Proofs of two conjectures of Gray involving the double suspension
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- by Stephen D. Theriault
- Proc. Amer. Math. Soc. 131 (2003), 2953-2962
- DOI: https://doi.org/10.1090/S0002-9939-03-06847-3
- Published electronically: January 28, 2003
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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 $p\geq 5$.References
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Bibliographic 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
- MR Author ID: 652604
- Email: st7b@virginia.edu, s.theriault@maths.abdn.ac.uk
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2953-2962
- MSC (2000): Primary 55P40; Secondary 55R35
- DOI: https://doi.org/10.1090/S0002-9939-03-06847-3
- MathSciNet review: 1974354