84 pages. Minor corrections (typos and inconsistant arity bounds in constructions of section 2 fixed in v2). Minor revisions, notation and bibliographical updates in v3-v5. To appear in J. Eur. Math. Soc. (JEMS)
We establish that $E_n$-operads satisfy a rational intrinsic formality theorem for $n\geq 3$. 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 $n$-Poisson operad $\mathsf{Pois}_n^c$, which represents the cohomology of the operad of little $n$-discs $\mathsf{D}_n$. We assume $n\geq 3$. We explicitly prove that a Hopf cooperad in cochain graded dg-modules $\mathsf{K}$ is weakly-equivalent (quasi-isomorphic) to $\mathsf{Pois}_n^c$ as a Hopf cooperad as soon as we have an isomorphism at the cohomology level $H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c$ when $4\nmid n$. We just need the extra assumption that $\mathsf{K}$ 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 $4\mid n$. We deduce from these results that any operad in simplicial sets $\mathsf{P}$ which satisfies the relation $H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c$ in rational cohomology (and an analogue of our extra involution requirement in the case $4\mid n$) is rationally weakly equivalent to an operad in simplicial sets $LG_{\bullet}(\mathsf{Pois}_n^c)$ which we determine from the $n$-Poisson cooperad $\mathsf{Pois}_n^c$. We also prove that the morphisms $ι: \mathsf{D}_m\rightarrow\mathsf{D}_n$, which link the little discs operads together, are rationally formal as soon as $n-m\geq 2$. 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 $n\geq 3$.
Symmetric multiplicative formality of the Kontsevich operad
在他著名的论文“Operads and motives in deformation quantization”中，Maxim Kontsevich构造了（为了证明小d盘操作的形式）一个拓扑操作，在文献中称为Kontsevich操作，在本文中记为Kd。这个操作数有一个很好的结构：它是一个乘法对称操作数，也就是说，它带有一个来自对称结合操作数的态射。文献中有许多关于Kd形式的结果。众所周知（由Kontsevich），Kd作为对称操作数在实数上是形式的。众所周知，Kd是形式上作为乘法非对称操作数的。本文证明了当d > 2时，Kontsevich算子在实数上是乘性对称算子。
An Obstruction Theory for the Existence of Maurer-Cartan Elements in curved $L_\infty$-algebras and an Application in Intrinsic Formality of $P_\infty$-Algebras
