Real moduli space of stable rational curves revised

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}$})$isaKoszulalgebra,andthatthemanifold$\overline{{\mathcal M}{0,{n+1}}}({\mathbb R})$isnotformalfor$n\geq 6$butisarational$K(\pi,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure Cactus groups.