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The top-cohomology of Artin groups with coefficients in rank-1 local systems over \(\mathbb{Z}\). (English) Zbl 0878.55003

Summary: Let \(W\) be a Coxeter group and let \(G_W\) be the associated Artin group. We consider the local system over \(k(G_W,1)\) with coefficients in \(R=\mathbb{Z}[q,q^{-1}]\) which associates to the standard generators of \(G_W\) the multiplication by \(q\). For the entire list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we compute is a sort of Alexander ideal for a hypersurface.
In case of the classical braid group \(\text{Br}_n\) this ideal is the principal ideal generated by the \(n\)th cyclotomic polynomial.
We use these results to calculate the topological category of \(k(G_W,1)\): we prove that it equals the obvious bound given by obstruction theory (so, in case of braid group \(\text{Br}_n\), it is exactly \(n\)).

MSC:

55N25 Homology with local coefficients, equivariant cohomology
20F36 Braid groups; Artin groups
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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