A conjecture of Gray and the 𝑝-th power map on Ω²𝑆^{2𝑛𝑝+1}

Abstract

For p ≥ 2 p\ge 2 , the p p -th power map [ p ] [p] on Ω 2 S 2 n p + 1 \Omega ^2 S^{2np+1} is homotopic to a composite Ω 2 S 2 n p + 1 ⟶ ϕ n S 2 n p − 1 ⟶ E 2 Ω 2 S 2 n p + 1 , \Omega ^2 S^{2np+1} \overset {\phi _n}{\longrightarrow } S^{2np-1} \overset {E^2}{\longrightarrow } \Omega ^2 S^{2np+1}, where the fiber of ϕ n \phi _n is B W n BW_n .