\(K(n)\) Chern approximations of some finite groups
For a complex oriented cohomology theory, \(E\), and a finite group \(G\), the \textit{Chern approximation} of \(E^*(BG)\) was introduced by \textit{N. P. Strickland} [Topology 40, No. 6, 1167--1216 (2001; Zbl 1005.55001)] and is built from the Chern classes of irreducible complex representations of \(G\), subject to relations obtained from the representation ring and the \(\lambda\)-operations.NEWLINENEWLINEThis paper studies the Chern approximation for Morava \(K\)-theory and certain \(2\)-groups and \(3\)-groups. In particular, for the dihedral and quaternion groups of order \(8\), and for quasidihedral groups, the Chern approximation is shown to be exact, i.e., isomorphic to the Morava \(K\)-theory of \(BG\). And for the nonabelian group of order \(27\) and exponent \(3\), it is shown that the Chern approximation is not exact, because the Chern classes fail to provide all the relations that hold in the Morava \(K\)-theory.
scientific article; zbMATH DE number 6068611
scientific article