Multiplicative formality of operads and Sinha’s spectral sequence for long knots

Non-Formality of the Odd Dimensional Framed Little Balls Operads

“…This fact and the formality of the rational singular chain coalgebra C * (S d−1 ) imply the formality of the rational singular chain multiplicative operad C * (S d ) . This multiplicative operad formality implies the Bousfield-Kan spectral sequence associated to the cosimplicial space S • d degenerates at E 2 -page, see the proof of Theorem 1.4 in [27]. This spectral sequence converges to the homology H * ( Tot(S • d )).…”

“…For general theory of model categories, see [6]. It is known that the category of chain operads has a model category structure where weak equivalences are the same as those given in section 2 (see [27,Theorem 2.1] or [16, Theorem 1.1], see also [5] for a model category of symmetric operads). 2)) is non-zero for this choise of ν and g. Let A ∞ be the Stasheff's associahedral chain operad.…”

“…is essentially a simple observation for the number of generators and degrees. Note that formality as a multiplicative operad implies degeneracy of the spectral sequence at E 2 -page as in [15,27,22], so in turn, non-degeneracy of the spectral sequence implies non-formality as a multiplicative operad. But it is weaker than non-formality as a non-symmetric operad.…”

On Cohen-Jones isomorphism in string topology

“…We use the case of n = 1 of this theorem to prove Theorem 1.0.1 (2). As the both E n -actions have many applications [14,20,17,39,37], this theorem will be useful in other context. Theorem 1.0.2 is not so trivial as it looks since the E n -operad actions on Tot and Tot are realized by different operads and there is no obvious morphism between them.…”

Deformation quantization and the Gerstenhaber structure on the homology of knot spaces

“…Theorem 1.1 (Songhafouo Tsopméné [29], Moriya [22]). For n ≥ 4 there is an isomorphism of Gerstenhaber algebras between the homology of the space of long knots modulo immersions and the Hochschild homology of Pois n , H * (Emb(R, R n ), Q) HH(Pois n ).…”