On the second homology of the genus 3 hyperelliptic Torelli group

TL;DR

This paper investigates the second homology group of the genus 3 hyperelliptic Torelli group, revealing a correspondence between certain cycles and algebraic splittings of homology, and establishing their linear independence.

Contribution

It characterizes simple abelian cycles in H_2 of the genus 3 hyperelliptic Torelli group and proves their linear independence, linking them to algebraic splittings of homology.

Findings

Simple abelian cycles correspond to orthogonal splittings of H_1(Σ_3;Z).

These cycles are linearly independent in H_2(𝕊𝕀_3;Z).

The work provides a new understanding of the second homology of the genus 3 hyperelliptic Torelli group.

Abstract

Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $\Sigma_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g)$ consisting of elements that act trivially on $\mathrm{H}_1(\Sigma_g;\mathbb{Z})$ 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 $\mathrm{H}_1(\Sigma_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$.