Mapping class group actions on configuration spaces and the Johnson filtration

TL;DR

This paper investigates how the Johnson filtration of the mapping class group acts on the homology of configuration spaces on surfaces, showing trivial action at certain levels and providing new insights into related groups.

Contribution

It proves the triviality of the Johnson filtration's action on homology at specific stages and offers new conceptual interpretations of Moriyama's group.

Findings

Johnson filtration acts trivially on homology at stage i

Nontrivial action of (2) on H_3 for g

New reinterpretations of Moriyama's group

Abstract

Let $F_n(\Sigma_{g,1})$ denote the configuration space of $n$ ordered points on the surface $\Sigma_{g,1}$ and let $\Gamma_{g,1}$ denote the mapping class group of $\Sigma_{g,1}$. We prove that the action of $\Gamma_{g,1}$ on $H_i(F_n(\Sigma_{g,1});\mathbb{Z})$ is trivial when restricted to the $i^{th}$ stage of the Johnson filtration $\mathcal{J}(i)\subset \Gamma_{g,1}$. We give examples showing that $\mathcal{J}(2)$ acts nontrivially on $H_3(F_3(\Sigma_{g,1}))$ for $g\ge 2$, and provide two new conceptual reinterpretations of a certain group introduced by Moriyama.