The rational homology of spaces of long knots in codimension > 2

We determine the rational homology of the space of long knots in Rd for d 4. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E1 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. 57Q45; 57R40, 55P62