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

Abstract

Let g 2 \mathfrak {g}_2 be the Hochschild complex of cochains on C ∞ ( R n ) C^\infty (\mathbb {R}^n) and let g 1 \mathfrak {g}_1 be the space of multivector fields on R n \mathbb {R}^n . In this paper we prove that given any G ∞ G_\infty -structure (i.e. Gerstenhaber algebra up to homotopy structure) on g 2 \mathfrak {g}_2 , and any C ∞ C_\infty -morphism φ \varphi (i.e. morphism of a commutative, associative algebra up to homotopy) between g 1 \mathfrak {g}_1 and g 2 \mathfrak {g}_2 , there exists a G ∞ G_\infty -morphism Φ \Phi between g 1 \mathfrak {g}_1 and g 2 \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 g 2 \mathfrak {g}_2 . We also show that any two of such G ∞ G_\infty -morphisms are homotopic.