# The intrinsic formality of $E_{n}$-operads

### Benoit Fresse

Université de Lille, France### Thomas Willwacher

ETH Zürich, Switzerland

## Abstract

We establish that $E_{n}$-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 $Pois_{n}$, which represents the cohomology of the operad of little $n$-discs $D_{n}$. We assume $n≥3$. We explicitly prove that a Hopf cooperad in cochain graded dg-modules $K$ is weakly-equivalent (quasi-isomorphic) to $Pois_{n}$ as a Hopf cooperad as soon as we have an isomorphism at the cohomology level $H_{∗}(K)≃Pois_{n}$ 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 $H_{∗}(P,Q)≃Pois_{n}$ 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 $LG_{∙}(Pois_{n})$ which we determine from the $n$-Poisson cooperad $Pois_{n}$. We also prove that the morphisms $ι:D_{m}→D_{n}$, which link the little discs operads together, are rationally formal as soon as $n−m≥2$.

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 $n≥3$.

## Cite this article

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

DOI 10.4171/JEMS/961