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Proofs of two conjectures of Gray involving the double suspension


Author: Stephen D. Theriault
Journal: Proc. Amer. Math. Soc. 131 (2003), 2953-2962
MSC (2000): Primary 55P40; Secondary 55R35
Published electronically: January 28, 2003
MathSciNet review: 1974354
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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 $p$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

\begin{displaymath}{\Omega{S^{2n+1}\{p\}}}\rightarrow{BW_{n}}.\end{displaymath}

We show that this is true as well when $p\geq 5$.


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Additional Information

Stephen D. Theriault
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Address at time of publication: Department of Mathematical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom
Email: st7b@virginia.edu, s.theriault@maths.abdn.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06847-3
Keywords: $p^{th}$-power map, double suspension
Received by editor(s): September 28, 2001
Received by editor(s) in revised form: April 2, 2002
Published electronically: January 28, 2003
Communicated by: Paul Goerss
Article copyright: © Copyright 2003 American Mathematical Society