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

TL;DR

This paper proves the formality of Sinha's cosimplicial model for long knots spaces, simplifies the proof of spectral sequence collapse, and establishes a Poisson algebra structure on homology for N > 3.

Contribution

It provides a direct proof of formality for Sinha's cosimplicial space and demonstrates the Poisson algebra structure of the homology of long knots.

Findings

Formality of Sinha's cosimplicial space established

Spectral sequence collapses at page 2 for N > 2

Homology is isomorphic to a Poisson algebra for N > 3

Abstract

Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this allows us to construct a very short proof of the collapsing result for N > 2. Moreover, we prove that the isomorphism between the page 2 and the rational homology of the space of long knots modulo immersions is of Poisson algebras, when N > 3.