The genus of the configuration spaces for Artin groups of affine type (Q467374)

For a locally trivial fibration \(f:Y\to X\), let \(g(f)\) denote the Schwartz genus given by the minimum cardinality of an open covering \(\mathcal{U}\) of \(X\) such that for each open set \(U\in \mathcal{U}\) there is a section of \(f\) over \(U\). For a finite Coxeter group \(\mathbf{W}\) of rank \(n\), it is known that there is a configuration space \(\mathbf{Y}\) with a natural free action of \(\mathbf{W}\), and let \(f_{\mathbf{W}}:\mathbf{W}\to\mathbf{Y}_{\mathbf{W}}\) be the regular covering given by the natural quotient, where we set \(\mathbf{Y}_{\mathbf{W}}:=\mathbf{Y}/\mathbf{W}\). Note that the Schwartz genus \(g(f_{\mathbf{W}})\) is a natural topological invariant, and it is known that it was already determined for all finite type Artin groups except the case of type \(A_n\).NEWLINENEWLINE In this paper the authors generalize the above known result by computing the Schwartz genus for a class of Artin groups which includes the affine type Artin groups. More precisely, for a Coxeter system \((\mathbf{W},S)\) let \(K=K(\mathbf{W},S)\) denote the simplicial scheme of all subsets \(J\subset S\) such that the parabolic group \(\mathbf{W}_J\) is finite. In particular, they show that \(g(f_{\mathbf{W}})\) is always the maximum possible for such class of groups and they prove the equality \(g(f_{\mathbf{W}})=vd(\mathbf{G}_{\mathbf{W}})+1\), where \(\mathbf{G}_{\mathbf{W}}= \pi_1(\mathbf{Y}_{\mathbf{W}})\) and \(vd(\mathbf{G}_{\mathbf{W}})\) denotes the virtual dimension of the group \(\mathbf{G}_{\mathbf{W}}\).