Non-trivial action of the Johnson filtration on the homology of configuration spaces

TL;DR

This paper demonstrates that the Johnson filtration's action on the homology of configuration spaces of surfaces is non-trivial, revealing complex interactions between surface mapping class groups and configuration space topology.

Contribution

It proves the non-triviality of the Johnson filtration's action on homology for surfaces with boundary and extends the result to closed surfaces, advancing understanding of surface group actions.

Findings

Action is non-trivial on the $(n-1)$st stage of the Johnson filtration

Results hold for all $n \,\geq\, 1$ and genus $g \geq 2$

Extension of results to closed surfaces

Abstract

We let the mapping class group $\Gamma_{g,1}$ of a genus $g$ surface $\Sigma_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(\Sigma_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.