Formality of Sinha's cosimplicial model for long knots spaces and the gerstenhaber algebra structure of homology
(2013) Algebraic & Geometric Topology — Vol. 13, n° 4, p. 2193-2205 (2013)
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- Sinha constructed a cosimplicial space K<inf>N</inf> N that gives a model for the space of long knots modulo immersions in ℝ<sup>N</sup>, N ≥ 4. On the other hand, Lambrechts, Turchin and Volić showed that for N ≥ 4 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space K<inf>N</inf> collapses at the E<sup>2</sup> page rationally. Their approach consists in first proving the formality of some other diagrams approximating K<inf>N</inf> 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 ≥ 3. Moreover, we prove that the isomorphism between the E<sup>2</sup> page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when N ≥ 4.
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Songhafouo Tsopméné, P. A. (2013). Formality of Sinha’s cosimplicial model for long knots spaces and the gerstenhaber algebra structure of homology. Algebraic & Geometric Topology, 13(4), 2193-2205. https://doi.org/10.2140/agt.2013.13.2193 (Original work published 2013)