A nonshrinkable decomposition of 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

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662066-7

MathSciNet review:
662066

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a decomposition of into points and a null sequence of cellular arcs such that is not a manifold; furthermore, the union of the nondegenerate elements from lies in a -cell in and the image in of this union has 0-dimensional closure. Examples of nonshrinkable decompositions with a null sequence of cellular arcs have been constructed in the case 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 , and then produce higher dimensional examples by spinning .

**[**W. R. Alford,**Al**]*Some “nice” wild 2-spheres in 𝐸³*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 29–33. MR**0141091****[**Steve Armentrout,**Ar**]*Cellular decompositions of 3-manifolds that yield 3-manifolds*, American Mathematical Society, Providence, R. I., 1971. Memoirs of the American Mathematical Society, No. 107. MR**0413104****[**Steve Armentrout and Thomas M. Price,**AP**]*Decompositions into compact sets with 𝑈𝑉 properties*, Trans. Amer. Math. Soc.**141**(1969), 433–442. MR**0244994**, https://doi.org/10.1090/S0002-9947-1969-0244994-3**[**R. H. Bing,**B**]*Upper semicontinuous decompositions of 𝐸³*, Ann. of Math. (2)**65**(1957), 363–374. MR**0092960**, https://doi.org/10.2307/1969968**[**R. H. Bing,**B**]*A wild surface each of whose arcs is tame*, Duke Math. J.**28**(1961), 1–15. MR**0123302****[**R. H. Bing,**B**]*Point-like decompositions of 𝐸³*, Fund. Math.**50**(1961/1962), 431–453. MR**0137104****[**R. H. Bing and Michael Starbird,**BS**]*A decomposition of 𝑆³ with a null sequence of cellular arcs*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 3–21. MR**537722****[**Marston Morse,**Br**]*A reduction of the Schoenflies extension problem*, Bull. Amer. Math. Soc.**66**(1960), 113–115. MR**0117694**, https://doi.org/10.1090/S0002-9904-1960-10420-X

Morton Brown,*A proof of the generalized Schoenflies theorem*, Bull. Amer. Math. Soc.**66**(1960), 74–76. MR**0117695**, https://doi.org/10.1090/S0002-9904-1960-10400-4**[**J. W. Cannon,**C**]*Taming cell-like embedding relations*, Geometric topology (Proc. Conf., Park City, Utah, 1974) Springer, Berlin, 1975, pp. 66–118. Lecture Notes in Math., Vol. 438. MR**0391104****[**J. W. Cannon and R. J. Daverman,**CD**]*Cell-like decompositions arising from mismatched sewings: applications to 4-manifolds*, Fund. Math.**111**(1981), no. 3, 211–233. MR**611761****[**Robert J. Daverman,**D**]*A nonshrinkable decomposition of 𝑆ⁿ determined by a null sequence of cellular sets*, Proc. Amer. Math. Soc.**75**(1979), no. 1, 171–176. MR**529236**, https://doi.org/10.1090/S0002-9939-1979-0529236-5**[**-,**D**]*Sewing of closed**-cell-complements*, Trans. Amer. Math. Soc. (to appear).**[**W. T. Eaton,**Ea**]*The sum of solid spheres*, Michigan Math. J.**19**(1972), 193–207. MR**0309116****[**Robert D. Edwards,**Ed**]*The topology of manifolds and cell-like maps*, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 111–127. MR**562601****[**David S. Gillman,**G**]*Note concerning a wild sphere of Bing*, Duke Math. J.**31**(1964), 247–254. MR**0161317****[**D. S. Gillman and J. M. Martin,**GM**]*Countable decompositions of**into points and point-like arcs*, Notices Amer. Math. Soc.**10**(1963), 74-75.**[**R. C. Lacher,**L**]*Cell-like mappings. I*, Pacific J. Math.**30**(1969), 717–731. MR**0251714****[**L. C. Siebenmann,**S**]*Approximating cellular maps by homeomorphisms*, Topology**11**(1972), 271–294. MR**0295365**, https://doi.org/10.1016/0040-9383(72)90014-6**[**Michael Starbird,**St**]*Null sequence cellular decompositions of 𝑆³*, Fund. Math.**112**(1981), no. 2, 81–87. MR**619484****[**David G. Wright,**W**]*A decomposition of 𝐸ⁿ (𝑛≥3) into points and a null sequence of cellular sets*, General Topology Appl.**10**(1979), no. 3, 297–304. MR**546103**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662066-7

Keywords:
Upper semicontinuous decomposition,
cellular,
shrinkable,
crumpled -cube,
Disjoint Disks Property,
Boundary Mismatch Property,
spun decomposition,
eyebolt

Article copyright:
© Copyright 1982
American Mathematical Society