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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Kahn-Priddy Theorem and the homotopy of the three-sphere
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by Piotr Beben and Stephen Theriault PDF
Proc. Amer. Math. Soc. 141 (2013), 711-723 Request permission

Abstract:

Let $p$ be an odd prime. The least nontrivial $p$-torsion homotopy group of $S^{3}$ occurs in dimension $2p$ and is of order $p$. This induces a map $f\colon P^{2p+1}(p)\rightarrow S^{3}$, where $P^{2p+1}(p)$ is a mod-$p$ Moore space. An important conjecture related to the Kahn-Priddy Theorem is that the double loops on the three-connected cover of $f$ has a right homotopy inverse. We prove a weaker but still useful property: if $X$ is the cofiber of $f$, then the double loop on the three-connected cover of the inclusion $S^{3}\rightarrow X$ is null homotopic.
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Additional Information
  • Piotr Beben
  • Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
  • Email: bebenp@unbc.ca
  • Stephen Theriault
  • Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
  • MR Author ID: 652604
  • Email: s.theriault@abdn.ac.uk
  • Received by editor(s): July 1, 2011
  • Published electronically: June 12, 2012
  • Communicated by: Brooke Shipley
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 711-723
  • MSC (2010): Primary 55P35, 55Q40
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11337-1
  • MathSciNet review: 2996976