Cohomology of affine artin groups and applications

Callegaro F, Moroni D, Salvetti M

Twisted cohomology  Group representations  Mathematics - Algebraic Topology  20J06  FOS: Mathematics  20F36  Applied Mathematics  Affine Artin groups  Algebraic Topology (math.AT)  General Mathematics 

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[q±1, t±1]. 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(p, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

Source: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 360 (issue 8), pp. 4169-4188

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