Cohomology of affine Artin groups and applications

TL;DR

This paper explicitly computes the cohomology of certain Artin groups with non-trivial local coefficients, providing new algebraic and topological insights into their structure and associated spaces.

Contribution

It offers the first explicit cohomology calculations for Artin groups of types B_n and affine ilde{A}_n with specific local coefficients, extending known results.

Findings

Cohomology of type B_n Artin group computed over \\Q[q^{\\pm 1},t^{\\pm 1}]

Rational cohomology of affine type \\tilde{A}_n and classical braid groups obtained

Explicit CW-complexes constructed as K(\\pi,1) spaces with computed Euler characteristics

Abstract

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 \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 {Br}_{n} with coefficients in the n-dimensional representation presented in \cite{tong}. 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.