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The Sphericity of Higher Dimensional Knots
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1973
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