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

Lambrechts, Pascal¹ ; Turchin, Victor² ; Volić, Ismar³

¹ Université Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
² Department of Mathematics, Kansas State University, Manhattan KS 66506, USA
³ Department of Mathematics, Wellesley College, 106 Central St, Wellesley MA 02481, USA

Geometry & topology, Tome 14 (2010) no. 4, pp. 2151-2187.

Résumé

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.

Détail

DOI : 10.2140/gt.2010.14.2151

Keywords: knot spaces, embedding calculus, formality, operads, Bousfield–Kan spectral sequence

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