On the top homology group of Johnson kernel

Abstract

The action of the mapping class group ${\mathrm{Mod}}_g$ of an oriented surface ${\Sigma }_g$ on the lower central series of ${\pi }_1({\Sigma }_g)$ defines the descending filtration in ${\mathrm{Mod}}_g$ called the Johnson filtration. The first two terms of it are the Torelli group ${\mathcal{I}}_g$ and the Johnson kernel ${\mathcal{K}}_g$. By a fundamental result of Johnson (1985), ${\mathcal{K}}_g$ is the subgroup of ${\mathrm{Mod}}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group ${\mathcal{K}}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}({\mathcal{K}}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}({\mathcal{K}}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}({\mathcal{K}}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[{\mathcal{I}}_g]$.

13 pages

References in corpus (5)

• On finite generation of the Johnson filtrations
• Torelli group, Johnson kernel and invariants of homology spheres
• Problems, Questions, and Conjectures about Mapping Class Groups
• The Dimension of the Torelli group
• On infinitely generated homology of Torelli groups

Cited by in corpus (2)

• On the structure of the top homology group of the Johnson kernel
• The top homology group of the genus 3 Torelli group