Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type $A_n, B_n$ and $\tilde{A}_{n}$ with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type $B_n$ with coefficients over the module $\mathbb{Q}[q^{\pm 1},t^{\pm 1}].$ Here the first $n-1$ standard generators of the group act by $(-q)$-multiplication, while the last one acts by $(-t)$-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type $\tilde{A}_{n}$ as well as the cohomology of the classical braid group $\mathrm{Br}_{n}$ with coefficients in the $n$-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be $K(\pi,1)$ spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.