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.

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.