The 5-primary homotopy exponent of the exceptional Lie group $E_8$

Abstract

We construct a new homotopy fibration at the prime 5, involving $E_{8}$ and Harper’s rank two finite mod-5 $H$-space. We then use this to show that the 5-primary homotopy exponent of $E_{8}$ is bounded above by $5^{31}$, which is at most one power of 5 from being optimal.