Proofs of two conjectures of Gray involving the double suspension

Abstract:

In proving that the fiber of the double suspension has a classifying space, Gray constructed fibrations \[ {S^{2n-1}}\xrightarrow {E^{2}}{\Omega ^{2} S^{2n+1}}\xrightarrow {f} {BW_{n}}\] and \[ {BW_{n}}\rightarrow {\Omega S^{2np+1}}\xrightarrow {\phi }{S^{2np-1}}.\] He conjectured that $E^{2}\circ \phi$ is homotopic to the $p^{th}$-power map on $\Omega ^{2} S^{2np+1}$ when $p$ is an odd prime. Harper proved this is true when looped once. We remove the loop when $p\geq 5$. Gray also conjectured that at odd primes $f$ factors through a map \[ {\Omega {S^{2n+1}\{p\}}}\rightarrow {BW_{n}}.\] We show that this is true as well when $p\geq 5$.