Résumé
Let g2\mathfrak {g}_2 be the Hochschild complex of cochains on C∞(Rn)C^\infty (\mathbb {R}^n) and let g1\mathfrak {g}_1 be the space of multivector fields on Rn\mathbb {R}^n. In this paper we prove that given any G∞G_\infty-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g2\mathfrak {g}_2, and any C∞C_\infty-morphism φ\varphi (i.e. morphism of a commutative, associative algebra up to homotopy) between g1\mathfrak {g}_1 and g2\mathfrak {g}_2, there exists a G∞G_\infty-morphism Φ\Phi between g1\mathfrak {g}_1 and g2\mathfrak {g}_2 that restricts to φ\varphi. We also show that any L∞L_\infty-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G∞G_\infty-morphism, using Tamarkin’s method for any G∞G_\infty-structure on g2\mathfrak {g}_2. We also show that any two of such G∞G_\infty-morphisms are homotopic.