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 $C_{\bullet }({\Sigma }_{g,1}):={\coprod }_{n\ge 0}{C}_n({\Sigma }_{g,1})$ with coefficients in ${\mathbb{F}}_p$. We describe $H_{\ast }(C_{\bullet }({\Sigma }_{g,1});{\mathbb{F}}_p)$ as a bigraded module over the Pontryagin ring $H_{\ast }(C_{\bullet }(D);{\mathbb{F}}_p)$, 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 $H_{\ast }(C_{\bullet }({\Sigma }_{g,1};{\mathbb{F}}_p))$.

55 pages, 5 figures, to appear in Journal für die reine und angewandte Mathematik