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Linking spheres

Published online by Cambridge University Press:  24 October 2008

D. B. A. Epstein
D. B. A. Epstein
Affiliation:
Trinity CollegeCambridge
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Extract

Andrews and Curtis have shown (1) that one can embed two Sn's in En+2 for n = 2, in such a way that one sphere cannot be shrunk to a point in the residue space of the other. In this paper the result is shown to be true for any n ≥ 1. (The result is obvious for n = 1.) The method is to calculate the appropriate homotopy group of the residue space of one sphere, and to show that the embedding of the other sphere represents a non-zero element of the group. The two spheres can both be embedded analytically.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 3 , July 1960 , pp. 215 - 219
Copyright
Copyright © Cambridge Philosophical Society 1960

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References

REFERENCES

(1)Andrews, J. J. and Curtis, M. L. Knotted 2-spheres in the 4-sphere (to appear).Google Scholar
(2)Artin, E.Abh. Math. Sem. Univ. hamburg 4 (1925), 174–7.CrossRefGoogle Scholar
(3)Papakyrlakopoulos, C. D.On Dehn's lemma and the asphericity of knots. Ann. Math., Princeton, 66 (1957), 1–26.CrossRefGoogle Scholar