\(K(n)\) Chern approximations of some finite groups
type: Item
instance of: scholarly article
title: \(K(n)\) Chern approximations of some finite groups
zbMATH Open document ID: 1254.55002
author: Björn Schuster
published in: Algebraic \& Geometric Topology
publication date: 2012-08-20T00:00:00Z
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55T25
zbMATH DE Number: 6068611
zbMATH Keywords: Morava \(K\)-theory
zbMATH Keywords: Chern approximation
reviewed by: Martin D. Crossley
MaRDI profile type: Publication
full work available at URL: agt.2012.12.1695
OpenAlex ID: W2049619667
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cites work: Q3827226
cites work: Q5753214
DOI: 10.2140/AGT.2012.12.1695
review text: 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.
description: scientific article; zbMATH DE number 6068611