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3-primary exponents

Published online by Cambridge University Press:  24 October 2008

Joseph A. Neisendorfer
Joseph A. Neisendorfer
Affiliation:
Institute for Advanced Study, Princeton
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Extract

The purpose of this paper is to show that 3n annihilates the 3-primary component of the homotopy groups of the 2n + 1-dimensional sphere. In the terminology of (2) and (3), S2n+1 has exponent 3n at 3.

In fact, a stronger result is proved. Localize at 3 and let Ω2nS2n + 1〈2n + 1〉 denote the 2n-fold loop space of the 2n + 1-connected cover of S2n+1. Then Ω2nS2n + 1〈2n + 1〉 has a null homotopic 3n-th power map.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 63 - 83
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Adams, J. F.The sphere, considered as an H-space mod p. Quart. J. Math. Oxford Ser. (2) 12 (1961), 52–60.CrossRefGoogle Scholar
(2)Cohen, F. R., Moore, J. C. and Neisendorfer, J. A.Torsion in homotopy groups. Ann. of Math. 109 (1979), 121–168.CrossRefGoogle Scholar
(3)Cohen, F. R., Moore, J. C. and Neisendorfer, J. A.The double suspension and exponents of the homotopy groups of spheres. Ann. of Math. 110 (1979), 549–565.CrossRefGoogle Scholar
(4)Cohen, F. R., Moore, J. C. and Neisendorfer, J. A. Moore spaces have exponents. (To appear.)Google Scholar
(5)Husemoller, D.Fibre Bundles, 2nd ed. (Springer-Verlag, 1966).CrossRefGoogle Scholar
(6)Liulevicius, A.The factorization of cyclic reduced powers by secondary cohomology operations. Mem. Amer. Math. Soc., no. 42, 1962.CrossRefGoogle Scholar
(7)Milnor, J. W. and Moore, J. C.On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211–264.CrossRefGoogle Scholar
(8)Moore, J. C. Algèbre homologique et nomologie des espaces classifiants, exposé 7. Séminaire H. Cartan-J. C. Moore, 1959/1960. Périodicité des groupes d'homotopie stable des groupes classiques, d'après Bott (Secrétariat Mathématique, Paris, 1961).Google Scholar
(9)Neisendorfer, J. A.Primary Homotopy Theory. Mem. Amer. Math. Soc., no. 232, 1980.CrossRefGoogle Scholar
(10)Selick, P. S.Odd primary torsion in πk(S 3). Topology 17 (1978), 407–412.CrossRefGoogle Scholar
(11)Shimada, N. and Yamanoshita, T.On triviality of the mod p Hopf invariant. Jap. J. Math. 31 (1961), 1–25.CrossRefGoogle Scholar
(12)Toda, H.p-primary components of homotopy groups II, mod p Hopf invariant. Mem. Coll. Sci. Univ. Kyoto, Ser. A. 31 (1958), 143–160.Google Scholar