Moscow Mathematical Journal

Volume 22, Issue 1, January–March 2022 pp. 83–102.

On the Top Homology Group of the Johnson Kernel

Authors: Alexander A. Gaifullin (1)
Author institution:(1) Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia,
Skolkovo Institute of Science and Technology, Skolkovo, Russia,
Lomonosov Moscow State University, Moscow, Russia,
Institute for Information Transmission Problems (Kharkevich Institute), Moscow, Russia

Summary:

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 that 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 $\mathbb{Q}[\mathcal{I}_g]$.

2020 Math. Subj. Class. Primary: 20F34; Secondary: 57M07, 20J05

Keywords: Johnson kernel, Torelli group, homology of groups, complex of cycles, Casson invariant, abelian cycle.

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