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10 pages, case of $C_\infty$-morphisms is studied, existence of lift is proved in that case, final version, to appear in Proc. of the A.M.SLet $\g_2$ 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 $\varphi$ ({\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 $\varphi$. 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

Lifts of 𝐶_{∞}- and 𝐿_{∞}-morphisms to 𝐺_{∞}-morphisms