Lifts of C_∞- and L_∞-morphisms to G _∞-morphisms

Let g₂ be the Hochschild complex of cochains on C ^∞(ℝⁿ) and let g₁ be the space of multivector fields on ℝⁿ. In this paper we prove that given any G_∞-structure (i-e. Gerstenhaber algebra up to homotopy structure) on g₂, and any C_∞-morphism φ (i.e. morphism of a commutative, associative algebra up to homotopy) between g ₁ and g₂, there exists a G_∞-morphism φ between g₁ and g₂ that restricts to φ. We also show that any L_∞-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G_∞-morphism, using Tamarkin's method for any G _∞-structure on g₂. We also show that any two of such G_∞-morphisms are homotopic.