On 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 ${\mathcal{S}\mathcal{I}}_g$ 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 ${\mathrm{H}}_2({\mathcal{S}\mathcal{I}}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in ${\mathrm{H}}_2({\mathcal{S}\mathcal{I}}_3;\mathbb{Z})$; 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 ${\mathrm{H}}_2({\mathcal{S}\mathcal{I}}_3;\mathbb{Z})$.

16 pages, 3 figures