Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55P35, 55Q40

Retrieve articles in all journals with MSC (2010): 55P35, 55Q40


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
Email: s.theriault@abdn.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11337-1
PII: S 0002-9939(2012)11337-1
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