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

TL;DR

This paper proves that the complement of a specific affine complex arrangement is a K(, 1) space and computes the cohomology of the associated affine Artin group with various local system coefficients, revealing new algebraic topological properties.

Contribution

It establishes the K(, 1) property for the affine complex arrangement of type B_n and computes the cohomology of the affine Artin group with novel local system coefficients.

Findings

Complement is a K(, 1) space.

Cohomology computed with local systems involving q and t.

Derived cohomology with trivial coefficients.

Abstract

In this paper we prove that the complement to the affine complex arrangement of type \widetilde{B}_n is a K(\pi, 1) space. We also compute the cohomology of the affine Artin group G of type \widetilde{B}_n with coefficients over several interesting local systems. In particular, we consider the module Q[q^{\pm 1}, t^{\pm 1}], where the first n-standard generators of G act by (-q)-multiplication while the last generator acts by (-t)-multiplication. Such representation generalizes the analog 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 with trivial coefficients is derived from the previous one.