Istituto di Scienza e Tecnologie dell'Informazione     
Callegaro F., Moroni D., Salvetti M. Cohomology of affine artin groups and applications. In: Transactions of the American Mathematical Society, vol. 360 (8) pp. 4169 - 4188. American Mathematical Society, 2008.
The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and A^~_n with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module Q[q1, t1]. Here the first n − 1 standard generators of the group act by (−q)-multiplication, while the last one acts by (−t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type A^_n as well as the cohomology of the classical braid group Br_n with coefficients in the n-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.
URL: http://www.ams.org/journals/tran/2008-360-08/
DOI: 10.1090/S0002-9947-08-04488-7
Subject Affine Artin groups
Twisted cohomology
Group representations
I.1 Symbolic and Algebraic Manipulation
20J06 Group theory and generalizations. Cohomology of groups
20F36 Group theory and generalizations. Braid groups
20F36 Group theory and generalizations. Artin groups

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