The $K(\pi ,1)$ problem for the affine Artin group of type ${\tilde{B}}_n$ and its cohomology

Abstract

We prove that the complement to the affine complex arrangement of type ${\tilde{B}}_n$ is a $K(\pi ,1)$ space. We also compute the cohomology of the affine Artin group $G_{{\tilde{B}}_n}$ (of type ${\tilde{B}}_n$) with coefficients in interesting local systems. In particular, we consider the module $Q[q^{\pm 1},t^{\pm 1}]$, where the first $n$ standard generators of $G_{{\tilde{B}}_n}$ act by $(-q)$-multiplication while the last generator acts by $(-t)$ multiplication. Such a representation generalizes the analogous 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of $G_{{\tilde{B}}_n}$ with trivial coefficients is derived from the previous one.