Lifts of C∞- and L∞-morphisms to G∞-morphisms

Let g(2) be the Hochschild complex of cochains on C-infinity(R-n) and let g(1) be the space of multivector fields on R-n. In this paper we prove that given any G(infinity)-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g(2), and any C infinity-morphism phi (i.e. morphism of a commutative, associative algebra up to homotopy) between g(1) and g(2), there exists a G infinity-morphism F between g(1) and g(2) that restricts to phi We also show that any L infinity-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G infinity-morphism, using Tamarkin's method for any G infinity-structure on g(2). We also show that any two of such G infinity-morphisms are homotopic.