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We compute the Poincare polynomial and the cohomology algebra with rationalcoefficeints of the manifold M_n of real points of the moduli space ofalgebraic curves of genus 0 with n labeled points. This cohomology is aquadratic algebra, and we conjecture that it is Koszul. We also compute the2-local torsion in the cohomology of M_n. As was shown by E. Rains inarXiv:math/0610743 the cohomology of M_n does not have odd torsion, so that theabove determines the additive structure of the integral homology andcohomology. Further, we prove that the rational homology operad of M_n is theoperad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachsoperad of 2-Lie algebras (generated by a ternary bracket). Finally, usingDrinfeld's theory of quantization of coboundary Lie quasibialgebras, we showthat a large series of representations of the quadratic dual Lie algebra L_n ofH^*(M_n,Q) (associated to such quasibialgebras) factors through the the naturalprojection of L_n to the associated graded Lie algebra of the prounipotentcompletion of the fundamental group of M_n. This leads us to conjecture thatthe said projection is an isomorphism, which would imply a formula for lowercentral series ranks of the fundamental group. On the other hand, we show thatthe spaces M_n are not formal starting from n=6.

The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points