2008

Journal article
Open Access

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 360 (2008): 4169–4188. doi:10.1090/S0002-9947-08-04488-7

**Publisher: **American Mathematical Society., Providence, R.I. [etc.], Stati Uniti d'America

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@article{oai:it.cnr:prodotti:44156, title = {Cohomology of affine artin groups and applications}, author = {Callegaro F. and Moroni D. and Salvetti M.}, publisher = {American Mathematical Society., Providence, R.I. [etc.], Stati Uniti d'America}, doi = {10.1090/s0002-9947-08-04488-7 and 10.48550/arxiv.0705.2823}, journal = {Transactions of the American Mathematical Society}, volume = {360}, pages = {4169–4188}, year = {2008} }