## 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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### References:

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