Homology of configuration spaces of surfaces modulo an odd prime

TL;DR

This paper investigates the homology of unordered configuration spaces on surfaces with boundary modulo an odd prime, describing their algebraic structure and the action of the mapping class group.

Contribution

It provides a detailed description of the homology as a module over the Pontryagin ring and identifies the mod-p Johnson kernel as the kernel of the mapping class group action.

Findings

Homology described as a bigraded module over the Pontryagin ring.

Computed the bigraded dimension over _p.

Identified the mod-p Johnson kernel as the kernel of the mapping class group action.

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 $H_*(C_\bullet(\Sigma_{g,1};\mathbb{F}_p))$.