# 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

### Mario Salvetti

Università di Pisa, Italy### Filippo Callegaro

Scuola Normale Superiore, Pisa, Italy### Davide Moroni

National Research Council of Italy (CNR), Pisa, Italy

## Abstract

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. 1–22

DOI 10.4171/JEMS/187