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

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.

12 pages. The paper was published in AGT, 2013, but I never updated the title on arXiv. Now the title is updated

References in corpus (4)

• Long knots and maps between operads
• Homotopy theory of non-symmetric operads
• Poisson structures on the homology of the space of knots
• Delooping totalization of a multiplicative operad

Cited by in corpus (7)

• Deformation quantization and the Gerstenhaber structure on the homology of knot spaces
• The rational homology of spaces of long links
• Configuration space integrals and the topology of knot and link spaces
• Non-formality of the odd dimensional framed little balls operads
• Embedding calculus and grope cobordism of knots
• On Cohen-Jones isomorphism in string topology
• Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad