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I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the classyfing spaces of Braid groups, i.e. of the configuration space of $n$ unordered points in $\mathbb{R}^2$. (See for e.g. this survey paper on the $K(\pi,1)$ conjecture). If My understanding is correct (let me know if I got it wrong), the classyfing space of $B_n$ can be given as a CW-complex with a cell of dimension $k$ for each subsets of size $k-1$ of $\{1,\dots,n-1\}$.

I have two related question about this:

  1. First a reference request: Is there a references that explicitly describe these CW-complexes in the special cases of Braid groups without going though the general case of an Artin group ? or at least that spell out explicitly the description in the case of Braid groups ?

  2. Are there higher dimensional version of this ? i.e. "nice and explicit" CW-presentations of the configuration spaces of $n$ unordered points in $\mathbb{R}^d$ ?

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  • $\begingroup$ Technically the Salvetti complex is for ordered configuration spaces. I would recommend looking at the Fox Neuwirth Fuks cells: these aren't quite a CW decomposition of the configuration space, but they are a stratification into contractible pieces-- which is just as good for homotopical purposes. The combinatorics of the Fox-Neuwirth-Fuks stratification is essentially the same as the quotient of Salvetti by S_n. Also the stratification generalizes to configurations in R^d, and is closely related to Joyal's Theta_d category. $\endgroup$
    – Phil Tosteson
    Commented Dec 7, 2020 at 16:55
  • $\begingroup$ @PhilTosteson : That seems to be exactly what I'm after. Do you have a recommendation for a reference on the topic ? For the ordered vs unordered, yes of course: that's why I said "using the Salvetti complex", and not that the Salvetti complex was such a presentation. $\endgroup$
    – Simon Henry
    Commented Dec 7, 2020 at 18:03
  • $\begingroup$ I don't know a canonical source, but I think that this paper: arxiv.org/abs/1110.4137 and the references in it could be a good place to start, in particular Ayala and Hepworth's paper. $\endgroup$
    – Phil Tosteson
    Commented Dec 7, 2020 at 18:32
  • $\begingroup$ Thank you very much. You should write this as an answer ! $\endgroup$
    – Simon Henry
    Commented Dec 7, 2020 at 18:44
  • $\begingroup$ When the number of points $n$ is small compared to the dimension $d$ there are some very cute (and explicit) CW-decompositions coming from "electrostatic potential functions", i.e. take the function that is the sum of the inverse of the pairwise distances between the points. You need to replace $\mathbb R^d$ with $\mathbb D^d$. These functions give you Bott-style Morse functions that decompose these configuration spaces into a union of disc bundles over submanifolds. What do you want to use these CW-decompositions for? $\endgroup$
    – Ryan Budney
    Commented Dec 7, 2020 at 18:50

2 Answers 2

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The Fox-Neuwirth-Fuks stratification of ${\rm Conf}_n ~ \mathbb R^2$ is constructed by considering the projection map $\mathbb R^2 \to \mathbb R^1$. The image of a configuration under this projection is a subset of the real line. Considering the number of preimages of each element of this subset, we obtain an ordered integer partition of $n$ associated to each configuration. This construction defines a stratification of the configuration space into contractible pieces, and can be extended to higher dimensions by first considering the sequence of projection $$\mathbb R^d \to \mathbb R^{d-1} \to \mathbb R^{d-2} \to \dots .$$ The combinatorics that emerges is closely related to Joyal's category $\Theta_d$, roughly we can describe it as a "d-fold nested orderings of sets".

One source that has references to other literature is Giusti and Sinha's paper arxiv.org/abs/1110.4137.

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  • $\begingroup$ Thnaks ! I also found the paper by Ayala and Hepworth ( arxiv.org/abs/1202.2806) that you recommended in the comment very neat and perfectly fitting what I needed (and completely self contained). $\endgroup$
    – Simon Henry
    Commented Dec 8, 2020 at 16:35
  • $\begingroup$ If anyone looking at these ideas can find a simpler presentation of the cellular boundary which we give explicitly (we were the first to do so correctly, I think), let me know. $\endgroup$
    – Dev Sinha
    Commented Dec 8, 2020 at 19:54
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Here is the original paper by Fox and Neuwirth:

Fox, R.; Neuwirth, L. The braid groups. Math. Scand. 10 (1962), 119–126.

I remember reading this in the late 1970's in grad school, and found it clear enough that it was obvious to me how to generalize this to get a CW complex on all of the configuration spaces $B(\mathbb R^n,k)$.

Then there is Jeff Smith's thesis, eventually published:

Smith, Jeffrey Henderson Simplicial group models for $\Omega^n \Sigma^n X$. Israel J. Math. 66 (1989), no. 1-3, 330–350.

He gives an explicit simplicial $E_n$--operad.

This paper relates these:

Kashiwabara, Takuji On the homotopy type of configuration complexes. Algebraic topology (Oaxtepec, 1991), 159–170, Contemp. Math., 146, Amer. Math. Soc., Providence, RI, 1993.

Yes, all of these papers are pre ArXiv, but they should not be forgotten.

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