Document - Topology - The genus of the configuration spaces for Artin groups of affine type

2014

Journal article Open Access

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

Moroni D., Salvetti M., Villa A.

Configuration spaces Schwarz genus Mathematics - Algebraic Topology 55N25 20J06 FOS: Mathematics Cohomology of groups Artin groups Algebraic Topology (math.AT) General Mathematics

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=?1(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}.

Source: Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni (Testo stamp.) 25 (2014): 233–248. doi:10.4171/RLM/676

Publisher: EMS Publishing House, Roma

Citations

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@article{oai:it.cnr:prodotti:285052,
	title = {Topology - The genus of the configuration spaces for Artin groups of affine type},
	author = {Moroni D. and Salvetti M. and Villa A.},
	publisher = {EMS Publishing House, Roma },
	doi = {10.4171/rlm/676 and 10.48550/arxiv.1404.2392},
	journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni (Testo stamp.)},
	volume = {25},
	pages = {233–248},
	year = {2014}
}