A 2-sphere in 𝐸³ with vertically connected interior is tame

Cannon, J. W.; Loveland, L. D.

doi:10.1090/S0002-9947-1974-0343273-5

A $2$-sphere in $E^{3}$ with vertically connected interior is tame
HTML articles powered by AMS MathViewer

by J. W. Cannon and L. D. Loveland PDF

Trans. Amer. Math. Soc. 195 (1974), 345-355 Request permission

Abstract:

A set X in ${E^3}$ is said to have vertical number n if the intersection of each vertical line with X contains at most n components. The set X is said to have vertical order n if each vertical line intersects X in at most n points. A set with vertical number 1 is said to be vertically connected. We prove that a 2-sphere in ${E^3}$ with vertically connected interior is tame. This result implies as corollaries several previously known taming theorems involving vertical order and vertical number along with several more general and previously unknown results.

References

R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836
R. H. Bing, Upper semicontinuous decompositions of $E^3$, Ann. of Math. (2) 65 (1957), 363–374. MR 92960, DOI 10.2307/1969968
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
R. H. Bing, Each disk in $E^{3}$ contains a tame arc, Amer. J. Math. 84 (1962), 583–590. MR 146811, DOI 10.2307/2372864
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 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
James W. Cannon, $^{\ast }$-taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429–440. MR 282353, DOI 10.1090/S0002-9947-1971-0282353-7
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
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
W. T. Eaton, Cross sectionally simple spheres, Bull. Amer. Math. Soc. 75 (1969), 375–378. MR 239600, DOI 10.1090/S0002-9904-1969-12180-4
Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979–990. MR 27512, DOI 10.2307/1969408
Norman Hosay, A proof of the slicing theorem for $2$-spheres, Bull. Amer. Math. Soc. 75 (1969), 370–374. MR 239599, DOI 10.1090/S0002-9904-1969-12178-6
R. A. Jensen and L. D. Loveland, Surfaces of vertical order $3$ are tame, Bull. Amer. Math. Soc. 76 (1970), 151–154. MR 250281, DOI 10.1090/S0002-9904-1970-12409-0
L. D. Loveland, A $2$-sphere of vetical order $5$ bounds a $3$-cell, Proc. Amer. Math. Soc. 26 (1970), 674–678. MR 268871, DOI 10.1090/S0002-9939-1970-0268871-0
L. D. Loveland, The boundary of a vertically connected cube is tame, Rocky Mountain J. Math. 1 (1971), no. 3, 537–540. MR 283777, DOI 10.1216/RMJ-1971-1-3-537
M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge, at the University Press, 1951. 2nd ed. MR 0044820
Robert L. Moore, Concerning a set of postulates for plane analysis situs, Trans. Amer. Math. Soc. 20 (1919), no. 2, 169–178. MR 1501119, DOI 10.1090/S0002-9947-1919-1501119-X
Edwin E. Moise, Affine structures in $3$-manifolds. VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159–170. MR 61822, DOI 10.2307/1969837
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491