Abelian Cycles in the Homology of the Torelli group

TL;DR

This paper investigates the rational homology of the Torelli group using the Johnson homomorphism, revealing large quotients and the structure of abelian cycles, with implications for stable homology and subrepresentations.

Contribution

It computes large quotients of the Torelli group's homology via abelian cycles and analyzes their generation, advancing understanding of the group's homological structure.

Findings

Identifies large quotients of homology in the stable range.

Shows abelian cycles generate a proper subrepresentation.

Extends results to the Torelli group of a surface with a marked point.

Abstract

In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is the Torelli group of a closed, connected and oriented surface of genus $g$ with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding pair maps, in order to compute a large quotient of $H_n(\mathcal{I}_{g}^1,\mathbb{Q})$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal{I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n(\mathcal{I}_{g,1})$ for $n\ge 2$ and $g$ large enough.