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On torsion in the homology of the Torelli group

Andrei Vladimirov

TL;DR

The paper investigates torsion in the homology of the Torelli group $\mathcal{I}_g$ via abelian cycles generated by Dehn twists about disjoint separating curves. It shows that the subgroup generated by these abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space and proves finiteness results, notably that $H_2^{\mathrm{ab,sep}}(\mathcal{I}_g)$ is finite-dimensional for $g\ge4$, with a vanishing criterion tied to cutting along the curves $\delta_i$. The Birman–Craggs–Johnson homomorphism and BP-relations reduce abelian-cycle relations to linear-algebraic data in a small Boolean algebra, enabling a finite classification. The work extends these torsion phenomena to higher homology $H_k(\mathcal{I}_g)$, establishing $2$-torsion and finiteness statements for a broad range of $k$ and providing criteria for nontrivial cycles via wedge-products of BCJ images. Together, these results advance understanding of the (often torsion) structure of Torelli-group homology and illuminate finite-generation questions for $H_k(\mathcal{I}_g)$.

Abstract

Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act trivially on $H_1(S_g)$. For any collection of pairwise disjoint, separating simple closed curves on $S_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$, which is called an abelian cycle. We prove that the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space for all $k$, and that it is finite-dimensional for $k = 2$ and $g \geq 4$.

On torsion in the homology of the Torelli group

TL;DR

The paper investigates torsion in the homology of the Torelli group via abelian cycles generated by Dehn twists about disjoint separating curves. It shows that the subgroup generated by these abelian cycles is a -vector space and proves finiteness results, notably that is finite-dimensional for , with a vanishing criterion tied to cutting along the curves . The Birman–Craggs–Johnson homomorphism and BP-relations reduce abelian-cycle relations to linear-algebraic data in a small Boolean algebra, enabling a finite classification. The work extends these torsion phenomena to higher homology , establishing -torsion and finiteness statements for a broad range of and providing criteria for nontrivial cycles via wedge-products of BCJ images. Together, these results advance understanding of the (often torsion) structure of Torelli-group homology and illuminate finite-generation questions for .

Abstract

Let be a closed, oriented surface of genus , and let denote its mapping class group. The Torelli group is the subgroup of consisting of mapping classes that act trivially on . For any collection of pairwise disjoint, separating simple closed curves on , the corresponding Dehn twists pairwise commute and determine a homology class in , which is called an abelian cycle. We prove that the subgroup of generated by such abelian cycles is a -vector space for all , and that it is finite-dimensional for and .

Paper Structure

This paper contains 8 sections, 30 theorems, 100 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Basic relations in $\mathop{\mathrm{Mod}}\nolimits(S)$ and $\mathop{\mathrm{\mathcal{I}}}\nolimits(S)$
  3. Abelian cycles have order $2$ for $k = 2$ and $g \geq 4$
  4. The BP-relation in $H_k(\mathop{\mathrm{\mathcal{I}}}\nolimits_g)$
  5. Main relations in $H_k^{\mathrm{ab,sep}}(\mathop{\mathrm{\mathcal{I}}}\nolimits_g)$
  6. Birman--Craggs--Johnson homomorphism and relations in $H_2(\mathop{\mathrm{\mathcal{I}}}\nolimits_g)$
  7. Proof of Theorem \ref{['thm:main-2']}
  8. Abelian cycles in $H_k(\mathop{\mathrm{\mathcal{I}}}\nolimits_g)$

Key Result

Theorem 1.1

For $g \geq 3$ and $k \geq 2$, the following hold:

Figures (7)

  • Figure 1: Two views of the lantern relation.
  • Figure 2: The curves $\varepsilon, \varepsilon', \varepsilon"$ on $S_g$.
  • Figure 3: The surface $\Sigma_1$ and the curves on $\Sigma_1$.
  • Figure 4: The curves $\alpha, \alpha', \beta, \beta'$ on $S_g$.
  • Figure 5: The surface $\Sigma$ and the curves $\gamma_1, \gamma_2, \theta$ on $\Sigma$.
  • ...and 2 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3: Lantern relation
  • ...and 50 more