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.


[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.


1 Answer 1


I believe that $F_2 \Sigma_g$ is formal. Here's a sketch of an argument, unfortunately I haven't checked the details. Apply the results of Morgan, "The algebraic topology of smooth algebraic varieties". To be specific, $F_2 \Sigma_g$ is the complement of a smooth divisor in the compact Kähler manifold $\Sigma_g^2$. Then Morgan's results say that the de Rham algebra of $F_2 \Sigma_g$ is equivalent to the algebra of forms on $\Sigma_g^2$ with logarithmic singularities along $\Delta$, and the latter algebra is quasi-isomorphic to the $E_2$ term of the Leray spectral sequence for $F_2\Sigma_g \to \Sigma_g^2$. This can be represented by a cdga given by the mapping cone of $s^{-2}H \to H\otimes H$, where $H = H^\bullet(\Sigma_g,\mathbf Q)$. But now this mapping cone is in turn quasi-isomorphic to the cohomology of $F_2\Sigma_g$, via the natural projection $H^{\otimes 2} \to H^\bullet(F_2\Sigma_g,\mathbf Q)$.

A potential snag is that Morgan reasons a lot with minimal models in his paper. But I believe that he doesn't actually need any of that for what I said above (this part, I think, is only involved in putting mixed Hodge structures on various rational invariants). In particular the argument shouldn't require any simple connectedness/nilpotence.

  • 1
    $\begingroup$ You're welcome. Do let me know if you can't patch together a proof based on what I wrote here and I'll think it over some more and/or try to write more details. $\endgroup$ Commented Jun 22, 2018 at 19:03

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