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

Abstract

Let $(W,S)$ be a Coxeter system, $S$ finite, and let $G_W$ be the associated Artin group. One has configuration spaces $Y,Y_W,$ where $G_W={\pi }_1(Y_W),$ and a natural $W$-covering $f_W:Y\to Y_W.$ The Schwarz genus $g(f_W)$ is a natural topological invariant to consider. 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 $K=K(W,S)$ be the simplicial scheme of all subsets $J\subset S$ such that the parabolic group $W_J$ is finite. We introduce the class of groups for which $dim(K)$ equals the homological dimension of $K,$ and we show that $g(f_W)$ 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 $dim(X_W)+1,$ where $X_W\subset Y_W$ is a well-known $CW$-complex which has the same homotopy type as $Y_W.$

To appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl

References in corpus (1)

• Some topological problems on the configuration spaces of Artin and Coxeter groups