Symmetric multiplicative formality of the Kontsevich operad

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 \(d \ge 3\).