Formality of Sinha's cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Abstract

Sinha constructed a cosimplicial space ${\mathcal{K}}_N^{\bullet }$ that gives a model for the space of long knots modulo immersions in ${ℝ}^N$, $N\ge 4$. On the other hand, Lambrechts, Turchin and Volić showed that for $N\ge 4$ the homology Bousfield–Kan spectral sequence associated to Sinha’s cosimplicial space ${\mathcal{K}}_N^{\bullet }$ collapses at the $E^2$ page rationally. Their approach consists in first proving the formality of some other diagrams approximating ${\mathcal{K}}_N^{\bullet }$ and next deducing the collapsing result. In this paper, we prove directly the formality of Sinha’s cosimplicial space, which immediately implies the collapsing result for $N\ge 3$. Moreover, we prove that the isomorphism between the $E^2$ page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when $N\ge 4$.