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.

arXiv comment30 pages, 14 figures, to appear in Transactions of the American Mathematical Society