Real moduli space of stable rational curves revisited

Abstract

The real locus of the moduli space of stable genus zero curves with marked points, ${\overline{\mathcal{M}}}_{0,n+1}(\mathbb{R})$, is known to be a smooth manifold and is the Eilenberg–MacLane spaces for the so-called pure cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of ${\overline{\mathcal{M}}}_{0,n+1}(\mathbb{R})$. In particular, we show that the operad ${\overline{\mathcal{M}}}_{0,n+1}(\mathbb{R})$ is not formal. As an application of these operadic constructions, we prove that for each $n$, the cohomology ring $H^{\bullet }({\overline{\mathcal{M}}}_{0,n+1}(\mathbb{R});\mathbb{Q})$ is a Koszul algebra, and that the manifold ${\overline{\mathcal{M}}}_{0,n+1}(\mathbb{R})$ is not formal for $n\ge 6$ but is a rational $K(\pi ,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure cactus groups.