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On invariant knots

Published online by Cambridge University Press:  24 October 2008

Sławomir Kwasik  and
Pierre Vogel
Sławomir Kwasik
Affiliation:
Université de Nantes, Institut de Mathématiques et d'Informatique, 44072 Nantes cédex, France
Pierre Vogel
Affiliation:
Université de Nantes, Institut de Mathématiques et d'Informatique, 44072 Nantes cédex, France
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Extract

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 473 - 475
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Cappell, S. E. and Shaneson, J. L.. On four dimensional surgery and applications. Comment. Math. Helv. 46 (1971), 500–528.CrossRefGoogle Scholar
[2]Cappell, S. E. and Shaneson, J. L.. On topological knots and knot cobordism. Topology 12 (1973), 33–40.CrossRefGoogle Scholar
[3]Ktrby, R. C. and Siebenmann, L. C.. Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Math. Studies, no. 88, (Princeton University Press, 1977).Google Scholar
[4]Lashof, R.. A non-smoothable knot. Bull. Amer. Math. Soc. (N.S.) 77 (1971), 613–614.CrossRefGoogle Scholar
[5]Quinn, F.. Ends of Maps, in: Dimensions 4 and 5. J. Differential Geom. 17 (1982), 503–521.Google Scholar
[6]Shaneson, J. L.. Surgery on four-manifolds and topological transformation groups. Proc. 1971 Amherst Conf. on Transformation Groups, Springer Lect. Notes in Math. 298 (1971), 441–453.Google Scholar