JournalsjemsVol. 22, No. 7pp. 2047–2133

The intrinsic formality of EnE_n-operads

  • Benoit Fresse

    Université de Lille, France
  • Thomas Willwacher

    ETH Zürich, Switzerland
The intrinsic formality of $E_n$-operads cover
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We establish that EnE_n-operads satisfy a rational intrinsic formality theorem for n3n\geq 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 nn-Poisson operad Poisnc\mathsf{Pois}_n^c, which represents the cohomology of the operad of little nn-discs Dn\mathsf{D}_n. We assume n3n\geq 3. We explicitly prove that a Hopf cooperad in cochain graded dg-modules K\mathsf{K} is weakly-equivalent (quasi-isomorphic) to Poisnc\mathsf{Pois}_n^c as a Hopf cooperad as soon as we have an isomorphism at the cohomology level H(K)PoisncH^*(\mathsf{K})\simeq\mathsf{Pois}_n^c when 4n4\nmid n. We just need the extra assumption that K\mathsf{K} is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little nn-discs operad in order to extend this formality statement in the case 4n4\mid n. We deduce from these results that any operad in simplicial sets P\mathsf{P} which satisfies the relation H(P,Q)PoisncH^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c in rational cohomology (and an analogue of our extra involution requirement in the case 4n4\mid n) is rationally weakly equivalent to an operad in simplicial sets LG(Poisnc)\mathtt L \mathtt G_{\bullet}(\mathsf{Pois}_n^c) which we determine from the nn-Poisson cooperad Poisnc\mathsf{Pois}_n^c. We also prove that the morphisms ι:DmDn\iota: \mathsf{D}_m\rightarrow\mathsf{D}_n, which link the little discs operads together, are rationally formal as soon as nm2n-m\geq 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 n3n\geq 3.

Cite this article

Benoit Fresse, Thomas Willwacher, The intrinsic formality of EnE_n-operads. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2047–2133

DOI 10.4171/JEMS/961