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ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

Published online by Cambridge University Press:  20 November 2018

J. W. Cannon
J. W. Cannon*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey
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In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 31 - 73
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Aleksandrov, P. S., Combinatorial topology. Vol. 3 (OGIZ, Moscow, 1947; English transi., Graylock Press, Albany, New York, 1960).Google Scholar
2. Bing, R. H., Each disk in Ezcontains a tame arc, Amer. J. Math. 84 (1962), 583–590.Google Scholar
3. Bing, R. H., Approximating surfaces from the side, Ann. of Math. 77 (1963), 145–192.Google Scholar
4. Bing, R. H., Pushing a 2-sphere into its complement, Michigan Math. J. 11 (1964), 33–45.Google Scholar
5. Boyd, W. S. and Wright, A. H., An algebraic characterization of tameness for a graph in a 3- manifold(to appear).Google Scholar
6. Brown, E. M., Unknotting in M2 XI, Trans. Amer. Math. Soc. 123 (1966), 480–505.Google Scholar
7. Brown, E. M. and Crowell, R. H., Deformation retractions of S-manifolds into their boundaries, Ann. of Math. 82 (1965), 445–458.Google Scholar
8. Burgess, C. E., Characterizations of tame surfaces inE3, Trans. Amer. Math. Soc. 114- (1965), 80-97.Google Scholar
9. Burgess, C. E. and Cannon, J. W., Embeddings of surfaces in E3, Rocky Mountain J. Math. 1 (1971), 259–344.Google Scholar
10. Cannon, J. W., Sets which can be missed by side approximations to 2-spheres, Pacific J. Math. II (1970), 321-334.Google Scholar
11. Cannon, J. W., *-Taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429–440.Google Scholar
12. Cannon, J. W., *-Taming sets for crumpled cubes. II. Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441–446.Google Scholar
13. Cannon, J. W., Characterization of tame subsets of 2-spheres in Ez, Amer. J. Math. 94 (1972), 173–188.Google Scholar
14. Cannon, J. W., New proofs of Bing's approximation theorems for surfaces (to appear in Pacific J. Math.).Google Scholar
15. Daverman, R. J., A new proof for the Hosay-Lininger Theorem about crumpled cubes, Proc. Amer. Math. Soc. 23 (1969), 52–54.Google Scholar
16. Detmer, R., Characterization of sets which are tame in complexes in E3 (to appear).Google Scholar
17. Eaton, W. T., A note about locally spherical spheres, Can. J. Math. 21 (1969), 1001–1003.Google Scholar
18. Eilenberg, S. and Wilder, R. L., Uniform local connectedness and contractibility, Amer. J. Math. 64 (1942), 613–622.Google Scholar
19. Gillman, D. S., Free curves in E3, Pacific J. Math. 28 (1969), 533–542.Google Scholar
20. Harrold, O. G., Jr., Euclidean domains with uniformly Abelian local fundamental groups, Trans. Amer. Math. Soc. 67 (1949), 120–129.Google Scholar
21. Harrold, O. G., Jr., The enclosing of simple arcs and curves by polyhedra, Duke Math. J. 21 (1954), 615–621.Google Scholar
22. Harrold, O. G., Jr., Griffith, H. C., and Posey, E. E., A characterization of tame curves in three-space, Trans. Amer. Math. Soc. 79 (1955), 12–34.Google Scholar
23. Hosay, Norman, The sum of a real cube and a crumpled cube is Sz, Notices Amer. Math. Soc. 10 (1963), 666.Google Scholar
24. Hu, S.-T., Homotopy Theory (Academic Press, New York and London, 1959).Google Scholar
25. Lininger, L. L., Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534–549.Google Scholar
26. Loveland, L. D., Piercing locally spherical spheres with tame arcs, Illinois J. Math. 13 (1969), 327–330.Google Scholar
27. McMillan, D. R., Jr., Local properties of the embedding of a graph in a three-manifold,Can. J. Math. 18 (1966), 517–528.Google Scholar
28. Magnus, W., Karrass, A., and Solitar, D., Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Pure and Applied Mathematics, XIII, Interscience Publishers (John Wiley & Sons, Inc., New York, London, Sydney, 1966).Google Scholar
29. Massey, W. S., Algebraic Topology: An introduction (Harcourt, Brace & World, New York, 1967).Google Scholar
30. Nicholson, Victor A., Tame and nice are equivalent in 3-manifolds, Notices Amer. Math. Soc. 18 (1971), 834.Google Scholar
31. Papakyriakopoulos, C. D., On Dehn's lemma and the asphericity of knots, Ann. of Math. 66 (1957), 1–26.Google Scholar
32. Papakyriakopoulos, C. D., On solid tori, Proc. London Math. Soc. 7 (1957), 281–299.Google Scholar
33. Scott, W. R., Group Theory (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964).Google Scholar
34. Seifert, H. and Threlfall, W., Lehrbuch der Topologie(Chelsea Publishing Company, New York, 1947).Google Scholar
35. Shapiro, A. and J. Whitehead, H. C., A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174–178.Google Scholar
36. Spanier, E. H., Algebraic Topology (McGraw-Hill, New York, 1966).Google Scholar
37. Stallings, J. R., On the loop theorem, Ann. of Math. 72 (1960), 12–19.Google Scholar
38. Stallings, J. R., On fibering certain Z-manifolds, Topology of S-Manifolds and Related Topics (Prentice- Hall, Englewood Cliffs, N.J., 1961).Google Scholar
39. Waldhausen, F., EineVerallgemeinerung des Schleifensatzes, Topology 6 (1967), 501–504.Google Scholar
40. Waldhausen, F., GruppenmitZentrum und Z-dimensionaleMannigfaltigkeiten, Topology 6 (1967), 505–517.Google Scholar
41. Whyburn, G. T., Analytic Topology, Amer. Math. Soc. Colloq. Publ. Vol. 28, Amer. Math. Soc. (Providence, R.I., 1942).Google Scholar
42. Wilder, R. L., Topology of Manifolds, Amer. Math. Soc. Colloq. Publ., Vol. 32, Amer. Math. Soc. (Providence, R.I., 1963).Google Scholar
43. Bing, R. H., A surface is tame if its complement is 1 — ULC, Trans. Amer. Math. Soc. 101 (1961), 294–305.Google Scholar