On spectral sequence for the action of genus 3 Torelli group on the complex of cycles

TL;DR

This paper investigates the second homology of the genus 3 Torelli group using spectral sequences, providing partial evidence that the group is not finitely presented by showing certain homology terms are infinitely generated.

Contribution

It demonstrates that the spectral sequence term E^3_{0,2} remains infinitely generated, advancing the understanding of the Torelli group's algebraic properties.

Findings

E^3_{0,2} is infinitely generated

Supports the conjecture that _3 is not finitely presented

Progress towards proving _3's non-finite presentability

Abstract

The Torelli group of a genus $g$ oriented surface $S_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group $\mathrm{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$ is finitely presented or not. A possible approach to this problem relies upon the study of the second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain a partial result towards the conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence $\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is infinitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by images of the differentials $d^1$ and $d^2$. If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of $d^3$, he would complete the proof of the fact that $\mathcal{I}_3$ is not finitely presented.