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

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.