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The Kahn-Priddy Theorem and the homotopy of the three-sphere

Authors: Piotr Beben and Stephen Theriault
Journal: Proc. Amer. Math. Soc. 141 (2013), 711-723
MSC (2010): Primary 55P35, 55Q40
Published electronically: June 12, 2012
MathSciNet review: 2996976
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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

Stephen Theriault
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Keywords: Three-sphere, homotopy group
Received by editor(s): July 1, 2011
Published electronically: June 12, 2012
Communicated by: Brooke Shipley
Article copyright: © Copyright 2012 American Mathematical Society

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