JournalsrlmVol. 25, No. 3pp. 233–248

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

  • Davide Moroni

    National Research Council of Italy (CNR), Pisa, Italy
  • Mario Salvetti

    Università di Pisa, Italy
  • Andrea Villa

    ISI Foundation, Torino, Italy
The genus of the configuration spaces for Artin groups of affine type cover

Abstract

Let (W,S)(\mathbf W,S) be a Coxeter system, SS finite, and let GW\mathbf G_{\mathbf W} be the associated Artin group. One has {\it configuration spaces} Y, YW,\mathbf Y,\ \mathbf Y_{\mathbf W}, where GW=π1(YW),\mathbf G_{\mathbf W}=\pi_1(\mathbf Y_{\mathbf W}), and a natural W\mathbf W-covering fW: YYW.f_{\mathbf W}:\ \mathbf Y\to\mathbf Y_{\mathbf W}. The {\it Schwarz genus} g(fW)g(f_{\mathbf W}) is a natural topological invariant to consider. In \cite{salvdec2} it was computed for all finite-type Artin groups, with the exception of case AnA_n (for which see \cite{vassiliev},\cite{salvdecproc3}). 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)K=K(\mathbf W,S) be the simplicial scheme of all subsets JSJ\subset S such that the parabolic group WJ\mathbf W_J is finite. We introduce the class of groups for which dim(K)dim(K) equals the homological dimension of K,K, and we show that g(fW)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(XW)+1,dim(\mathbf X_{\mathbf W})+1, where XWYW\mathbf X_{\mathbf W}\subset \mathbf Y_{\mathbf W} is a well-known CWCW-complex which has the same homotopy type as $\mathbf Y_{\mathbf W}.

Cite this article

Davide Moroni, Mario Salvetti, Andrea Villa, The genus of the configuration spaces for Artin groups of affine type. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 3, pp. 233–248

DOI 10.4171/RLM/676