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Local Properties of the Embedding of a Graph in a Three-Manifold

Published online by Cambridge University Press:  20 November 2018

D. R. McMillan Jr.
D. R. McMillan Jr.*
Affiliation:
The University of Virginia
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Let G be a finite graph topologically embedded in the interior of a 3-manifold M. Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each point p ∈ G is necessary in order for the embedding of G to be tame:

For each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and if W is any connected open set such that p ∈ W ⊂ V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n — 1 generators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 517 - 528
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bing, R. H., Locally tame sets are tame, Ann. of Math., 59 (1954), 145–158.Google Scholar
2. Bing, R. H., A surface is tame if its complement is, 1-ULC, Trans. Amer. Math. Soc, 101 (1961), 294–305.Google Scholar
3. Debrunner, H. and Fox, R. H., A mildly wild imbedding of an n-frame, Duke Math. J., 27 (1960), 425–430.Google Scholar
4. Doyle, P. H., A wild triod in three-space, Duke Math. J., 26 (1959), 263–267.Google Scholar
5. Edwards, C. H. Jr., A characterization of tame curves in the 3-sphere, Abstract 573-32, Notices Amer. Math. Soc., 7 (1960), 875.Google Scholar
6. Fox, R. H., On the imbedding of polyhedra in 3-space, Ann. of Math., 49 (1948), 462–470.Google Scholar
7. Harrold, O. G. Jr., Euclidean domains with uniformly abelian local fundamental groups, Trans. Amer. Math. Soc, 67 (1949), 120–129.Google Scholar
8. Harrold, O. G. Jr., Euclidean domains with uniformly abelian local fundamental groups, II, Duke Math. J., 17 (1950), 269–272.Google Scholar
9. Harrold, O. G. Jr., Griffith, H. C., and Posey, E. E., A characterization of tame curves in 3-space, Trans. Amer. Math. Soc, 79 (1955), 12–35.Google Scholar
10. Harrold, O. G. Jr., and Moise, E. E., Almost locally polyhedral spheres, Ann. of Math., 57 (1953), 575–578.Google Scholar
11. Hempel, J. P. and McMillan, D. R. Jr., Locally nice embeddings of manifolds, to appear.Google Scholar
12. Moise, E. E., Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung, Ann. of Math., 56 (1952), 96–114.Google Scholar
13. Moise, E. E., Affine structures in 3-manifolds, VIII. Invariance of the knot-types; Local tame imbedding, Ann. of Math., 59 (1954), 159–170.Google Scholar
14. Papakyriakopoulos, C. D., On solid tori, Proc. London Math. Soc. (3), 7 (1957), 281–299.Google Scholar
15. Papakyriakopoulos, C. D., On Dehn's lemma and the asphericity of knots, Ann. of Math., 66 (1957), 1–26.Google Scholar
16. Stallings, J. R., On fibering certain 3-manifolds, University of Georgia Conference on Topology of Manifolds (1961), 95–100.Google Scholar