Stability for the cohomology of Artin groups
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.
scientific article; zbMATH DE number 1340113