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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

Sinha constructed a cosimplicial space KN that gives a model for the space of long knots modulo immersions in N, 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 KN collapses at the E2 page rationally. Their approach consists in first proving the formality of some other diagrams approximating KN 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 E2 page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when N 4.

multiplicative operads, model categories, long knots
Mathematical Subject Classification 2010
Primary: 57Q45, 57Q45
Secondary: 18D50, 55P48, 17B63
Received: 11 December 2012
Revised: 6 March 2013
Accepted: 6 March 2013
Published: 6 June 2013
Paul Arnaud Songhafouo Tsopméné
Université catholique de Louvain
Chemin du Cyclotron 2
B-1348 Louvain-la-Neuve