Repairing embeddings of cells with monotone maps of
Author:
William S. Boyd
Journal:
Trans. Amer. Math. Soc. 161 (1971), 123144
MSC:
Primary 54.78
MathSciNet review:
0282352
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Abstract: If is a 2sphere topologically embedded in Euclidean 3space and is the unit sphere about the origin, then there may not be a homeomorphism of onto itself carrying onto . We show here how to construct a map f of onto itself such that is a homeomorphism of onto , and is a compact continuum for each point x in . Similar theorems are obtained for 3cells and disks topologically embedded in .
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 Ralph J. Bean, Repairing embeddings and decompositions in , Duke Math. J. 36 (1969), 373385. MR 39 #4820. MR 0243499 (39:4820)
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 R. H. Bing, Each disk in contains a tame arc, Amer. J. Math. 84 (1962), 583590. MR 26 #4331. MR 0146811 (26:4331)
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 , Each disk in is pierced by a tame arc, Amer. J. Math. 84 (1962), 591599. MR 26 #4332. MR 0146812 (26:4332)
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 , Extending monotone decompositions of 3manifolds, Trans. Amer. Math. Soc. 149 (1970), 351369. MR 0263051 (41:7656)
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 , Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145158. MR 15, 816. MR 0061377 (15:816d)
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 , Pushing a 2sphere into its complement, Michigan Math. J. 11 (1964), 3345. MR 28 #3408. MR 0160194 (28:3408)
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 R. J. Daverman, A new proof of the HosayLininger Theorem about crumpled cubes, Proc. Amer. Math. Soc. 23 (1969), 5254. MR 39 #7578. MR 0246274 (39:7578)
 [8]
 R. J. Daverman and W. T. Eaton, An equivalence for the embeddings of cells in a 3manifold, Trans. Amer. Math. Soc. 145 (1969), 369381. MR 40 #3519. MR 0250280 (40:3519)
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 E. Dyer and M. E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103118. MR 19, 1187. MR 0092959 (19:1187e)
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 John Hempel, A surface in is tame if it can be deformed into each complementary domain, Trans. Amer. Math. Soc. 111 (1964), 273287. MR 28 #3409. MR 0160195 (28:3409)
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 N. Hosay, The sum of a real cube and a crumpled cube is , Notices Amer. Math. Soc. 10 (1963), 666; errata, ibid. 11 (1964), 152. Abstract #60717.
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 William Jaco and D. R. McMillan, Jr., Retracting threemanifolds onto finite graphs, Illinois J. Math. 14 (1970), 150158. MR 0256370 (41:1026)
 [13]
 H. W. Lambert, Mapping cubes with holes onto cubes with handles, Illinois J. Math. 13 (1969), 606615. MR 40 #2037. MR 0248787 (40:2037)
 [14]
 L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534549. MR 31 #2717. MR 0178460 (31:2717)
 [15]
 D. R. McMillan, Jr., Neighborhoods of surfaces in 3manifolds, Michigan Math. J. 14 (1967), 161170. MR 35 #3643. MR 0212778 (35:3643)
 [16]
 R. L. Moore, Concerning upper semicontinuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416428. MR 1501320
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102823525
PII:
S 00029947(1971)02823525
Keywords:
Wild sphere,
tame sphere,
monotone map,
upper semicontinuous decomposition,
crumpled cube,
repairing embeddings
Article copyright:
© Copyright 1971
American Mathematical Society
