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By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}.$$ These spaces are "formal" over the rationals, meaning that their cohomology ring completely encodes their rational homotopy type.

For $n = 2$ there is no need to involve operads, because there is a direct map $H^*(\operatorname{Conf}_k(\mathbb{C})) \xrightarrow{\sim} \Omega^*(\operatorname{Conf}_k(\mathbb{C}))$, sending the usual generators of the cohomology to $d\log(z_i - z_j)$ (Arnold, 1969). For $n \ge 3$ this doesn't work because the Arnold relations do not hold "on the nose" in $\Omega^*(\operatorname{Conf}_k(\mathbb{R}^n))$.

The proofs I know for $n \ge 3$ all involve operads. More precisely, the little disks operads, whose components are homotopy equivalent to the configuration spaces above. The theorem is that these operads are formal. Hence the configuration spaces are formal. See the results of Kontsevich (and Guillén Santos–Navarro–Pascual–Roig to descend to $\mathbb{Q}$), Tamarkin (for $n = 2$ actually), Lambrechts–Volić, Petersen, Fresse–Willwacher...

I am curious, is there a proof somewhere in the literature that has nothing to do with operads? I have not found any, but, well, I cannot be 100% sure...

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When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in degree $n-1$, which has a pure Hodge structure of weight $n$. Using a "purity implies formality" principle one can then deduce that $\mathrm{Conf}_k(\mathbf{R}^n)$ is formal. Part of this is certainly folklore since a long time, but to my knowledge the earliest explicit statement of a sufficiently general "purity implies formality" statement to deduce this is in a paper of Cirici-Horel from last year. See Proposition 8.2 of their paper for an explicit statement which includes as a special case formality of these configuration spaces.

But yes, I do believe that (remarkably enough) the first proof of formality of $\mathrm{Conf}_k(\mathbf R^n)$ for $n > 2$ is through the results of Kontsevich and Lambrechts-Volic.

A remark is that the phrase "formality of the little disks" could mean either "the dg operad of chains on $\mathcal D_n$ is formal (as a dg operad)" or "the dg Hopf co-operad of cochains on $\mathcal D_n$ is formal (as a dg Hopf co-operad)". The former statement does not at all imply formality of the individual spaces $\mathrm{Conf}_k(\mathbf R^n)$ since the formality is not required to have any compatibility with cup product. Tamarkin's proof, for instance, only gave formality in the weaker sense, and the proof in Lambrechts-Volic only gives Hopf co-operad formality when $n \geq 3$.

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  • $\begingroup$ Thanks! I hadn't thought of this whole purity business. I'm still amazed that parity can make such a difference. (Thanks also for the catch regarding chains vs Hopf. I was a bit imprecise... I meant Hopf, but it's true that then I have to qualify the initial results of Kontsevich and Tamarkin...) $\endgroup$ – Najib Idrissi Apr 26 '18 at 13:38

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