  We propose an explicit relation between the cohomology of compactified and
noncompactified moduli spaces of algebraic curves with punctures. This
relationship generalizes one between commutative algebras and Lie algebras
proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we
show that the cohomology of one moduli space is the graph complex decorated
with the cohomology of the other, which in its turn generalizes Getzler's
result for the genus zero moduli. We also show that a certain class of algebras
over the chain operad of compactified moduli spaces has a natural structure of
a commutative homotopy algebra. This may be regarded as a lifting of the dot
product of Lian-Zuckerman's BV algebra structure to the cochain level, thus
complementing the results of an earlier paper hep-th/9307114 of the authors
regarding the BV bracket.
