We prove that the complement to the afﬁne complex arrangement of type B__n is a K(π,1) space. We also compute the cohomology of the afﬁne Artin group G__B__n (of type B__n) with coefﬁcients in interesting local systems. In particular, we consider the module ℚ[q_±1 , t_±1], where the ﬁrst n standard generators of G__B__n act by (−_q)-multiplication while the last generator acts by (−_t) multiplication. Such a representation generalizes the analogous 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor ﬁbre. The cohomology of G__B__n with trivial coefﬁcients is derived from the previous one.
Cite this article
Mario Salvetti, Filippo Callegaro, Davide Moroni, The <em>K</em>(<em>π</em>,1) problem for the afﬁne Artin group of type <span style="text-decoration: overline;"><em>B</em></span><sub><em>n</em></sub> and its cohomology. J. Eur. Math. Soc. 12 (2010), no. 1, pp. 1–22DOI 10.4171/JEMS/187