The Homotopy Type of Artin Groups

Let $W$ be a group generated by reflections in $\Bbb R^n$. $W$ acts on the complement $Y \subset \Bbb C^n$ of the complexification of the reflection hyperplanes 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 $\Bbb R^n$ 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 $\{ 1,\dots ,n\}$ 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 $W$ for all the exceptional groups.

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Published 1 January 1994