Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11337-1
Published electronically: June 12, 2012
MathSciNet review: 2996976
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?)

  • [A] D. Anick, Differential algebras in topology, Research Notes in Math. 3, A K Peters, 1993. MR 1213682 (94h:55020)
  • [AG] D. Anick and B. Gray, Small $ H$-spaces related to Moore spaces, Topology 34 (1995), 859-881. MR 1362790 (97a:55011)
  • [BT] P. Beben and S. Theriault, Torsion in finite $ H$-spaces and the homotopy of the three-sphere, Homology, Homotopy Appl. 12 (2010), 25-37. MR 2721030
  • [C] F.R. Cohen, Two-primary analogues of Selick's theorem and the Kahn-Priddy theorem for the $ 3$-sphere, Topology 23 (1984), 401-421. MR 780733 (86e:55020)
  • [CM] F. Cohen and M. Mahowald, A remark on the self-maps of $ \Omega ^{2} S^{2n+1}$, Indiana Univ. Math. J. 30 (1981), 583-588. MR 620268 (82i:55013)
  • [CMN1] F.R. Cohen, J.C. Moore and J.A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), 121-168. MR 519355 (80e:55024)
  • [CMN2] F.R. Cohen, J.C. Moore and J.A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), 549-565. MR 554384 (81c:55021)
  • [G] B. Gray, On the iterated suspension, Topology 27 (1988), 301-310. MR 963632 (89h:55016)
  • [GT] B. Gray and S. Theriault, An elementary construction of Anick's fibration, Geom. Topol. 14 (2010), 243-275. MR 2578305 (2011a:55013)
  • [J] I.M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197. MR 0073181 (17:396b)
  • [KP] D.S. Kahn and S.B. Priddy, The transfer and stable homotopy theory, Math. Proc. Camb. Philos. Soc. 83 (1978), 103-111. MR 0464230 (57:4164b)
  • [N1] J.A. Neisendorfer, Properties of certain $ H$-spaces, Quart. J. Math. Oxford 34 (1983), 201-209. MR 698206 (84h:55007)
  • [N2] J.A. Neisendorfer, The exponent of a Moore space, Algebraic Topology and Algebraic $ K$-theory (Princeton, NJ, 1983), 72-100, Ann. of Math. Study 113, Princeton University Press, 1987. MR 921472 (89e:55029)
  • [N3] J.A. Neisendorfer, Algebraic methods in unstable homotopy theory, New Mathematical Monographs, vol. 12, Cambridge University Press, Cambridge, 2010. MR 2604913 (2011j:55026)
  • [S1] P.S. Selick, Odd primary torsion in $ \pi _{k}(S^{3})$, Topology 17 (1978), 407-412. MR 516219 (80c:55010)
  • [S2] P.S. Selick, Space exponents for loop spaces of spheres, Stable and unstable homotopy theory (Toronto, ON, 1996), 279-283, Fields Inst. Commun. 19, Amer. Math. Soc., 1998. MR 1622353 (99b:55013)
  • [T] S.D. Theriault, Anick's space and the double loops on odd primary Moore spaces, Trans. Amer. Math. Soc. 353 (2001), 1551-1566. MR 1709779 (2001f:55011)

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: https://doi.org/10.1090/S0002-9939-2012-11337-1
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

American Mathematical Society