Symmetric multiplicative formality of the Kontsevich operad

Abstract

In his famous paper entitled “Operads and motives in deformation quantization”, Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted $${\mathcal {K}}_d$$ in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature regarding the formality of $${\mathcal {K}}_d$$ . It is well known (by Kontsevich) that $${\mathcal {K}}_d$$ is formal over reals as a symmetric operad. It is also well known (independently by Syunji Moriya and the author) that $${\mathcal {K}}_d$$ is formal as a multiplicative nonsymmetric operad. In this paper, we prove that the Kontsevich operad is formal over reals as a multiplicative symmetric operad, when $$d \ge 3$$ .