Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T00:58:52.345Z Has data issue: false hasContentIssue false
  • English
  • Français

The Sphericity of Higher Dimensional Knots

Published online by Cambridge University Press:  20 November 2018

Eldon Dyer  and
A. T. Vasquez
Eldon Dyer
Affiliation:
The City University of New York, Graduate Center, New York, New York
A. T. Vasquez
Affiliation:
The City University of New York, Graduate Center, New York, New York
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1956 CD. Papakyriakopoulos showed [5] that the complement C of a 1-sphere S1 tamely imbedded in a 3-sphere S3 is aspherical; that is, that for all i ≧ 2, πi(C) = 0. In this note we show that for n ≧ 2 the complement C of an n-sphere Sn smoothly imbedded in Sn+2 is aspherical only if the fundamental group of C is infinite cyclic. Combined with results of J. Stallings [6] or of J. Levine [3], this implies that if the complement of an Sn smoothly imbedded in Sn+2 is aspherical, n ≥ 4 , then Sn is topologically unknotted in Sn+2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 6 , December 1973 , pp. 1132 - 1136
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Epstein, D. B. A., Linking spheres, Proc. Cambridge Philos. Soc. 56 (1960), 215–219.Google Scholar
2. Fox, R. H., Some problems in knot theory, Topology of 3-manifolds, (Prentice-Hall, 1962, p. 168–176).Google Scholar
3. Levine, J., Unknotting spheres in codimension two, Topology 4 (1965), 9–16.Google Scholar
4. Massey, W. S., On the normal bundle of a sphere imbedded in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), 959–964.Google Scholar
5. Papakyriakopoulos, C. D., On Dehns lemma and the asphericity of knots, Ann. of Math. 66 (1957), 1–26.Google Scholar
6. Stallings, J., On topological^ unknotted spheres, Ann. of Math. 77 (1963), 490–503..Google Scholar
7. Steenrod, N. E., Homology with local coefficients, Ann. of Math. 44 (1943), 610–627.Google Scholar
8. Swan, R. G., Groups of cohomological dimension one, J. Algebra 12 (1969), 586–610.Google Scholar