  The Johnson kernel is the subgroup $\mathcal{K}_g$ of the mapping class group
${\rm Mod}(\Sigma_{g})$ of a genus $g$ oriented closed surface $\Sigma_{g}$
generated by all Dehn twists about separating curves. In this paper we study
the structure of the top homology group ${\rm H}_{2g-3}(\mathcal{K}_g,
\mathbb{Z})$. For any collection of $2g-3$ disjoint separating curves on
$\Sigma_{g}$ one can construct the corresponding abelian cycle in the group
${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$; such abelian cycles will be called
simplest. In this paper we describe the structure of $\mathbb{Z}[{\rm
Mod}(\Sigma_{g})/ \mathcal{K}_g]$-module on the subgroup of ${\rm
H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$ generated by all simplest abelian cycles
and find all relations between them.
