Some topological problems on the configuration spaces of Artin and Coxeter groups

Moroni D., Salvetti M., Villa A.

Artin Groups  Group representations  Twisted cohomology  Schwartz genus  Coxeter groups 

This paper contains some results regarding some particular topological aspects in the theory of Artin and Coxeter groups. Among other things, we compute the Schwartz genus of the covering associated to the orbit space for all affine Artin groups. Our paper contains also a brief review of some results, concerning the topol- ogy of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated) which are essential to our computations. Other reviews, even if considering some interesting aspects of the theory, are not very satisfying about the topological underlying structure. Our review will still be very partial: the more than thirty years old literature on the subject would require a much longer paper. We concentrate essentially on a single line of research, which (in our opinion) gives the possibility to produce a clear and neat picture of some basic topological situation underlying all in very few pages. The new results that we present here concern some computations of: - the cohomology of the braid groups with non-abelian coefficients, coming from geometric representations of the braid groups into the homology of an orientable surfaces; -the computation of the Schwartz genus of the covering associated to the orbit space of an affine Artin group. We skip the details of the computations of the first application, while we give details for the second one. We introduce here a particularly interesting class of "sheaves over posets", which we call "weighted sheaves over posets". We use them for some explicit computations for the top cohomology of the affine group An tilde. A natural spectral sequence is associated to such sheaves. We are going to exploit this general construction in future works.

Source: ISTI Technical reports, 2011