JournalsjemsVol. 12, No. 1pp. 1–22

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
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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