On the structure of the top homology group of the Johnson kernel (Q6657450)

Let \(\Sigma_{g}\) be a compact oriented surface of genus \(g\) and \(\mathsf{Mod}(\Sigma_{g})= \pi_{0} \big ( \mathrm{Homeo}^{+}(\Sigma_{g}) \big )\) be the mapping class group of \(\Sigma_{g}\), where \(\mathrm{Homeo}^{+}(\Sigma_{g})\) is the group of orientation-preserving homeomorphisms of \(\Sigma_{g}\). The group \(\mathsf{Mod}(\Sigma_{g})\) has a surjective representation \(\mathsf{Mod}(\Sigma_{g}) \rightarrow \mathrm{Sp}(2g,\mathbb{Z})\) whose kernel \(\mathcal{I}_{g}\) is known as the Torelli group. The Johnson kernel \(\mathcal{K}_{g}\) is the subgroup of \(\mathcal{I}_{g}\) generated by all Dehn twists about separating curves. In the case \(g=1\) the representation \(\mathsf{Mod}(\Sigma_{1}) \rightarrow \mathrm{Sp}(2, \mathbb{Z})\) is an isomorphism, so \(\mathcal{I}_{1}\) is trivial. If \(g=2\), \textit{G. Mess}, in [Topology 31, No. 4, 775--790 (1992; Zbl 0772.57025)], proved that the group \(\mathcal{I}_{2}=\mathcal{K}_{2}\) is free with a countable number of generators.\N\NIn the paper under review the author deals with the case \(g \geq 3\) and, in particular, studies the structure of the top homology group \(\mathrm{H}_{2g-3}(\mathcal{K}_{g}, \mathbb{Z})\). For a collection of \(2g-3\) disjoint separating curves on \(\Sigma_{g}\), one can construct the corresponding abelian cycle in the group \(\mathrm{H}_{2g-3}(\mathcal{K}_{g}, \mathbb{Z})\), such abelian cycles will be called simple. In this paper, the author describes the structure of \(\mathbb{Z}[\mathsf{Mod}(\Sigma_{g})/\mathcal{K}_{g}]\)-module on the subgroup \(\mathrm{H}_{2g-3}(\mathcal{K}_{g}, \mathbb{Z})\), generated by all simple abelian cycles. Furthermore, he finds all relations between them.