Decomposition in Coxeter-chambers of the configuration space of $d$ marked points on the complex plane

N. C. Combe

Published 2018-08-24Version 1

In this paper we investigate a new decomposition in Coxeter-chambers of the $d$-th unordered configuration space of the complex plane space, using its natural relation with the space $Dpol_d$ of complex monic degree $d>0$ polynomials in one variable with simple roots. This decomposition relies on objects called {\it elementa} and which are the pieces of a good cover in the sense of Cech of $Dpol_{d}$. Each elementa is a set of polynomials indexed by a decorated graph, reminiscent of {\it Grothendieck's dessin's d'enfant}. The main result of this paper is that this new decomposition is invariant under a Coxeter group. This result has two major impacts. Using the good cover in elementa, the explicit calculation of Cech cohomology groups requires an exponential number of incidence relations to study. Therefore, using this new construction by chambers and galleries, the complexity is very much reduced. Secondly, the well known interpretation of the $d$-strand braid group $\mathcal{B}_{d}$, namely as the fundamental group of this configuration space, implies that any braid (which is a path in $\pi_{1}(Dpol_{d})$) can be obtained from a loop in one chamber.