JournalsjemsVol. 12 , No. 1DOI 10.4171/jems/187

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
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 cover

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.