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A certain triple Whitehead product

Published online by Cambridge University Press:  24 October 2008

P. J. Hilton
P. J. Hilton
Affiliation:
Pembroke CollegeCambridge
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Extract

The triple Whitehead product we consider in this note is [[ɩn, ɩn], ɩn] ∈ π3n−2(Sn), where ɩn generates πn(Sn). It follows from the Jacobi identity for Whitehead products

α ∈ πp(X), β ∈ πq(X), γ ∈ πr(X), that 3[[ɩn, ɩn], ɩn] = 0. Now, if n is odd, 2[ɩn, ɩn] = 0, so that 2[[ɩn, ɩn], ɩn] = 0, whence

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 189 - 197
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Barratt, M. G. and Hilton, P. J.On join operations in homotopy groups. Proc. Lond. math. Soc. (3), 1 (1953), 430–45.CrossRefGoogle Scholar
(2)Borel, A. and Serre, J.-P.Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409–48.CrossRefGoogle Scholar
(3)Chang, S. C.Some suspension theorems. Quart. J. Math. (2), 1 (1950), 310–17.CrossRefGoogle Scholar
(4)Chang, S. C. On πr(Sp V Sq). (In the Press.)Google Scholar
(5)Hilton, P. J.Suspension theorems and the generalized Hopf invariant. Proc. Lond. math. Soc. (3), 1 (1951), 462–93.CrossRefGoogle Scholar
(6)Hilton, P. J.The Hopf invariant and homotopy groups of spheres. Proc. Camb. phil. Soc. 48 (1952), 547–54.CrossRefGoogle Scholar
(7)Hilton, P. J. On the Hopf invariant of a composition element. J. Lond. math. Soc. (In the Press.)Google Scholar
(8)Hilton, P. J. and Whitehead, J. H. C.Note on the Whitehead product. Ann. Math., Princeton, 58 (1953), 429–42.CrossRefGoogle Scholar
(9)Serre, J.-P.Homologie singulière des espaces fibrés. Ann. Math., Princeton, 54 (1951), 425–505.CrossRefGoogle Scholar
(10)Serre, J.-P.Sur la suspension de Freudenthal. C.R. Acad. Sci., Paris, 234 (1952), 1340–2.Google Scholar
(11)Serre, J.-P.Groupes d'homotopie et classes de groupes Abéliens. Ann. Math., Princeton, 58 (1953), 258–94.CrossRefGoogle Scholar
(12)Steenrod, N. E.Cohomology invariants of mappings. Ann. Math., Princeton, 50 (1949), 954–88.CrossRefGoogle Scholar
(13)Whitehead, G. W.On the Freudenthal theorems. Ann. Math., Princeton, 57 (1953), 209–28.CrossRefGoogle Scholar