Stability for the cohomology of Artin groups
type: Item
instance of: scholarly article
title: Stability for the cohomology of Artin groups
zbMATH Open document ID: 0982.20036
author: Corrado De Concini
author: Mario Salvetti
published in: Advances in Mathematics
publication date: 2000-09-14T00:00:00Z
review text: The authors consider the braid groups \(B^{[n]}\) of type \(A_n\), \(B_n\) and \(C_n\). For each type one has the natural inclusion \(j_n\colon B^{[n]}\to B^{[n+1]}\). Let \(R_n(q)\) be the \(\mathbb{Z}[B_n]\) module obtained by having the generators of \(B^{[n]}\) act by multiplication by \(-q\), \(q\) is an invertible element in some domain with unit. Then \(j_n\) induces a homomorphism of \(r\)-th cohomology groups: \[ j_n^*\colon H^r(B^{[n+1]},R_{n+1}(q))\to H^r(B^{[n]},R_n(q)). \] The authors show that \(j_n^*\) is an isomorphism if \(n\geq 3(r+1)\). They call these stability properties. The proofs are given separately for each type. In some cases they give sharper bounds and show that their results are the sharpest possible.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 1340113
zbMATH Keywords: Artin braid groups
zbMATH Keywords: cohomology groups
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: aima.1998.1820
OpenAlex ID: W2049825411
cites work: The cohomology ring of the colored braid group
cites work: Q3941610
cites work: Cohomology of Artin groups
cites work: The top-cohomology of Artin groups with coefficients in rank-1 local systems over \(\mathbb{Z}\)
cites work: Q4185034
cites work: The homotopy type of Artin groups
DOI: 10.1006/AIMA.1998.1820
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description: scientific article; zbMATH DE number 1340113