Mapping class group actions on configuration spaces and the Johnson filtration

Bianchi, Andrea; Miller, Jeremy; Wilson, Jennifer

doi:10.1090/tran/8637

Mapping class group actions on configuration spaces and the Johnson filtration
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by Andrea Bianchi, Jeremy Miller and Jennifer C. H. Wilson PDF

Trans. Amer. Math. Soc. 375 (2022), 5461-5489 Request permission

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.

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