Cohomology of Artin groups of type tilde{A}_n, B_n and applications

Details Cohomology of Artin groups of type tilde{A}_n, B_n and applications Cohomology of Artin groups of type tilde{A}_n, B_n and applications

2009-04-02

arxiv.org/abs/0904.0457v1

Geom. Topol. Monogr. 13 (2008) 85-104

Mathematics

Group Theory

This is the version published by Geometry & Topology Monographs on 22 February 2008

Scientific paper

10.2140/gtm.2008.13.85

We consider two natural embeddings between Artin groups: the group G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} 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_{B_n} 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_{tilde{A}_n} 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(pi,1) spaces (in particular, for type tilde{A}_n). We point out that the above cohomology of G_{B_n} 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_{B_n},1) onto the 2-torus.

Affiliated with

Also associated with

No associations

Rating

Cohomology of Artin groups of type tilde{A}_n, B_n and applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cohomology of Artin groups of type tilde{A}_n, B_n and applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cohomology of Artin groups of type tilde{A}_n, B_n and applications will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-150093