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

TL;DR

This paper investigates the structure of the top homology group of the Johnson kernel, focusing on the module generated by simplest abelian cycles associated with disjoint separating curves.

Contribution

It describes the module structure and relations of the top homology group of the Johnson kernel generated by simplest abelian cycles.

Findings

Describes the module structure of H_{2g-3}(K_g, Z)

Identifies relations among simplest abelian cycles

Provides a detailed algebraic description of the top homology group

Abstract

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.