  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.
