If $X$ is a smooth manifold, we define its *kth ordered configuration space* as $$F_kX:=\{(x_1, \ldots,x_k) \; | \; x_i \neq x_j \,\, \mathrm{if} \, \, i \neq j\},$$
in other words, $F_kX = X^k - \Delta,$ where $\Delta$ is the big diagonal.

I'm aware of the following two results about the formality (in the sense of rational homotopy theory) of $F_kX$. In the sequel, $g$ will be an integer $\geq 1$.

Theorem 1 ([B94, p. 133])If $\Sigma_g$ is a closed Riemann surface of genus $g$, then $F_k \Sigma_g$ is not formal for $k>2$.

Theorem 2 ([LS04, p. 1048])If $X$ is a closed, connected formal space such that $$H^1(X, \, \mathbb{Q})=H^2(X, \, \mathbb{Q})=0,$$ then $F_2 X$ is formal.

Let us now consider $F_2 \Sigma_g =\Sigma_g \times \Sigma_g - \Delta$. Then Theorem 1 does not apply because $k=2$; on the other hand, Theorem 2 does not apply either, even if $\Sigma_g$ is formal (it is a compact Kähler manifold), because the assumptions on the cohomology are not satisfied. So let me ask the following

Question.Is the 2nd configuration space $F_2 \Sigma_g$ a formal topological space?

I am by no means an expert in rational homotopy theory (actually, at the moment I am using it as a black box allowing me to perform some group theoretical computations), so I apologize in advance if the answer turns out to be trivial for the experts in the field. Every reference to the relevant literature will be highly appreciated.

**Bibliography.**

**[B94]** R. Bezrukavnikov: Koszul DG-algebras arising from configuration spaces, *Geometric and functional analysis* **4** (1994), 119-135.

**[LS04]** P. Lambrechts, D. Stanley: The rational homotopy type of configuration spaces of two points, *Ann. Inst. Fourier* **54** (2004), 1029-1052.