An upper bound for the 3-primary homotopy exponent of the exceptional Lie group $E_7$

Abstract

A new homotopy fibration is constructed at the prime 3 which shows that the quotient group $E_{7}/F_{4}$ is spherically resolved. This is then used to show that the 3-primary homotopy exponent of $E_{7}$ is bounded above by $3^{23}$, which is at most four powers of 3 from being optimal.