Squaring operations in the 4-connective fibre spaces over the classifying spaces of the exceptional Lie groups
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 \(H^*(B\tilde G; {\mathbb{Z}}_ 2)\).
scientific article; zbMATH DE number 3980023
scientific article