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](https://arxiv.org/abs/1211.7339) 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 $\mathbb{R}^d$ ?



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


