Real moduli space of stable rational curves revisited

The real locus of the moduli space of stable genus zero curves with marked points, (Formula Present), is known to be a smooth manifold and is the Eilenberg–MacLane spaces for the socalled 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 (Formula Present). In particular, we show that the operad (Formula Present) is not formal. As an application of these operadic constructions, we prove that for each n, the cohomology ring H·(Formula Present) is a Koszul algebra, and that the manifold (Formula Present) is not formal for n ≥ 6 but is a rational K(π, 1)-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure cactus groups.