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

Abstract

Abstract. We determine the rational homology of the space of long knots in R d for d ≥ 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 a 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 GoodwillieWeiss 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.

On the homology of the space of knots

“…These results lead to some natural questions about the structure of the homology of the higher-dimensional embedding spaces K n;1 (n 4) studied recently by Sinha [37], Volic [41], Lambrechts [26] as well as others , CattaneoCotta-Ramusino-Riccardo [10], Goodwillie-Weiss [14], Kohno [24] and Sakai [33]). Constructions related to these questions are also addressed here.…”

The rational homology of spaces of long links

“…Few years later, the author [14] and Moriya [9] prove independently that the collapsing result still holds for d ≥ 3, and simplify the proof of the main result of [6]. Notice that, in this paper, we produce (using a completely different approach than that of [6,9,14]) another proof of the collapsing result.…”

“…Next, Lambrechts, Turchin and Volić prove [6] that the homology Bousfield-Kan spectral sequence associated to K • d collapses at the E 2 page rationally, when d ≥ 4. Few years later, the author [14] and Moriya [9] prove independently that the collapsing result still holds for d ≥ 3, and simplify the proof of the main result of [6].…”

Topology of knot spaces in dimension 3

“…Recently Lambrechts, Tourtchine and Volic [27] have reduced the problem of computing the rational homology of such knot spaces to computing the homology of a rather explicit DGA. They do this by showing the rational collapse of the Vassiliev spectral sequence.…”