Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:58:18.031Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

1. Steve, Armentrout, Cellular decompositions of 3-manifolds that yield 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 453–456.Google Scholar
2. Bing, R. H., Extending monotone decompositions of 3-manifolds, Trans. Amer. Math. Soc. 149 (1970), 351–369.Google Scholar
3. Bing, R. H., Locally tame sets are tame, Ann. of Math. 59 (1954), 145–158.Google Scholar
4. William S., Boyd, Jr., Repairing embeddings of 3-cells with monotone maps of Ez, Trans. Amer. Math. Soc. 161 (1971), 123–144.Google Scholar
5. Wolfgang, Haken, Trivial loops in homotopy 3-spheres, Illinois J. Math. 11 (1967), 547–554.Google Scholar
6. Hempel, J. P., A surface is tame if it can be deformed into each complementary domain, Trans. Amer. Math. Soc. III (1964), 273–287.Google Scholar
7. Lister, F. M., Simplifying intersections of disks in Binges approximation theorem, Pacific J. Math. 22 (1967), 281–295.Google Scholar
8. McMillan, D. R., Jr., Acyclicity in 3-manifolds, Bull. Amer. Math. Soc. 76 (1970), 942–964.Google Scholar
9. McMillan, D. R., Jr., A criterion for cellularity in a manifold. II, Trans. Amer. Math. Soc. 126 (1967), 217–224.Google Scholar
10. Moise, E. E., Affine structures in 3-manifolds. VIII, Invariance of the knot-type; local tame imbedding, Ann. of Math. 59 (1954), 159–170.Google Scholar
11. Smythe, N., Handlebodies in 3-manifolds, Proc. Amer. Math. Soc. 26 (1970), 534–538.Google Scholar
12. Alden, Wright, Mappings from 3-manifolds onto 3-manifolds, Trans. Amer. Math. Soc. 167 (1972), 479–501.Google Scholar