Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A nonshrinkable decomposition of $ S\sp{n}$ involving a null sequence of cellular arcs

Authors: R. J. Daverman and J. J. Walsh
Journal: Trans. Amer. Math. Soc. 272 (1982), 771-784
MSC: Primary 57N60; Secondary 54B15
MathSciNet review: 662066
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a decomposition $ G$ of $ S^n(n\ge 3)$ into points and a null sequence of cellular arcs such that $ S^n/G$ is not a manifold; furthermore, the union of the nondegenerate elements from $ G$ lies in a $ 2$-cell in $ S^n$ and the image in $ S^n/G$ of this union has 0-dimensional closure. Examples of nonshrinkable decompositions with a null sequence of cellular arcs have been constructed in the case $ n=3$ by D. S. Gillman and J. M. Martin and by R. H. Bing and M. Starbird. We construct another example in this dimension, for which all the arcs lie in the boundary of a crumpled cube $ C$, and then produce higher dimensional examples by spinning $ C$.

References [Enhancements On Off] (What's this?)

  • [Al] W. R. Alford, Some ``nice'' wild $ 2$-spheres in $ E^3$, Topology of $ 3$-Manifolds and Related Topics (M. K. Fort, Jr., ed.),Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 29-33. MR 0141091 (25:4504)
  • [Ar] S. Armentrout, Cellular decompositions of $ 3$-manifolds that yield $ 3$-manifolds, Mem. Amer. Math. Soc. no. 107 (1971). MR 0413104 (54:1225)
  • [AP] S. Armentrout and T. M. Price, Decompositions into compact sets with UV properties, Trans. Amer. Math. Soc. 141 (1969), 433-442. MR 0244994 (39:6307)
  • [B$ _{1}$] R. H. Bing, Upper semicontinuous decompositions of $ E^3$, Ann. of Math. (2) 65 (1957), 363-374. MR 0092960 (19:1187f)
  • [B$ _{2}$] -, A wild surface each of whose arcs is tame, Duke Math. J. 28 (1961), 1-15. MR 0123302 (23:A630)
  • [B$ _{3}$] -, Point-like decompositions of $ E^3$, Fund. Math. 50 (1962), 431-453. MR 0137104 (25:560)
  • [BS] R. H. Bing and M. Starbird, A decomposition of $ S^3$ with a null sequence of cellular arcs, Geometric Topology (J. C. Cantrell, ed.), Academic Press, New York, 1979, pp. 3-21. MR 537722 (80i:57008)
  • [Br] Morton Brown, A proof of the generalized Shoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 0117695 (22:8470b)
  • [C] J. W. Cannon, Taming cell-like embedding relations, Geometric Topology (L. C. Glaser and T. B. Rushing, eds.), Lecture Notes in Math., vol. 438, Springer-Verlag, New York, 1975, pp. 66-118. MR 0391104 (52:11926)
  • [CD] J. W. Cannon and R. J. Daverman, Cell-like decompositions arising from mismatched sewings: applications to $ 4$-manifolds, Fund. Math. 111 (1981), 211-233. MR 611761 (82m:57007)
  • [D$ _{1}$] R. J. Daverman, A nonshrinkable decomposition of $ S^n$ determined by a null sequence of cellular sets, Proc. Amer. Math. Soc. 75 (1979), 171-176. MR 529236 (80j:57015)
  • [D$ _{2}$] -, Sewing of closed $ n$-cell-complements, Trans. Amer. Math. Soc. (to appear).
  • [Ea] W. T. Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972), 193-207. MR 0309116 (46:8227)
  • [Ed] R. D. Edwards, The topology of manifolds and cell-like maps, Proc. Internat. Congr. Math. (Helsinki, 1978), Academia Scientarium Fennica, Helsinki, 1980, pp. 111-127. MR 562601 (81g:57010)
  • [G] D. S. Gillman, Note concerning a wild sphere of Bing, Duke Math. J. 31 (1964), 247-254. MR 0161317 (28:4525)
  • [GM] D. S. Gillman and J. M. Martin, Countable decompositions of $ E^3$ into points and point-like arcs, Notices Amer. Math. Soc. 10 (1963), 74-75.
  • [L] R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717-731. MR 0251714 (40:4941)
  • [S] L. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-295. MR 0295365 (45:4431)
  • [St] M. Starbird, Null sequence cellular decompositions of $ S^3$, Fund. Math. 112 (1981), 81-87. MR 619484 (82j:57007)
  • [W] D. G. Wright, A decomposition of $ E^n(n\ge 3)$ into points and a null sequence of cellular sets, General Topology Appl. 10 (1979), 297-304. MR 546103 (81a:57014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N60, 54B15

Retrieve articles in all journals with MSC: 57N60, 54B15

Additional Information

Keywords: Upper semicontinuous decomposition, cellular, shrinkable, crumpled $ n$-cube, Disjoint Disks Property, Boundary Mismatch Property, spun decomposition, eyebolt
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society