Cohomology of affine Artin groups and applications

Abstract

Abstract. The result of this paper is the determination of the cohomology of Artin groups of type A n , B n andà 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 ±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 typeà 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(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

Combinatorics and topology of toric arrangements defined by root systems

“…In this section we prove that the Euler characteristic of R, denoted by χ Φ , is equal to (−1) n |W |. This may also be seen as a consequence of [4,Prop. 5.3].…”

Combinatorial methods for the twisted cohomology of Artin groups

“…In all cases, the topology of the orbit space Y W := Y/W is interesting to study. Its cellular description is given in [23] (see also [6][7][8]21]), where a CW-complex X W which is a deformation retract is explicitly described.…”

K(π,1) conjecture for Artin groups

“…We show that Sal(Γ) coincides with Sal(A), where A is the Coxeter arrangement of (W, S) (see Theorem 3.3). Moreover, we prove that the homotopy equivalence Sal(Γ) → M (W, S) is equivariant under the action of W and induces a homotopy equivalence [7,8,9,10,11,12,20,21,22,23,24,45,46,48,49]). …”