A Model for Configuration Spaces of Closed Manifolds

Published .

Last week I was at the Max Planck Institute for the Conference for Young researchers in homotopy theory and categorical structures (which was, by the way, a great conference -- thanks to the organizers), and I gave yet another talk about the Lambrechts--Stanley model for configuration spaces. So maybe it's time I write a little bit about it on this blog. I'll write a first post about the model itself, and later I will explain how the Fulton--MacPherson operad is involved in all this.

Consider a manifold MM. Then given some integer k0k \ge 0, one can build the configuration space of kk points in MM:

Confk(M)={(x1,,xk)M×kxixj  ij}.\mathrm{Conf}_k(M) = \{ (x_1, \dots, x_k) \in M^{\times k} \mid x_i \neq x_j \; \forall i \neq j \}.

Many people are interested in configuration spaces for many different reasons. One could wonder: what does the homotopy type of MM tell us about the homotopy type of Confk(M)\mathrm{Conf}_k(M)? More concretely, does the homotopy type of Confk(M)\mathrm{Conf}_k(M) depend exclusively on the homotopy type of MM?

Of course, if we don't know anything about MM, then this is blatantly false. Indeed, consider that Conf2(R)\mathrm{Conf}_2(\mathbb{R}) is homotopy equivalent to S0S^0, whereas Conf2(R2)\mathrm{Conf}_2(\mathbb{R}^2) is homotopy equivalent to S1S^1, even though RR2\mathbb{R} \simeq \mathbb{R}^2. The main defect, here, is that MM is not a closed manifold, so we might as well restrict ourselves to studying closed manifolds.

But even then, the question could appear hopeless. Indeed, [Longoni--Salvatore, 2005], found a counterexample: they showed that some 33-dimensional lens spaces are homotopy equivalent and yet their configuration spaces are not homotopy equivalent. But not all hope is lost! Indeed, these lens spaces are not simply connected. So maybe if we restrict to simply connected manifolds, things will work out.

I'll admit it, I'm only comfortable over fields of characteristic zero. And over ℚ, we can use Sullivan's framework of rational models to study the real homotopy type of spaces. So let us take a rational model AA of MM, and try to build a rational model of Confk(M)\mathrm{Conf}_k(M) from it.

If you recall, we are looking at a closed manifold MM. This feature appears on the level of cohomology in the form of Poincaré duality. Well, it also appears on the level of rational models! [Lambrechts--Stanley, 2008] showed that any simply-connected closed manifold has a rational model which is a Poincaré duality CDGA. Roughly speaking, it means that it has a non-degenerate dg-pairing with itself, of formal dimension nn.

Intuitively, we can look at the configuration space Confk(M)\mathrm{Conf}_k(M) as the product M×kM^{\times k} from which we removed the fat diagonal ijΔij\bigcup_{i \neq j} \Delta_{ij}, where Δij={xM×kxi=xj}\Delta_{ij} = \{ x \in M^{\times k} \mid x_i = x_j \}. We can then reuse the ideas of Poincaré--Lefschetz duality and say that, morally, a model of Confk(M)\mathrm{Conf}_k(M) should be given by a model of M×kM^{\times k} in which we kill the classes that are Poincaré dual to homology classes from the fat diagonal. We can take AkA^{\otimes k} to be the model of M×kM^{\times k}, and the Poincaré duality allows to build explicit representatives of the fundamental classes of the diagonals Δij\Delta_{ij}.

This is precisely what [Lambrechts--Stanley, 2008] did, and they considered a CDGA GA(k)\mathtt{G}_A(k) given by:

GA(k)=(AkS(ωij)/relations,dωij=[Δij]).\mathtt{G}_A(k) = \bigl( A^{\otimes k} \otimes S(\omega_{ij}) / \text{relations}, d \omega_{ij} = [\Delta_{ij}] \bigr).

