Mapping class group actions on configuration spaces and the Johnson filtration

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.