Istituto di Scienza e Tecnologie dell'Informazione     
Moroni D., Salvetti M., Villa A. Topology The genus of the configuration spaces for Artin groups of affine type. In: RENDICONTI LINCEI - MATEMATICA E APPLICAZIONI, vol. 25 (3) pp. 233 - 248. European Mathematical Society Publishing House, 2014.
Let (W,S) be a Coxeter system, S finite, and let GW be the associated Artin group. One has {it configuration spaces} Y, YW, where GW=π1(YW), and a natural W-covering fW: Y→YW. The {it Schwarz genus} g(fW) is a natural topological invariant to consider. In cite{salvdec2} it was computed for all finite-type Artin groups, with the exception of case An (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) be the simplicial scheme of all subsets J⊂S such that the parabolic group WJ is finite. We introduce the class of groups for which dim(K) equals the homological dimension of K, and we show that g(fW) 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, where XW⊂YW is a well-known CW-complex which has the same homotopy type as $mathbf Y_{mathbf W}.
URL: http://www.ems-ph.org/journals/show_abstract.php?issn=1120-6330&vol=25&iss=3&rank=2
DOI: 10.4171/RLM/676
Subject Configuration spaces
Schwarz genus
Artin groups
cohomology of groups

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