On group homomorphisms inducing mod-\(p\) cohomology isomorphisms
type: Item
instance of: scholarly article
title: On group homomorphisms inducing mod-\(p\) cohomology isomorphisms
zbMATH Open document ID: 0713.55009
DOI: 10.1007/BF02566619
author: Guido Mislin
published in: Commentarii Mathematici Helvetici
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140206
review text: In this beautiful paper the author gives a necessary and sufficient condition for for a homomorphism \(\rho:G\to H\) between compact Lie groups to induce an isomorphism on \(Z_ p\) cohomology of the classifying spaces. The following are equivalent: A) \(\rho^*:H^*(BH;Z_ p)\to H^*(BG;Z_ P)\) is an isomorphism. B) \(Rep(\rho):Rep(\pi,G)\to Rep(\pi,H)\) is a bijection for every finite \(p\)-group \(\pi\), where \(Rep(\pi,G)\) is the set of all \(H\)-conjugacy classes of homomorphisms of \(\pi\) into \(G\). As corollaries, we have S. Jackowski's result that if \(G\) and \(H\) are finite groups so that a homomorphism \(\rho\) induces an isomorphism on mod \(p\) cohomology, then the order of the kernel is prime to \(p\) and the index of the image is prime to \(p\). If \(\rho\) induces an isomorphism and is a homomorphism between compact Lie groups, then for every homomorphism \(\phi:\pi\to G\) the induced map of Weyl-groups \(\rho_*: W(\phi)\to W(\rho\phi)\) is a group isomorphism when \(\pi\) is a finite \(p\)-group. Here the Weyl-group of a homomorphism is defined to be the quotient of the normalizer by the centralizer of the image of \(\rho\).
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 4173908
zbMATH Keywords: representation
zbMATH Keywords: compact Lie groups
zbMATH Keywords: cohomology
zbMATH Keywords: classifying spaces
zbMATH Keywords: \(p\)-group
zbMATH Keywords: Weyl-group
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4173908