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

Abstract

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

23 pages, minor corrections

Cited by in corpus (1)

• The top homology group of the genus 3 Torelli group