctures affect the homology of the twist subgroup?

Key findings

• For g ≥ 7, the first homology group H₁(𝒯(N), ℤ) is trivial, i.e., 𝒯(N) is perfect.
• For g = 3 and s = 0, H₁(𝒯(N), ℤ) ≅ ℤ₁₂, generated by the Dehn twist about a specific two-sided curve.
• For g = 3 and s ≥ 1, H₁(𝒯(N), ℤ) ≅ ℤ₂₄ × ℤ₂^{s−1}, with the torsion arising from relations involving boundary twists.
• For g = 4 and s = 0, H₁(𝒯(N), ℤ) ≅ ℤ₂ × ℤ, with the ℤ factor arising from a non-torsion generator.
• For g = 4 and s ≥ 1, H₁(𝒯(N), ℤ) ≅ ℤ₂^s × ℤ, reflecting contributions from both boundary and non-separating curves.
• For g = 5 or 6, H₁(𝒯(N), ℤ) ≅ ℤ₂, indicating a single nontrivial torsion class in the abelianization.