On the Stiefel-Whitney class of the adjoint representation of $E_8$

Abstract

Let $\widetilde{E}_8$ be the 3-connected covering space of the 1-connected, compact exceptional group $E_8$, which is regarded as the loop space of the homotopy fibre $B\widetilde{E}_8$ of a map from $BE_8$, the classifying space of $E_8$, to an Eilenberg-MacLane space. The Stiefel-Whitney classes of the adjoint representation of $E_8$ induce elements of the mod 2 cohomology of $B\widetilde{E}_8$. These images are computed.