On group homomorphisms inducing mod-\(p\) cohomology isomorphisms
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 \(\rho\).
scientific article; zbMATH DE number 4173908