Skip to main content
Log in

A new proof of the odd primary homotopy exponent of spheres

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

An Erratum to this article was published on 10 July 2013

Abstract

We give a different proof of Cohen, Moore and Neisendorfer’s theorem stating that for odd primes p the homotopy exponent of S 2n+1 is p n. This is done using methods recently introduced by Gray and the author to give a new construction of Anick’s fibration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anick, D.: Differential Algebras in Topology. Research Notes in Mathematics, vol. 3. AK Peters, Wellesley (1993)

  2. Cohen F.R., Moore J.C., Neisendorfer J.A.: Torsion in homotopy groups. Ann. Math. 109, 121–168 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen F.R., Moore J.C., Neisendorfer J.A.: The double suspension and exponents of the homotopy groups of spheres. Ann. Math. 110, 549–565 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ganea T.: A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39, 295–322 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gray B.: On Toda’s fibration. Math. Proc. Camb. Phil. Soc. 97, 289–298 (1985)

    Article  MATH  Google Scholar 

  6. Gray B.: On the iterated suspension. Topology 27, 301–310 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gray, B.: Homotopy commutativity and the EHP sequence. In: Proc. Internat. Conf., 1988, Contemp. Math., Vol. 1370. American Mathematical Society, Providence, pp. 181–188 (1989)

  8. Gray B., Theriault S.: On the double suspension and the mod-p Moore space. Contemp. Math. 399, 101–121 (2006)

    Article  MathSciNet  Google Scholar 

  9. Gray B., Theriault S.: An elementary construction of Anick’s fibration. Geom. Topol. 14, 243–275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mather M.: Pull-backs in homotopy theory. Can. J. Math. 28, 225–263 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Neisendorfer J.A.: 3-Primary exponents. Math. Proc. Camb. Phil. Soc. 90, 63–83 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Neisendorfer J.A.: Properties of certain H-spaces. Quart. J. Math. Oxford 34, 201–209 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Selick P.: Odd primary torsion in π k (S 3). Topology 17, 407–412 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Selick P.: A spectral sequence concerning the double suspension. Invent. Math. 64, 15–24 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Theriault S.D.: Proofs of two conjectures of Gray involving the double suspension. Proc. Am. Math. Soc. 131, 2953–2962 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Theriault S.D.: The 3-primary classifying space of the fiber of the double suspension. Proc. Am. Math. Soc. 136, 1489–1499 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Theriault, S.D.: 2-Primary Anick fibrations. J. Topol. (accepted)

  18. Toda H.: On the double suspension E 2. J. Inst. Polytech. Osaka City Univ. 7, 103–145 (1956)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stephen D. Theriault.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Theriault, S.D. A new proof of the odd primary homotopy exponent of spheres. manuscripta math. 139, 137–151 (2012). https://doi.org/10.1007/s00229-011-0507-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-011-0507-0

Mathematics Subject Classification (2000)

Navigation