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The
rational homology of spaces of long knots in codimension $\gt
2$
Pascal Lambrechts, Victor Turchin and Ismar Volić |
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Geometry & Topology 14 (2010) 2151–2187
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Abstract |
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We determine the rational homology of the space of long knots in for . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the 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 , 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 –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. |
Keywords
knot spaces, embedding calculus, formality, operads,
Bousfield–Kan spectral sequence
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Mathematical Subject Classification 2000
Primary: 57Q45
Secondary: 57R40, 55P62
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References |
Publication
Received: 26 November 2009
Accepted: 11 August 2010
Published: 9 October 2010
Proposed: Haynes Miller
Seconded: Ralph Cohen, Bill Dwyer
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Authors |
| Pascal Lambrechts | |
| Université Catholique de
Louvain 2 Chemin du Cyclotron B-1348 Louvain-la-Neuve Belgium |
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| http://milnor.math.ucl.ac.be/plwiki | |
| Victor Turchin | |
| Department of Mathematics Kansas State University Manhattan KS 66506 USA |
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| http://www.math.ksu.edu/~turchin/ | |
| Ismar Volić | |
| Department of Mathematics Wellesley College 106 Central St Wellesley MA 02481 USA |
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| http://palmer.wellesley.edu/~ivolic | |
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