The intrinsic formality of $E_n$-operads

International audience ; We establish that En-operads satisfy a rational intrinsic formality theorem for n ≥ 3. 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 n-Poisson operad Pois c n , which represents the cohomology of the operad of little n-discs Dn. We assume n ≥ 3. We explicitly prove that a Hopf cooperad in cochain graded dg-modules K is weakly-equivalent (quasi-isomorphic) to Pois c n as a Hopf cooperad as soon as we have an isomorphism at the cohomology level H * (K) ≃ Pois c n when 4 ∤ n. We just need the extra assumption that K is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little n-discs operad in order to extend this formality statement in the case 4 | n. We deduce from these results that any operad in simplicial sets P which satisfies the relation H * (P, Q) ≃ Pois c n in rational cohomology (and an analogue of our extra involution requirement in the case 4 | n) is rationally weakly equivalent to an operad in simplicial sets L G•(Pois c n) which we determine from the n-Poisson cooperad Pois c n. We also prove that the morphisms ι : Dm → Dn, which link the little discs operads together, are rationally formal as soon as nm ≥ 2. 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 n ≥ 3.