Weyl-Coxeter chambers and the decomposition of the n-th (unordered) configuration space of $\mathbb{C}$

The objective of this talk is to introduce a new topological stratification of the n-th (unordered) configuration space of the complex plane,  using objects reminiscent of Grothendieck’s dessins d’enfant.   
This topological stratification, is used to construct a good cover in the sense of Cech, allowing explicit cohomological computations of this space with coefficients in any sheaf.  As an auxiliary result, we show that this stratification forms a decomposition invariant under a Coxeter group.