A sphere in with vertically connected interior is tame
Authors:
J. W. Cannon and L. D. Loveland
Journal:
Trans. Amer. Math. Soc. 195 (1974), 345355
MSC:
Primary 57A10
MathSciNet review:
0343273
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Abstract: A set X in 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 2sphere in 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.
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R. Fox and E. Artin, Some wild cells and spheres in threedimensional space, Ann. of Math. (2) 49 (1948), 979990. MR 10, 317. MR 0027512 (10:317g)
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Norman Hosay, A proof of the slicing theorem for 2spheres, Bull. Amer. Math. Soc. 75 (1969), 370374. MR 39 #956. MR 0239599 (39:956)
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R. A. Jensen and L. D. Loveland, Surfaces of vertical order 3 are tame, Bull. Amer. Math. Soc. 76 (1970), 151154. MR 40 #3520. MR 0250281 (40:3520)
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L. D. Loveland, A 2sphere of vertical order 5 bounds a 3cell, Proc. Amer. Math. Soc. 26 (1970), 674678. MR 42 #3768. MR 0268871 (42:3768)
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E. E. Moise, Affine structures in 3manifolds. VIII. Invariance of the knottypes; local tame imbedding, Ann. of Math. (2) 59 (1954), 159170. MR 15, 889. MR 0061822 (15:889g)
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R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, R.I., 1949. MR 10, 614. MR 0029491 (10:614c)
 [1]
 R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145158. MR 15, 816. MR 0061377 (15:816d)
 [2]
 , Upper semicontinuous decompositions of , Ann. of Math. (2) 65 (1957), 363374. MR 19, 1187. MR 0092960 (19:1187f)
 [3]
 , Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456483. MR 19, 300. MR 0087090 (19:300f)
 [4]
 , A surface is tame if its complement is 1ULC, Trans. Amer. Math. Soc. 101 (1961), 294305. MR 24 #A1117. MR 0131265 (24:A1117)
 [5]
 , Each disk in contains a tame arc, Amer. J. Math. 84 (1962), 583590. MR 26 #4331. MR 0146811 (26:4331)
 [6]
 C. E. Burgess, Characterizations of tame surfaces in , Trans. Amer. Math. Soc. 114 (1965), 8097. MR 31 #728. MR 0176456 (31:728)
 [7]
 C. E. Burgess and J. W. Cannon, Embeddings of surfaces in , Rocky Mountain J. Math. 1 (1971), no. 2, 259344. MR 43 #4008. MR 0278277 (43:4008)
 [8]
 J. W. Cannon, taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429440. MR 43 #8065. MR 0282353 (43:8065)
 [9]
 , taming sets for crumpled cubes. II. Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441446. MR 43 #8066. MR 0282354 (43:8066)
 [10]
 , taming sets for crumpled cubes. III. Horizontal sections in 2spheres, Trans. Amer. Math. Soc. 161 (1971), 447456. MR 43 #8067. MR 0282355 (43:8067)
 [11]
 P. H. Doyle and J. G. Hocking, Some results on tame disks and spheres in , Proc. Amer. Math. Soc. 11 (1960), 832836. MR 23 #A4133. MR 0126839 (23:A4133)
 [12]
 W. T. Eaton, Cross sectionally simple spheres, Bull. Amer. Math. Soc. 75 (1969), 375378. MR 39 #957. MR 0239600 (39:957)
 [13]
 R. Fox and E. Artin, Some wild cells and spheres in threedimensional space, Ann. of Math. (2) 49 (1948), 979990. MR 10, 317. MR 0027512 (10:317g)
 [14]
 Norman Hosay, A proof of the slicing theorem for 2spheres, Bull. Amer. Math. Soc. 75 (1969), 370374. MR 39 #956. MR 0239599 (39:956)
 [15]
 R. A. Jensen and L. D. Loveland, Surfaces of vertical order 3 are tame, Bull. Amer. Math. Soc. 76 (1970), 151154. MR 40 #3520. MR 0250281 (40:3520)
 [16]
 L. D. Loveland, A 2sphere of vertical order 5 bounds a 3cell, Proc. Amer. Math. Soc. 26 (1970), 674678. MR 42 #3768. MR 0268871 (42:3768)
 [17]
 , The boundary of a vertically connected cube is tame, Rocky Mountain J. Math. 1 (1971), no. 3, 537540. MR 44 #1007. MR 0283777 (44:1007)
 [18]
 M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge Univ. Press, Cambridge, 1937. MR 0044820 (13:483a)
 [19]
 R. L. Moore, Concerning upper semicontinuous collections of continua, Trans. Amer. Math. Soc. 20 (1919), 169178. MR 1501119
 [20]
 E. E. Moise, Affine structures in 3manifolds. VIII. Invariance of the knottypes; local tame imbedding, Ann. of Math. (2) 59 (1954), 159170. MR 15, 889. MR 0061822 (15:889g)
 [21]
 R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, R.I., 1949. MR 10, 614. MR 0029491 (10:614c)
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DOI:
http://dx.doi.org/10.1090/S00029947197403432735
PII:
S 00029947(1974)03432735
Keywords:
Taming sets,
taming sets,
vertical order,
vertical number,
vertically connected,
2spheres in ,
surfaces in 3manifolds
Article copyright:
© Copyright 1974
American Mathematical Society
