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On the Asphericity of Knot Complements

Published online by Cambridge University Press:  20 November 2018

Vo Thanh Liem  and
Gerard A. Venema
Vo Thanh Liem
Affiliation:
Princeton University, Princeton, New Jersey 08544, U.S.A.
Gerard A. Venema
Affiliation:
Princeton University, Princeton, New Jersey 08544, U.S.A.
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Abstract

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Two examples of topological embeddings of S2 in S4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.

Two theorems are proved. The first answers the question of just when good π1 implies the vanishing of the higher homotopy groups for knot complements in S4. The second theorem characterizes local flatness for 2-spheres in S4 in terms of a local π1 condition.

Keywords

57Q4557N45knot complementfundamental groupasphericity1 –alglocal flatness2–sphere
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 2 , 01 April 1993 , pp. 340 - 356
Copyright
Copyright © Canadian Mathematical Society 1993

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