Mathematical Research Letters

Volume 1, Issue 5, September 1994 pp. 565-577.

The Homotopy Type of Artin Groups

Authors: Mario Salvetti
Author institution: Universita' di Pisa

Summary: Let $\bold W$ be a group generated by reflections in $\Bbb R^n$.\ $\bold W$ acts on the complement $\bold Y \subset \Bbb C^n$ of the complexification of the reflection hyperplanes of $\bold W$. The fundamental group of the orbit space $\bold Y /\bold W$ is the so called $Artin\ group$ of type $\bold W$. Here we give a new description of the homotopy type of $\bold Y / \bold W$ in terms of a convex polyhedrum in $\Bbb R^n$ with identifications on the faces. Such identifications are quite easy to describe and are naturally connected to the combinatorics of $\bold W$. We derive an associated algebraic complex which computes the cohomology of local systems on $\bold Y / \bold W$: its $k^{\text{th}}$-module is freely generated by the $k$-$subsets$ of $\{ 1,\dots ,n\}$ 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 $\bold W$ for all the exceptional groups.

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