The K(pi, 1) problem for the affine Artin group of type (B)over-tilde(n) and its cohomology

Callegaro F, Moroni D, Salvetti M

Affine Artin groups  Group representations  Twisted cohomology  Computational Algebraic Topology 

We prove that the complement to the affine complex arrangement of type (B) over tilde (n) is a K(pi, 1) space. We also compute the cohomology of the affine Artin group G (B) over tilde (n) ( of type (B) over tilde (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) over tilde (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) over tilde (n) with trivial coefficients is derived from the previous one.

Source: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, vol. 12, pp. 1-22

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