Symmetric multiplicative formality of the Kontsevich operad (Q1744187)

In [Lett. Math. Phys. 48, No. 1, 35--72 (1999; Zbl 0945.18008)], \textit{M. Kontsevich} constructed a topological operad \(K_d\) in order to prove the formality of the little \(d\)-disks operad. It was proved by \textit{D. Sinha} [J. Am. Math. Soc. 19, No. 2, 461--486 (2006; Zbl 1112.57004)] that the operad \(K_d\), called the Kontsevich operad, is a multiplicative symmetric operad. Thus \(K_d\) is a symmetric operad and there is a morphism of operads from the symmetric associative operad to \(K_d\) that respects the symmetric structures. A topological operad \(O\) is said to be formal over a field \(\mathbb K\) if there exists a zigzag \[ C_\ast(O;\mathbb K)\overset{\sim}{\longleftarrow}\cdots \overset{\sim}{\longrightarrow} H_\ast(O;\mathbb K) \] of quasi-isomorphisms between the singular chain of \(O\) and its homology. The formality properties of \(K_d\) have been studied extensively. By Kontsevich [loc. cit.], \(K_d\) is formal over the reals as a symmetric operad. It was proved independently by \textit{S. Moriya} [Kyoto J. Math. 55, No. 1, 17--27 (2015; Zbl 1312.55018)] and the author [Algebr. Geom. Topol. 13, No. 4, 2193--2205 (2013; Zbl 1275.57032)], that \(K_d\) is formal over the reals as a multiplicative nonsymmetric operad. In the paper under review, the author proves that \(K_d\) is formal over the reals as a multiplicative symmetric operad, for \(d\geq 3\). This means that the above zigzag exists in the category of multiplicative symmetric operads.