Geometry & Topology Monographs 13 (2008) 85–104

DOI: 10.2140/gtm.2008.13.85

arXiv: 0904.0457

Abstract

We consider two natural embeddings between Artin groups: the group G_Ãn-1 of type Ã_n-1 embeds into the group G_Bn of type B_n; G_Bn in turn embeds into the classical braid group Br_n+1:=G_An of type A_n. 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 G_Bn 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 G_Ãn as well as the cohomology of Br_n+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(π,1) spaces (in particular, for type Ã_n).

We point out that the above cohomology of G_Bn 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(G_Bn,1) onto the 2–torus.

Keywords

affine Artin groups, twisted cohomology, group representations

Mathematical Subject Classification

Primary: 20J06

Secondary: 20F36

References

Publication

Received: 30 May 2006
Revised: 17 January 2007
Accepted: 24 January 2007
Published: 22 February 2008

Authors