For small kk, this CDGA is particularly simple:

  • GA(0)\mathtt{G}_A(0) is isomorphic to ℝ, which is indeed a model for Conf0(M)={}\mathrm{Conf}_0(M) = \{ \varnothing \};
  • GA(1)\mathtt{G}_A(1) is isomorphic to AA, which is a model for Conf1(M)=M\mathrm{Conf}_1(M) = M;
  • and GA(2)\mathtt{G}_A(2) is the mapping cone of AAA,a(a1)[Δ]A \to A \otimes A, \, a \mapsto (a \otimes 1) [\Delta]; it is quasi-isomorphic to A2/(Δ)A^{\otimes 2} / (\Delta).

This model has a long history, which I'll try to summarize here (hopefully without forgetting anything):

  • Around 1969, Arnold (for n=2n = 2) and Cohen (for a general nn) described the cohomology of Confk(Rn)\mathrm{Conf}_k(\mathbb{R}^n); it is suspiciously similar to GA(k)\mathtt{G}_A(k) where we take AA to be H(Rn)H^*(\mathbb{R}^n) and the diagonal classes vanish.
  • In 1978, Cohen and Taylor built a spectral sequence which converges to H(Confk(M))H^*(\mathrm{Conf}_k(M)) and whose E2E^2 term is precisely given by GH(M)(k)\mathtt{G}_{H^*(M)}(k).
  • In 1991, Bendersky and Gitler built another spectral sequence which converges instead to the homology of Confk(M)\mathrm{Conf}_k(M).
  • Around 1994, two independent results were found in the case where MM is a smooth projective complex manifold. Note that in this case, MM has to be a Kähler manifold, and it is thus formal by a result of [Deligne--Griffiths--Morgan--Sullivan, 1975], i.e. H(M)H^*(M) is a rational model of MM.
    • Totaro showed that the Cohen--Taylor spectral sequence collapses at the E2E^2 page, and therefore that H(GH(M)(k))H^*(\mathtt{G}_{H^*(M)}(k)) is isomorphic to H(Confk(M))H^*(\mathrm{Conf}_k(M)) as a graded algebra (by a result of Deligne);
    • Kriz showed that in this case, GH(M)(k)\mathtt{G}_{H^*(M)}(k) is, in fact, a rational model for Confk(M)\mathrm{Conf}_k(M).
  • In 2004, Lambrechts and Stanley showed that A2/(Δ)A^{\otimes 2} / (\Delta) is indeed a model for Conf2(M)\mathrm{Conf}_2(M) when MM is 22-connected.
  • Also around 2004, Félix and Thomas, and Berceanu, Markl and Papadima, showed independently that the dual of GH(M)(k)\mathtt{G}_{H^*(M)}(k) is isomorphic to the dual of the Bendersky--Gitler spectral sequence.
  • In 2008, Lambrechts and Stanley showed that GA(k)\mathtt{G}_A(k) has the same cohomology as Confk(M)\mathrm{Conf}_k(M) as a graded Σk\Sigma_k-vector space. This is, to my knowledge, the first time that the model GH(M)(k)\mathtt{G}_{H^*(M)}(k) was built out of a general Poincaré duality CDGA (and not just the cohomology of MM), hence my choice for the name of the model.
  • Finally, in 2015, Cordova Bulens showed that A2/(Δ)A^{\otimes 2} / (\Delta) is a rational model for Conf2(M)\mathrm{Conf}_2(M) when MM is even dimensional and simply connected.

We now get to my contribution:

Theorem. Let MM be a smooth, simply connected manifold of dimension at least 44. Then GA(k)\mathtt{G}_A(k) is a real model for Confk(M)\mathrm{Conf}_k(M), for all k0k \ge 0.

Unlike the theorems mentioned before, this describes a model over ℝ, not over ℚ. Descent would be an interesting (but difficult problem). Note that since MM is at least 33-dimensional in the theorem, using the Fadell--Neuwirth fibrations one can show that all the Confk(M)\mathrm{Conf}_k(M) remain simply connected, hence AA completely determines the real homotopy type of Confk(M)\mathrm{Conf}_k(M).

I actually showed more than that: the model is compatible with the action of the Fulton--MacPherson operad, by using a proof inspired by Kontsevich's proof of the formality of the little disks operad. What does this mean? How does the proof go? Stay tuned! (Alternatively, you can read my article :wink:.)