Cohomology of Affine Artin Groups and Applications

The result of this paper is the determination of the cohomology of Artin groups of type An, Bn and Ãn 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 affine type Ãn as well as the cohomology of the classical braid group Brn 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(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.