On the Stiefel-Whitney class of the adjoint representation of \(E_8\)
Let \(\widetilde{E_8}\) be the \(3\)-connected covering space of the simply connected Lie group \(E_8\). Let \(\pi_8: \widetilde{E_8} \to E_8\) be the covering map and \(\pi_8^*: H^*(Be_8) \to H^*(B\widetilde{E_8})\) be the induced map between the \(\mathbb{Z}_2\) cohomology of the classifying spaces. The main result (Theorem 4) computes the images of the \(2^i\)th Stiefel--Whitney class of the adjoint representation of \(E_8\) under \(\pi_8^*\) for \(i\leq7\). The article contains only a sketch of proof; the author postpones the proof to another paper. This sketch is an explicit calculation using an explicit expression for the cohomology rings and some maps, and applying the Steenrod operations.
scientific article; zbMATH DE number 2151800