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arXiv · 1210.2561

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

Paul Arnaud Songhafouo Tsopméné
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Abstract

Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this allows us to construct a very short proof of the collapsing result for N > 2. Moreover, we prove that the isomorphism between the page 2 and the rational homology of the space of long knots modulo immersions is of Poisson algebras, when N > 3.

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Paul Arnaud Songhafouo Tsopméné. 2018-09-26. Formality of Sinha's cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology. https://doi.org/10.2140/agt.2013.13.2193

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