Squaring operations in the 4-connective fibre spaces over the classifying spaces of the exceptional Lie groups
type: Item
instance of: scholarly article
title: Squaring operations in the 4-connective fibre spaces over the classifying spaces of the exceptional Lie groups
zbMATH Open document ID: 0606.55013
author: Akira Kono
author: Kiminao Ishitoya
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1985-01-01T00:00:00Z
review text: Let G be a compact, connected and simply connected Lie group of the exceptional type. Denote by \(B\tilde G\) the homotopy fiber of the map \(y_ 4: BG\to K({\mathbb{Z}},4)\) determined by a generator \(y_ 4\in H^ 4(BG; {\mathbb{Z}})\). The authors compute the action of the Steenrod algebra on \(H^*(B\tilde G; {\mathbb{Z}}_ 2)\). This is done first for the groups \(E_ 6\), \(E_ 7\) and \(E_ 8\), and then the cases for \(G_ 2\) and \(F_ 4\) are deduced therefrom. Their method is to define a space \(B\tilde T\) and a map \({\tilde\lambda}: B\tilde T\to B\tilde G\) in analogy with the map defined by a maximal torus T in G. The induced morphism \({\tilde\lambda}^*\) is almost injective, and after a study of \({\tilde\lambda}^*\) by means of the Weyl group of G relative to T they are able to determine the action of the Steenrod algebra on \(H^*(B\tilde G; {\mathbb{Z}}_ 2)\).
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 3980023
zbMATH Keywords: compact simply connected Lie group
zbMATH Keywords: cohomology of the classifying space
zbMATH Keywords: exceptional type
zbMATH Keywords: action of the Steenrod algebra
zbMATH Keywords: maximal torus
zbMATH Keywords: Weyl group
reviewed by: S. Y. Husseini
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178517
OpenAlex ID: W1969066411
Wikidata QID: Q115224804
cites work: On the cohomology of irreducible symmetric spaces of exceptional type
cites work: Homotopy Groups of Compact Lie Groups E6, E7 and E8
cites work: Q4283556
cites work: Q4076525
cites work: The integral cohomology ring of the symmetric space EVII
DOI: 10.2977/PRIMS/1195178517
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description: scientific article; zbMATH DE number 3980023