On the Stiefel-Whitney classes of the adjoint representation of $E_8$
Copyright policy )On the Stiefel-Whitney classes 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.
Semisimple Lie groups and their representations, Homology and cohomology of Lie groups, adjoint representation, Characteristic classes and numbers in differential topology, 57R20, 55R40, Stiefel--Whitney class, Stiefel-Whitney class, exceptional Lie group, 55S05, classifying space
Semisimple Lie groups and their representations, Homology and cohomology of Lie groups, adjoint representation, Characteristic classes and numbers in differential topology, 57R20, 55R40, Stiefel--Whitney class, Stiefel-Whitney class, exceptional Lie group, 55S05, classifying space
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