A nonshrinkable decomposition of 𝑆ⁿ involving a null sequence of cellular arcs

Daverman, R. J.; Walsh, J. J.

doi:10.1090/S0002-9947-1982-0662066-7

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

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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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