The Homotopy Type of Artin Groups

Abstract

Let W be a group generated by reflections in R n. W acts on the complement Y ⊂ C n of the complexification of the reflection hy-perplanes of W. The fundamental group of the orbit space Y/W is the so called Artin group of type W. Here we give a new description of the homotopy type of Y/W in terms of a convex polyhedrum in R n with identifications on the faces. Such identifications are quite easy to describe and are naturally connected to the combinatorics of W. We derive an associated algebraic complex which computes the cohomology of local systems on Y/W: its k th-module is freely generated by the k-subsets of {1,. .. , n} and the coboundary is explicitly given by a formula involving the Poincaré series of the group. In particular, we are able to compute the cohomology of the Artin group associated to W for all the exceptional groups.