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Taming Wild Simple Closed Curves with Monotone Maps

Published online by Cambridge University Press:  20 November 2018

W. S. Boyd  and
A. H. Wright
W. S. Boyd
Affiliation:
Western Michigan University, Kalamazoo, Michigan
A. H. Wright
Affiliation:
Western Michigan University, Kalamazoo, Michigan
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Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.

It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).

In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 768 - 788
Copyright
Copyright © Canadian Mathematical Society 1972

References

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