Lift of $C\_\infty$ and $L\_\infty$ morphisms to $G\_\infty$ morphisms

Abstract

Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G\_\infty$-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any $C\_\infty$-morphism $\phi$ ({\rm i.e.} morphism of commutative, associative algebra up to homotopy) between $\g\_1$ and $\g\_2$, there exists a $G\_\infty$-morphism $\Phi$ between $\g\_1$ and $\g\_2$ that restricts to $\phi$. We also show that any $L\_\infty$-morphism ({\rm i.e.} morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G\_\infty$-morphism, using Tamarkin's method for any $G\_\infty$-structure on $\g\_2$. We also show that any two of such $G\_\infty$-morphisms are homotopic.