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 (A) over tilde (n) with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type Bn with coefficients over the module Q[q(+/- 1), t(+/- 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 a. ne type (A) over tilde (n) as well as the cohomology of the classical braid group 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.