$^{\ast }$-taming sets for crumpled cubes. I. Basic properties
HTML articles powered by AMS MathViewer
- by James W. Cannon
- Trans. Amer. Math. Soc. 161 (1971), 429-440
- DOI: https://doi.org/10.1090/S0002-9947-1971-0282353-7
- PDF | Request permission
Abstract:
Is a surface in a 3-manifold tame if it is tame modulo a tame set? This question was answered by the author through the introduction and characterization of taming sets. The purpose of this paper is to introduce and establish the basic properties of the more general and more flexible, but closely related, $^ \ast$-taming set.References
- R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465–483. MR 87090
- R. H. Bing, A surface is tame if its complement is $1$-ULC, Trans. Amer. Math. Soc. 101 (1961), 294–305. MR 131265, DOI 10.1090/S0002-9947-1961-0131265-1
- H. G. Bothe, Differenzierbare Flächen sind zahm, Math. Nachr. 43 (1970), 161–180 (German). MR 267593, DOI 10.1002/mana.19700430110
- C. E. Burgess, Properties of certain types of wild surfaces in $E^{3}$, Amer. J. Math. 86 (1964), 325–338. MR 163295, DOI 10.2307/2373168
- C. E. Burgess, Characterizations of tame surfaces in $E^{3}$, Trans. Amer. Math. Soc. 114 (1965), 80–97. MR 176456, DOI 10.1090/S0002-9947-1965-0176456-2
- C. E. Burgess, Pairs of $3$-cells with intersecting boundaries in $E^{3}$, Amer. J. Math. 88 (1966), 309–313. MR 196723, DOI 10.2307/2373194
- C. E. Burgess and J. W. Cannon, Tame subsets of spheres in $E^{3}$, Proc. Amer. Math. Soc. 22 (1969), 395–401. MR 242135, DOI 10.1090/S0002-9939-1969-0242135-5
- C. E. Burgess and J. W. Cannon, Embeddings of surfaces in $E^{3}$, Rocky Mountain J. Math. 1 (1971), no. 2, 259–344. MR 278277, DOI 10.1216/RMJ-1971-1-2-259
- J. W. Cannon, Characterization of taming sets on $2$-spheres, Trans. Amer. Math. Soc. 147 (1970), 289–299. MR 257996, DOI 10.1090/S0002-9947-1970-0257996-6
- J. W. Cannon, Sets which can be missed by side approximations to spheres, Pacific J. Math. 34 (1970), 321–334. MR 267545, DOI 10.2140/pjm.1970.34.321
- James W. Cannon, $^{\ast }$-taming sets for crumpled cubes. II. Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441–446. MR 282354, DOI 10.1090/S0002-9947-1971-0282354-9
- James W. Cannon, $^{\ast }$-taming sets for crumpled cubes. III. Horizontal sections in $2$-spheres, Trans. Amer. Math. Soc. 161 (1971), 447–456. MR 282355, DOI 10.1090/S0002-9947-1971-0282355-0
- Robert J. Daverman, A new proof for the Hosay-Lininger theorem about crumpled cubes, Proc. Amer. Math. Soc. 23 (1969), 52–54. MR 246274, DOI 10.1090/S0002-9939-1969-0246274-4
- P. H. Doyle and J. G. Hocking, Some results on tame disks and spheres in $E^{3}$, Proc. Amer. Math. Soc. 11 (1960), 832–836. MR 126839, DOI 10.1090/S0002-9939-1960-0126839-2
- Samuel Eilenberg and R. L. Wilder, Uniform local connectedness and contractibility, Amer. J. Math. 64 (1942), 613–622. MR 7100, DOI 10.2307/2371708
- O. G. Harrold Jr., The enclosing of simple arcs and curves by polyhedra, Duke Math. J. 21 (1954), 615–621. MR 68208
- O. G. Harrold Jr., H. C. Griffith, and E. E. Posey, A characterization of tame curves in three-space, Trans. Amer. Math. Soc. 79 (1955), 12–34. MR 91457, DOI 10.1090/S0002-9947-1955-0091457-4
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- L. D. Loveland, Tame subsets of spheres in $E^{3}$, Pacific J. Math. 19 (1966), 489–517. MR 225309, DOI 10.2140/pjm.1966.19.489 —, A 2-sphere of vertical order five bounds a 3-cell, Notices Amer. Math. Soc. 17 (1970), 472. Abstract #70T-G54. —, The boundary of a linearly connected crumpled cube is tame, Notices Amer. Math. Soc. 17 (1970), 581. Abstract #70T-G69.
- D. R. McMillan Jr., Some topological properties of piercing points, Pacific J. Math. 22 (1967), 313–322. MR 216486, DOI 10.2140/pjm.1967.22.313
- C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
- Raymond Louis Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications, Vol. XXXII, American Mathematical Society, Providence, R.I., 1963. MR 0182958
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 161 (1971), 429-440
- MSC: Primary 54.78
- DOI: https://doi.org/10.1090/S0002-9947-1971-0282353-7
- MathSciNet review: 0282353