Geometry & Topology

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

Pascal Lambrechts, Victor Turchin, and Ismar Volić

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We determine the rational homology of the space of long knots in d for d4. 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 d1, 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.

Article information

Geom. Topol., Volume 14, Number 4 (2010), 2151-2187.

Received: 26 November 2009
Accepted: 11 August 2010
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
Secondary: 57R40: Embeddings 55P62: Rational homotopy theory

knot spaces embedding calculus formality operads Bousfield–Kan spectral sequence


Lambrechts, Pascal; Turchin, Victor; Volić, Ismar. The rational homology of spaces of long knots in codimension $\gt 2$. Geom. Topol. 14 (2010), no. 4, 2151--2187. doi:10.2140/gt.2010.14.2151.

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