CW-presentation of configurations of points in plane and space 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:

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*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 ?


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