The rational homology of spaces of long knots in codimension $\gt 2$

Abstract

We determine the rational homology of the space of long knots in ${ℝ}^d$ for $d\ge 4$. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the $E^1$ page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree $d-1$, which can be obtained as the homology of an explicit graph complex and is in theory completely computable.

Our proof is a combination of a relative version of Kontsevich’s formality of the little $d$–disks operad and of Sinha’s cosimplicial model for the space of long knots arising from Goodwillie–Weiss embedding calculus. As another ingredient in our proof, we introduce a generalization of a construction that associates a cosimplicial object to a multiplicative operad. Along the way we also establish some results about the Bousfield–Kan spectral sequences of a truncated cosimplicial space.