Cohomology of affine Artin groups and applications
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- by Filippo Callegaro, Davide Moroni and Mario Salvetti PDF
- Trans. Amer. Math. Soc. 360 (2008), 4169-4188 Request permission
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.References
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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
- 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%
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 4169-4188
- MSC (2000): Primary 20J06, 20F36
- DOI: https://doi.org/10.1090/S0002-9947-08-04488-7
- MathSciNet review: 2395168