Homology of configuration spaces of surfaces modulo an odd prime

Abstract

For a compact orientable surface $\Sigma_{g,1}$ of genus $g$ with one boundary component and for an odd prime number $p$, we study the homology of the unordered configuration spaces $C_\bullet(\Sigma_{g,1}):=\coprod_{n\ge0}C_n(\Sigma_{g,1})$ with coefficients in $\mathbb{F}_p$. We describe $H_*(C_\bullet(\Sigma_{g,1});\mathbb{F}_p)$ as a bigraded module over the Pontryagin ring $H_*(C_\bullet(D);\mathbb{F}_p)$, where $D$ is a disc, and compute in particular the bigraded dimension over $\mathbb{F}_p$. We also consider the action of the mapping class group $\Gamma_{g,1}$, and prove that the mod-$p$ Johnson kernel $\mathcal{K}_{g,1}(p)\subseteq\Gamma_{g,1}$ is the kernel of the action on $H_*(C_\bullet(\Sigma_{g,1};\mathbb{F}_p))$.