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On the top homology group of Johnson kernel

Alexander A. Gaifullin

Abstract

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $Σ_g$ on the lower central series of $π_1(Σ_g)$ 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 $\mathbb{Q}[\mathcal{I}_g]$.

On the top homology group of Johnson kernel

Abstract

The action of the mapping class group of an oriented surface on the lower central series of defines the descending filtration in called the Johnson filtration. The first two terms of it are the Torelli group and the Johnson kernel . By a fundamental result of Johnson (1985), is the subgroup of generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group has cohomological dimension . We prove that the top homology group is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space is infinite-dimensional. Moreover, we prove that is not finitely generated as a module over the group ring .

Paper Structure

This paper contains 4 sections, 14 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Casson invariant and cohomology class $\Phi$
  3. Complex of cycles and Cartan--Leray spectral sequence
  4. Proof of Theorem \ref{['theorem_Abelian']}

Key Result

Theorem 1.1

Suppose that $g\ge 3$. Then the group $H_{2g-3}(\mathcal{K}_g)$ contains a free abelian subgroup of infinite rank (hence, is not finitely generated). Equivalently, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional.

Figures (1)

  • Figure 1: Surface $\Sigma_g$ and curves $\delta_1,\ldots,\delta_g,\varepsilon_2,\ldots,\varepsilon_{g-2}$

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['theorem_main2']}
  • Proposition 2.1
  • Theorem 2.2: Morita Mor91
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1: Bestvina, Bux, Margalit BBM07
  • ...and 12 more