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:

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 $\mathbb{R}^d$ ?

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