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

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 $H_{\ast }(F_n({\Sigma }_{g,1});\mathbb{Q})$ 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.