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

TL;DR

This paper simplifies the proof of the collapse of Sinha's spectral sequence for long knots using operad formality, and extends results to the case of d=3 with rational coefficients, also clarifying algebraic extensions.

Contribution

It provides a simplified proof of spectral sequence collapse for d≥4 and introduces new results for the case d=3, utilizing a model category approach.

Findings

Spectral sequence collapses at E^2 for d≥4 with rational coefficients.

Collapse of the spectral sequence also holds for d=3.

No non-trivial extensions in the Gerstenhaber algebra structure.

Abstract

Lambrechts, Turchin and Voli\'c proved the Bousfield-Kan type rational homology spectral sequence associated to the $d$-th Kontsevich operad collapses at $E^2$-page if $d\geq 4$. The key of their proof is formality of the operad. In this paper, we simplify their proof using a model category of operads. As byproducts 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 non-trivial extension for the Gerstenhaber algebra structure on the spectral sequence.