A counter-example to the admissibility of the \(\gamma\)-filtration on 2- groups
The main result in this paper is that Atiyah's conjecture on the equivalence of the topological and \(\gamma\)-filtrations on the representation ring \(R(G)\) fails when \(G = D^r\), \(r\geq 3\), the central product of \(r\) copies of the dihedral group of order 8 (extra special). The method is to show that the \(\gamma\)-filtration is not `admissible' in the sense that it behaves badly with respect to induction. In passing it is also shown that for \(r = 1\) or 2 we do have admissibility. As with all the counterexamples now known, the filtrations are equivalent infinitely often, i.e. \(R^\gamma_{2k} = R^{\text{top}}_{2k}\), provided certain congruence conditions on \(k\) are satisfied. It would be good to have a systematic explanation of this phenomenon.
central product of \(r\) copies of the dihedral group of order 8
scientific article; zbMATH DE number 614221