## Geometry & Topology

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

#### Abstract

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.

#### Article information

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

Dates
Accepted: 11 August 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883516

Digital Object Identifier
doi:10.2140/gt.2010.14.2151

Mathematical Reviews number (MathSciNet)
MR2740644

Zentralblatt MATH identifier
1222.57020

#### Citation

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. https://projecteuclid.org/euclid.gt/1513883516

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