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Rendiconti Lincei - Matematica e Applicazioni

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Volume 25, Issue 3, 2014, pp. 233–248
DOI: 10.4171/RLM/676

Published online: 2014-08-31

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

Davide Moroni[1], Mario Salvetti[2] and Andrea Villa[3]

(1) National Research Council of Italy (CNR), Pisa, Italy
(2) Università di Pisa, Italy
(3) ISI Foundation, Torino, Italy

Let $(\mathbf W,S)$ be a Coxeter system, $S$ finite, and let $\mathbf G_{\mathbf W}$ be the associated Artin group. One has {\it 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 {\it Schwarz genus} $g(f_{\mathbf W})$ is a natural topological invariant to consider. In \cite{salvdec2} it was computed for all finite-type Artin groups, with the exception of case $A_n$ (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 $K=K(\mathbf W,S)$ 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}.

Keywords: Configuration spaces, Schwarz genus, Artin groups, cohomology of groups

Moroni Davide, Salvetti Mario, Villa Andrea: The genus of the configuration spaces for Artin groups of affine type. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), 233-248. doi: 10.4171/RLM/676