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On the second homology of the genus 3 hyperelliptic Torelli group

Igor Spiridonov

TL;DR

This work analyzes the second homology ${\rm H}_2({\mathcal{SI}}_3,\mathbb{Z})$ of the genus 3 hyperelliptic Torelli group, focusing on simple abelian cycles generated by disjoint symmetric separating curves. It establishes a precise correspondence between these cycles and symmetric orthogonal splittings of ${\rm H}_1(\Sigma_3;\mathbb{Z})$ satisfying an Arf-type condition, and proves their linear independence modulo a sign relation. The paper develops a refined Euclidean algorithm, verifies injectivity and image structure for the simple cycles, and uses Morita invariants, the complex of symmetric cycles, and a group-action spectral sequence to prove independence. The results clarify the topology of ${\mathcal{SI}}_3$ and provide a concrete, infinite family of independent elements in ${\rm H}_2({\mathcal{SI}}_3,\mathbb{Z})$, contributing to the understanding of Torelli-type groups at low genus and suggesting directions for broader genus behavior.

Abstract

Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $Σ_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g)$ consisting of elements that act trivially on $\mathrm{H}_1(Σ_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(Σ_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$.

On the second homology of the genus 3 hyperelliptic Torelli group

TL;DR

This work analyzes the second homology of the genus 3 hyperelliptic Torelli group, focusing on simple abelian cycles generated by disjoint symmetric separating curves. It establishes a precise correspondence between these cycles and symmetric orthogonal splittings of satisfying an Arf-type condition, and proves their linear independence modulo a sign relation. The paper develops a refined Euclidean algorithm, verifies injectivity and image structure for the simple cycles, and uses Morita invariants, the complex of symmetric cycles, and a group-action spectral sequence to prove independence. The results clarify the topology of and provide a concrete, infinite family of independent elements in , contributing to the understanding of Torelli-type groups at low genus and suggesting directions for broader genus behavior.

Abstract

Let be a fixed hyperelliptic involution of the closed, oriented genus surface . The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on and commute with . It is generated by Dehn twists about -invariant separating curves, and its cohomological dimension is . In this paper we study the top homology group . For each pair of disjoint -invariant separating curves there is a naturally associated abelian cycle in ; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in .
Paper Structure (19 sections, 18 theorems, 66 equations, 3 figures)

This paper contains 19 sections, 18 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hyperelliptic Torelli Group
  3. Results
  4. History
  5. Structure of the paper
  6. Acknowledgements
  7. Preliminaries on symmetric mapping class groups
  8. Birman-Hilden theorem
  9. Birman-Craggs homomorphisms
  10. Classification of the simple abelian cycles
  11. A refined Euclidean algorithm
  12. Proof of Proposition \ref{['inj']}
  13. Proof of Proposition \ref{['image']}
  14. Proof of the linear independence
  15. Morita homomorphisms
  16. ...and 4 more sections

Key Result

Proposition 1.1

Let $(\gamma, \delta)$ and $(\gamma', \delta')$ be two symmetric separating pairs, such that the corresponding orthogonal splittings of ${\rm H}$ coincide. Then $(\gamma, \delta)$ and $(\gamma', \delta')$ are $\mathcal{SI}_3$-equivalent.

Figures (3)

  • Figure 1: The surface $\Sigma_3$, the involution $s$, and the curves $\alpha_i$ and $\beta_i$.
  • Figure 2:
  • Figure 3:

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Lemma 3.1
  • ...and 20 more