The genus of the configuration spaces for Artin groups of affine type

Abstract

Let $(\mathbf{W},S)$ be a Coxeter system, $S$ finite, and let ${\mathbf{G}}_{\mathbf{W}}$ be the associated Artin group. One has _ configuration spaces_ $\mathbf{Y},{\mathbf{Y}}_{\mathbf{W}},$ where ${\mathbf{G}}_{\mathbf{W}}={\pi }_1({\mathbf{Y}}_{\mathbf{W}}),$ and a natural $\mathbf{W}$-covering $f_{\mathbf{W}}:\mathbf{Y}\to {\mathbf{Y}}_{\mathbf{W}}.$ The_Schwarz genus_ $g(f_{\mathbf{W}})$ is a natural topological invariant to consider. In [DS00] it was computed for all finite-type Artin groups, with the exception of case $A_n$ (for which see [Vas92],[DPS04]). In this paper we generalize this result by computing the Schwarz genus for a class of Artin groups, which includes the affine-type Artin groups. Let $K=K(\mathbf{W},S)$ be the simplicial scheme of all subsets $J\subset S$ such that the parabolic group ${\mathbf{W}}_J$ is finite. We introduce the class of groups for which $\dim (K)$ equals the homological dimension of $K,$ and we show that $g(f_{\mathbf{W}})$ is always the maximum possible for such class of groups. For affine Artin groups, such maximum reduces to the rank of the group. In general, it is given by $\dim ({\mathbf{X}}_{\mathbf{W}})+1,$ where ${\mathbf{X}}_{\mathbf{W}}\subset {\mathbf{Y}}_{\mathbf{W}}$ is a well-known $CW$-complex which has the same homotopy type as ${\mathbf{Y}}_{\mathbf{W}}$.