We compute the Poincare polynomial and the cohomology algebra with rational
coefficeints of the manifold M_n of real points of the moduli space of algebraic curves of
genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture
that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was
shown by E. Rains in arXiv:math/0610743 the cohomology of M_n does not have odd torsion, so
that the above determines the additive structure of the integral homology and cohomology.
Further, we prove that the rational homology operad of M_n is the operad of 2-Gerstenhaber
algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated
by a ternary bracket). Finally, using Drinfeld's theory of quantization of coboundary Lie
quasibialgebras, we show that a large series of representations of the quadratic dual Lie
algebra L_n of H^*(M_n,Q) (associated to such quasibialgebras) factors through the the
natural projection of L_n to the associated graded Lie algebra of the prounipotent
completion of the fundamental group of M_n. This leads us to conjecture that the said
projection is an isomorphism, which would imply a formula for lower central series ranks of
the fundamental group. On the other hand, we show that the spaces M_n are not formal
starting from n=6.