#### Volume 13, issue 4 (2013)

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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
##### Abstract

Sinha constructed a cosimplicial space ${\mathsc{K}}_{N}^{\bullet }$ that gives a model for the space of long knots modulo immersions in ${ℝ}^{N}$, $N\ge 4$. On the other hand, Lambrechts, Turchin and Volić showed that for $N\ge 4$ the homology Bousfield–Kan spectral sequence associated to Sinha’s cosimplicial space ${\mathsc{K}}_{N}^{\bullet }$ collapses at the ${E}^{2}$ page rationally. Their approach consists in first proving the formality of some other diagrams approximating ${\mathsc{K}}_{N}^{\bullet }$ 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 $N\ge 3$. 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 $N\ge 4$.

##### Keywords
multiplicative operads, model categories, long knots
##### Mathematical Subject Classification 2010
Primary: 57Q45, 57Q45
Secondary: 18D50, 55P48, 17B63