On the structure of the top homology group of the Johnson kernel

The Johnson kernel of a genus $g$ oriented surface $\Sigma_{g}$ is a subgroup $\mathcal{K}(\Sigma_{g})$ of the mapping class group $\mathrm{Mod}(\Sigma_{g})$ generated by all Dehn twists along separating curves. Given a family of $2g-3$ pairwise disjoint separating curves on $\Sigma_{g}$ one can construct the corresponding abelian cycle in the top homology group $H_{2g-3}(\mathcal{K}(\Sigma_{g}), \mathbb{Z})$; such abelian cycles we call primitive. We will discuss the structure of the subgroup of $H_{2g-3}(\mathcal{K}(\Sigma_{g}), \mathbb{Z})$ generated by all primitive abelian cycles. In particular, we will describe the relations between them.