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We establish that -operads satisfy a rational intrinsic formality theorem for . We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in spaces. We consider, in this context, the dual cooperad of the -Poisson operad , which represents the cohomology of the operad of little -discs . We assume . We explicitly prove that a Hopf cooperad in cochain graded dg-modules is weakly-equivalent (quasi-isomorphic) to as a Hopf cooperad as soon as we have an isomorphism at the cohomology level when . We just need the extra assumption that is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little -discs operad in order to extend this formality statement in the case . We deduce from these results that any operad in simplicial sets which satisfies the relation in rational cohomology (and an analogue of our extra involution requirement in the case ) is rationally weakly equivalent to an operad in simplicial sets which we determine from the -Poisson cooperad . We also prove that the morphisms , which link the little discs operads together, are rationally formal as soon as .
These results enable us to retrieve the (real) formality theorems of Kontsevich by a new approach, and to sort out the question of the existence of formality quasi-isomorphisms defined over the rationals (and not only over the reals) in the case of the little discs operads of dimension .
Cite this article
Benoit Fresse, Thomas Willwacher, The intrinsic formality of -operads. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2047–2133DOI 10.4171/JEMS/961