Towards representation stability for the second homology of the Torelli group

Abstract

We show for $g\ge 7$ that the second homology group of the Torelli group, $H_2\left({\mathcal{ℐ}}_{g,1};\mathbb{Q}\right)$, is generated as an $Sp\left(2g,\mathbb{Z}\right)$–module by the image of $H_2\left({\mathcal{ℐ}}_{6,1};\mathbb{Q}\right)$ under the stabilization map. In the process we also show that the quotient $B\left(F_{g,i};i\right)∕{\mathcal{ℐ}}_{g,i}$ by the Torelli group of the complex of arcs with identity permutation is $\left(g-2\right)$–connected for $i=1,2$.