Multiplicative formality of operads and Sinha’s spectral sequence for long knots

Abstract

Lambrechts, Turchin, and Volić proved the Bousfield–Kan-type rational homology spectral sequence associated to the $d$th Kontsevich operad collapses at $E^2$-page if $d\ge 4$. The key of their proof is formality of the operad. In this paper, we simplify their proof by using a model category of operads. As by-products we obtain two new consequences. One is collapse of the spectral sequence in the case of $d=3$ (and the coefficients being rational numbers). The other says there is no nontrivial extension for the Gerstenhaber algebra structure on the spectral sequence.