Homology operations for gravity algebras

Abstract

Let $\mathcal{M}_{0,n+1}$ be the moduli space of genus zero Riemann surfaces with $n+1$ marked points. In this paper we compute $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p)$ and $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p(\pm 1))$ for any $n\in\mathbb{N}$ and any prime $p$, where $\mathbb{F}_p(\pm 1)$ denotes the sign representation of the symmetric group $\Sigma_n$. The interest in these homology groups is twofold: on the one hand classes in these equivariant homology groups parametrize homology operations for gravity algebras. On the other hand the homotopy quotient $(\mathcal{M}_{0,n+1})_{\Sigma_n}$ is a model for the classifying space for $B_n/Z(B_n)$, the quotient of the braid group $B_n$ by its center.