A nonshrinkable decomposition of whose nondegenerate elements are contained in a cellular arc
Authors:
W. H. Row and John J. Walsh
Journal:
Trans. Amer. Math. Soc. 289 (1985), 227252
MSC:
Primary 57N10; Secondary 54B10, 57N30, 57N60
MathSciNet review:
779062
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Abstract: A decomposition of is constructed with the following properties: (1) The set of all nondegenerate elements consists of a null sequence of arcs and is a simple closed curve. (2) Each arc contained in is cellular. (3) is the boundary of a disk that is locally flat except at points of . (4) The decomposition is not shrinkable; that is, the decomposition space is not homeomorphic to .
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 [Ba]
 C. D. Bass, Squeezing cells to cells in , Fund. Math. 110 (1980), 3550. MR 600578 (82a:57011)
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 R. H. Bing, Pointlike decompositions of Fund. Math. 50 (1962), 431453. MR 0137104 (25:560)
 [BS]
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 [Da ]
 R. J. Daverman, A nonshrinkable decomposition of determined by a null sequence of cellular sets, Proc. Amer. Math. Soc. 75 (1979), 171176. MR 529236 (80j:57015)
 [Da ]
 , Shrinking certain closed dimensional decompositions of manifolds, Houston J. Math. 5 (1979), 4147. MR 533638 (80j:57014)
 [Da ]
 , On cells in Euclidean space that cannot be squeezed, Rocky Mountain J. Math. 5 (1975), 8793. MR 0370598 (51:6825)
 [DE ]
 R. J. Daverman and W. T. Eaton, An equivalence for the embeddings of cells in a manifold, Trans. Amer. Math. Soc. 145 (1969), 369381. MR 0250280 (40:3519)
 [DE ]
 , A dense set of sewings of two crumpled cubes yield , Fund. Math. 65 (1969), 5160. MR 0243500 (39:4821)
 [DW]
 R. J. Daverman and J. J. Walsh, A nonshrinkable decomposition of involving a null sequence of cellular sets, Trans. Amer. Math. Soc. 272 (1982), 771784. MR 662066 (83m:57012)
 [Ed]
 R. D. Edwards, Approximating certain celllike maps by homeomorphisms, Notices Amer. Math. Soc. 24 (1977), A649.
 [Fr]
 M. Freedman, The topology of fourdimensional manifolds, J. Differential Geom. 17 (1982), 357453. MR 679066 (84b:57006)
 [GM]
 D. S. Gillman and J. M. Martin, Countable decompositions of into points and pointlike arcs, Notices Amer. Math. Soc. 10 (1963), 7475.
 [Mc]
 D. R. McMillan, Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327337. MR 0161320 (28:4528)
 [Wr]
 D. G. Wright, A decomposition of into points and a null sequence of cellular sets, General Topology Appl. 10 (1979), 297304. MR 546103 (81a:57014)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507790624
PII:
S 00029947(1985)07790624
Keywords:
Cellular decomposition,
cellular arc,
nonshrinkable,
manifold
Article copyright:
© Copyright 1985
American Mathematical Society
