Homology of configuration spaces of surfaces modulo an odd prime

Abstract

For a compact orientable surface Σ g , 1 \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 ∙ ( Σ g , 1 ) : = ∐ n ≥ 0 C n ( Σ g , 1 ) C_{\bullet}(\Sigma_{g,1}):=\coprod_{n\geq 0}C_{n}(\Sigma_{g,1}) with coefficients in F p \mathbb{F}_{p} . We describe H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ) ; F p ) H_{*}(C_{\bullet}(\Sigma_{g,1});\mathbb{F}_{p}) as a bigraded module over the Pontryagin ring H ∗ ⁢ ( C ∙ ⁢ ( D ) ; F p ) H_{*}(C_{\bullet}(D);\mathbb{F}_{p}) , where 𝐷 is a disc, and compute in particular the bigraded dimension over F p \mathbb{F}_{p} . We also consider the action of the mapping class group Γ g , 1 \Gamma_{g,1} and prove that the mod-𝑝 Johnson kernel K g , 1 ⁢ ( p ) ⊆ Γ g , 1 \mathcal{K}_{g,1}(p)\subseteq\Gamma_{g,1} is the kernel of the action on H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ; F p ) ) H_{*}(C_{\bullet}(\Sigma_{g,1};\mathbb{F}_{p})) .