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Proofs of two conjectures of Gray involving the double suspension

Authors:	Stephen D Theriault

Institution:	Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Abstract:	In proving that the fiber of the double suspension has a classifying space, Gray constructed fibrations $\begin{displaymath}{S^{2n-1}}\xrightarrow{E^{2}}{\Omega^{2} S^{2n+1}}\xrightarrow{f} {BW_{n}}\end{displaymath}$ and $\begin{displaymath}{BW_{n}}\rightarrow{\Omega S^{2np+1}}\xrightarrow{\phi}{S^{2np-1}}.\end{displaymath}$ He conjectured that $E^{2}\circ\phi$ is homotopic to the $p^{th}$ -power map on $\Omega^{2} S^{2np+1}$ when 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 factors through a map $\begin{displaymath}{\Omega{S^{2n+1}\{p\}}}\rightarrow{BW_{n}}.\end{displaymath}$ We show that this is true as well when $p\geq 5$ .

Keywords:	$p^{th}$-power map double suspension

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