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

Pascal Lambrechts, Victor Turchin and Ismar Volić

Geometry & Topology 14 (2010) 2151–2187

We determine the rational homology of the space of long knots in d 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.

knot spaces, embedding calculus, formality, operads, Bousfield–Kan spectral sequence
Mathematical Subject Classification 2000
Primary: 57Q45
Secondary: 57R40, 55P62
Received: 26 November 2009
Accepted: 11 August 2010
Published: 9 October 2010
Proposed: Haynes Miller
Seconded: Ralph Cohen, Bill Dwyer
Pascal Lambrechts
Université Catholique de Louvain
2 Chemin du Cyclotron
B-1348 Louvain-la-Neuve
Victor Turchin
Department of Mathematics
Kansas State University
Manhattan KS 66506
Ismar Volić
Department of Mathematics
Wellesley College
106 Central St
Wellesley MA 02481