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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cohomology of affine Artin groups and applications


Authors: Filippo Callegaro, Davide Moroni and Mario Salvetti
Journal: Trans. Amer. Math. Soc. 360 (2008), 4169-4188
MSC (2000): Primary 20J06, 20F36
Published electronically: March 11, 2008
MathSciNet review: 2395168
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Abstract: The result of this paper is the determination of the cohomology of Artin groups of type $ A_n, B_n$ and $ \tilde{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 $ \mathbb{Q}[q^{\pm 1},t^{\pm 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 $ \tilde{A}_{n}$ as well as the cohomology of the classical braid group $ \mathrm{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(\pi,1)$ spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.


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Additional Information

Filippo Callegaro
Affiliation: Scuola Normale Superiore, P.za dei Cavalieri, 7, Pisa, Italy
Email: f.callegaro@sns.it

Davide Moroni
Affiliation: Dipartimento di Matematica “G.Castelnuovo”, P.za A. Moro, 2, Roma, Italy – and – ISTI-CNR, Via G. Moruzzi, 3, Pisa, Italy
Email: davide.moroni@isti.cnr.it

Mario Salvetti
Affiliation: Dipartimento di Matematica “L.Tonelli”, Largo B. Pontecorvo, 5, Pisa, Italy
Email: salvetti@dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04488-7
PII: S 0002-9947(08)04488-7
Keywords: Affine Artin groups, twisted cohomology, group representations
Received by editor(s): June 20, 2006
Published electronically: March 11, 2008
Additional Notes: The third author is partially supported by M.U.R.S.T. 40%
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.