An upper bound for the 3-primary homotopy exponent of the exceptional Lie group \(E_7\) (Q932934)

The author shows that the quotient \(E_7/F_4\) of the exceptional compact Lie groups \(E_7\) and \(F_4\) is spherically resolved. More precisely, there is a \(3\)-primary homotopy fibration \(S^{19}\to E_7/F_4 \to B\) to a space \(B\) which itself admits a homotopy fibration \(S^{27} \to B \to S^{35}\). This spherical resolution is used to prove that \(3^{23}\) annihilates the \(3\)-torsion in \(\pi_{\ast}(E_7)\); note that \textit{D. M. Davis} [New York J. Math. 4, 185--221, electronic only (1998; Zbl 0904.55007)] has shown that the homotopy exponent \(\mathrm{exp}(E_7)\) is at least \(3^{19}\), i.e., \(3^m\) does not annihilate \(3\)-torsion if \(m \leq 18\). It remains an open problem which of the two bounds for \(\mathrm{exp}(E_7)\) is optimal.