Repairing embeddings of -cells with monotone maps of

Author:
William S. Boyd

Journal:
Trans. Amer. Math. Soc. **161** (1971), 123-144

MSC:
Primary 54.78

DOI:
https://doi.org/10.1090/S0002-9947-1971-0282352-5

MathSciNet review:
0282352

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If is a 2-sphere topologically embedded in Euclidean 3-space 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 3-cells and disks topologically embedded in .

**[1]**Ralph J. Bean,*Repairing embeddings and decompositions in*, Duke Math. J.**36**(1969), 373-385. MR**39**#4820. MR**0243499 (39:4820)****[2]**R. H. Bing,*Each disk in**contains a tame arc*, Amer. J. Math.**84**(1962), 583-590. MR**26**#4331. MR**0146811 (26:4331)****[3]**-,*Each disk in**is pierced by a tame arc*, Amer. J. Math.**84**(1962), 591-599. MR**26**#4332. MR**0146812 (26:4332)****[4]**-,*Extending monotone decompositions of*3-*manifolds*, Trans. Amer. Math. Soc.**149**(1970), 351-369. MR**0263051 (41:7656)****[5]**-,*Locally tame sets are tame*, Ann. of Math. (2)**59**(1954), 145-158. MR**15**, 816. MR**0061377 (15:816d)****[6]**-,*Pushing a*2-*sphere into its complement*, Michigan Math. J.**11**(1964), 33-45. MR**28**#3408. MR**0160194 (28:3408)****[7]**R. J. Daverman,*A new proof of the Hosay-Lininger Theorem about crumpled cubes*, Proc. Amer. Math. Soc.**23**(1969), 52-54. MR**39**#7578. MR**0246274 (39:7578)****[8]**R. J. Daverman and W. T. Eaton,*An equivalence for the embeddings of cells in a*3-*manifold*, Trans. Amer. Math. Soc.**145**(1969), 369-381. MR**40**#3519. MR**0250280 (40:3519)****[9]**E. Dyer and M. E. Hamstrom,*Completely regular mappings*, Fund. Math.**45**(1958), 103-118. MR**19**, 1187. MR**0092959 (19:1187e)****[10]**John Hempel,*A surface in**is tame if it can be deformed into each complementary domain*, Trans. Amer. Math. Soc. 111 (1964), 273-287. MR**28**#3409. MR**0160195 (28:3409)****[11]**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 #607-17.**[12]**William Jaco and D. R. McMillan, Jr.,*Retracting three-manifolds onto finite graphs*, Illinois J. Math.**14**(1970), 150-158. MR**0256370 (41:1026)****[13]**H. W. Lambert,*Mapping cubes with holes onto cubes with handles*, Illinois J. Math.**13**(1969), 606-615. MR**40**#2037. MR**0248787 (40:2037)****[14]**L. Lininger,*Some results on crumpled cubes*, Trans. Amer. Math. Soc.**118**(1965), 534-549. MR**31**#2717. MR**0178460 (31:2717)****[15]**D. R. McMillan, Jr.,*Neighborhoods of surfaces in*3-*manifolds*, Michigan Math. J.**14**(1967), 161-170. MR**35**#3643. MR**0212778 (35:3643)****[16]**R. L. Moore,*Concerning upper semi-continuous collections of continua*, Trans. Amer. Math. Soc.**27**(1925), 416-428. MR**1501320**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54.78

Retrieve articles in all journals with MSC: 54.78

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0282352-5

Keywords:
Wild sphere,
tame sphere,
monotone map,
upper semicontinuous decomposition,
crumpled cube,
repairing embeddings

Article copyright:
© Copyright 1971
American Mathematical Society