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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
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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

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