Homotopy G-algebras and moduli space operad

Abstract

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $Ω^\bullet X$ of a Kähler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $Ω^\bullet X$.