## A nonshrinkable decomposition of $S^{n}$ involving a null sequence of cellular arcs

HTML articles powered by AMS MathViewer

- by R. J. Daverman and J. J. Walsh
- Trans. Amer. Math. Soc.
**272**(1982), 771-784 - DOI: https://doi.org/10.1090/S0002-9947-1982-0662066-7
- PDF | Request permission

## 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

- W. R. Alford,
*Some βniceβ wild $2$-spheres in $E^{3}$*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp.Β 29β33. MR**0141091** - Steve Armentrout,
*Cellular decompositions of $3$-manifolds that yield $3$-manifolds*, Memoirs of the American Mathematical Society, No. 107, American Mathematical Society, Providence, R.I., 1971. MR**0413104** - Steve Armentrout and Thomas M. Price,
*Decompositions into compact sets with $UV$ properties*, Trans. Amer. Math. Soc.**141**(1969), 433β442. MR**244994**, DOI 10.1090/S0002-9947-1969-0244994-3 - R. H. Bing,
*Upper semicontinuous decompositions of $E^3$*, Ann. of Math. (2)**65**(1957), 363β374. MR**92960**, DOI 10.2307/1969968 - R. H. Bing,
*A wild surface each of whose arcs is tame*, Duke Math. J.**28**(1961), 1β15. MR**123302** - R. H. Bing,
*Point-like decompositions of $E^{3}$*, Fund. Math.**50**(1961/62), 431β453. MR**137104**, DOI 10.4064/fm-50-4-431-453 - R. H. Bing and Michael Starbird,
*A decomposition of $S^{3}$ with a null sequence of cellular arcs*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp.Β 3β21. MR**537722** - Marston Morse,
*A reduction of the Schoenflies extension problem*, Bull. Amer. Math. Soc.**66**(1960), 113β115. MR**117694**, DOI 10.1090/S0002-9904-1960-10420-X - J. W. Cannon,
*Taming cell-like embedding relations*, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp.Β 66β118. MR**0391104** - J. W. Cannon and R. J. Daverman,
*Cell-like decompositions arising from mismatched sewings: applications to $4$-manifolds*, Fund. Math.**111**(1981), no.Β 3, 211β233. MR**611761**, DOI 10.4064/fm-111-3-211-233 - Robert J. Daverman,
*A nonshrinkable decomposition of $S^{n}$ determined by a null sequence of cellular sets*, Proc. Amer. Math. Soc.**75**(1979), no.Β 1, 171β176. MR**529236**, DOI 10.1090/S0002-9939-1979-0529236-5
β, - W. T. Eaton,
*The sum of solid spheres*, Michigan Math. J.**19**(1972), 193β207. MR**309116**, DOI 10.1307/mmj/1029000892 - Robert D. Edwards,
*The topology of manifolds and cell-like maps*, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp.Β 111β127. MR**562601** - David S. Gillman,
*Note concerning a wild sphere of Bing*, Duke Math. J.**31**(1964), 247β254. MR**161317**
D. S. Gillman and J. M. Martin, - R. C. Lacher,
*Cell-like mappings. I*, Pacific J. Math.**30**(1969), 717β731. MR**251714**, DOI 10.2140/pjm.1969.30.717 - L. C. Siebenmann,
*Approximating cellular maps by homeomorphisms*, Topology**11**(1972), 271β294. MR**295365**, DOI 10.1016/0040-9383(72)90014-6 - Michael Starbird,
*Null sequence cellular decompositions of $S^{3}$*, Fund. Math.**112**(1981), no.Β 2, 81β87. MR**619484**, DOI 10.4064/fm-112-2-81-87 - David G. Wright,
*A decomposition of $E^{n}$ $(n\geq 3)$ into points and a null sequence of cellular sets*, General Topology Appl.**10**(1979), no.Β 3, 297β304. MR**546103**, DOI 10.1016/0016-660X(79)90042-4

*Sewing of closed*$n$

*-cell-complements*, Trans. Amer. Math. Soc. (to appear).

*Countable decompositions of*$E^3$

*into points and point-like arcs*, Notices Amer. Math. Soc.

**10**(1963), 74-75.

## Bibliographic Information

- © Copyright 1982 American Mathematical Society
- 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