Homology operations for gravity algebras

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 $H_{\ast }^{{\mathrm{\Sigma }}_n}({\mathcal{M}}_{0,n+1};{\mathbb{F}}_p)$ and $H_{\ast }^{{\mathrm{\Sigma }}_n}({\mathcal{M}}_{0,n+1};{\mathbb{F}}_p(\pm 1))$ for any $n\in \mathbb{N}$ and any prime $p$, where ${\mathbb{F}}_p(\pm 1)$ denotes the sign representation of the symmetric group ${\mathrm{\Sigma }}_n$. 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 $({\mathcal{M}}_{0,n+1}{)}_{{\mathrm{\Sigma }}_n}$ is a model for the classifying space for $B_n\mathrm{/}Z(B_n)$, the quotient of the braid group $B_n$ by its center.