Cohomology of Artin groups of type Ãn, Bn and applications

We consider two natural embeddings between Artin groups: the group GÃn-1 of type Ãn-1 embeds into the group GBn of type Bn; GBn in turn embeds into the classical braid group Brn+1:=GAn of type An. The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of GBn with coefficients over the module Q[q±1,t±1], where the action is (-q)-multiplication for the standard generators associated to the first n-1 nodes of the Dynkin diagram, while is (-t)-multiplication for the generator associated to the last node. As a corollary we obtain the rational cohomology for GÃn as well as the cohomology of Brn+1 with coefficients in the (n+1)-dimensional representation obtained by Tong, Yang and Ma. We stress the topological significance, recalling some constructions of explicit finite CW-complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case. For affine groups, some of these orbit spaces are known to be K(?,1) spaces (in particular, for type Ãn). We point out that the above cohomology of GBn gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of K(GBn,1) onto the 2-torus.