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On invariant knots

  • Sławomir Kwasik (a1) and Pierre Vogel (a1)

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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 S3S5 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.

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[1]Cappell, S. E. and Shaneson, J. L.. On four dimensional surgery and applications. Comment. Math. Helv. 46 (1971), 500528.
[2]Cappell, S. E. and Shaneson, J. L.. On topological knots and knot cobordism. Topology 12 (1973), 3340.
[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).
[4]Lashof, R.. A non-smoothable knot. Bull. Amer. Math. Soc. (N.S.) 77 (1971), 613614.
[5]Quinn, F.. Ends of Maps, in: Dimensions 4 and 5. J. Differential Geom. 17 (1982), 503521.
[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), 441453.

On invariant knots

  • Sławomir Kwasik (a1) and Pierre Vogel (a1)

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