A nonshrinkable decomposition of $S^{n}$ 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

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

MathSciNet review:
662066

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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 $C$.

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*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.

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Additional Information

Keywords:
Upper semicontinuous decomposition,
cellular,
shrinkable,
crumpled <IMG WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img36.gif" ALT="$n$">-cube,
Disjoint Disks Property,
Boundary Mismatch Property,
spun decomposition,
eyebolt

Article copyright:
© Copyright 1982
American Mathematical Society