Formality of Sinha’s cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Songhafouo Tsopméné, Paul Arnaud

doi:10.2140/agt.2013.13.2193

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Formality of Sinha's cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Paul Arnaud Songhafouo Tsopméné

Algebraic & Geometric Topology 13 (2013) 2193–2205

DOI: 10.2140/agt.2013.13.2193

Abstract

Sinha constructed a cosimplicial space $K_{N}^{∙}$ that gives a model for the space of long knots modulo immersions in $ℝ^{N}$ , $N \geq 4$ . On the other hand, Lambrechts, Turchin and Volić showed that for $N \geq 4$ the homology Bousfield–Kan spectral sequence associated to Sinha’s cosimplicial space $K_{N}^{∙}$ collapses at the $E^{2}$ page rationally. Their approach consists in first proving the formality of some other diagrams approximating $K_{N}^{∙}$ and next deducing the collapsing result. In this paper, we prove directly the formality of Sinha’s cosimplicial space, which immediately implies the collapsing result for $N \geq 3$ . Moreover, we prove that the isomorphism between the $E^{2}$ page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when $N \geq 4$ .

Keywords

multiplicative operads, model categories, long knots

Mathematical Subject Classification 2010

Primary: 57Q45, 57Q45

Secondary: 18D50, 55P48, 17B63

References

Publication

Received: 11 December 2012

Revised: 6 March 2013

Accepted: 6 March 2013

Published: 6 June 2013

Authors

Paul Arnaud Songhafouo Tsopméné
Université catholique de Louvain Chemin du Cyclotron 2 B-1348 Louvain-la-Neuve Belgium

Volume 13, issue 4 (2013)