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2017 A hierarchy for closed $n$-cell complements
Robert J. Daverman, Shijie Gu
Rocky Mountain J. Math. 47(7): 2133-2166 (2017). DOI: 10.1216/RMJ-2017-47-7-2133

Abstract

Let $C$ and $D$ be a pair of crumpled $n$-cubes and $h$ a homeo\-morphism of Bd$C$ to Bd$D$ for which there exists a map $f_h: C\to D$ such that $f_h\mid$ Bd$C=h$ and $f_{h}^{-1}$Bd$D$=Bd$C $. In our view, the presence of such a triple $(C,D,h)$ suggests that $C$ is ``at least as wild as" $D$. The collection $\mathscr {W}_n$ of all such triples is the subject of this paper. If $(C,D,h)\in \mathscr {W}_n$, but there is no homeomorphism such that $D$ is at least as wild as $C$, we say that $C$ is ``strictly wilder than" $D$. The latter concept imposes a partial order on the collection of crumpled $n$-cubes. Here, we study features of these wildness comparisons, and we present certain attributes of crumpled cubes that are preserved by the maps arising when $(C,D,h) \in \mathscr {W}_n$. This effort may be viewed as an initial way of classifying the wildness of crumpled cubes.

Citation

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Robert J. Daverman. Shijie Gu. "A hierarchy for closed $n$-cell complements." Rocky Mountain J. Math. 47 (7) 2133 - 2166, 2017. https://doi.org/10.1216/RMJ-2017-47-7-2133

Information

Published: 2017
First available in Project Euclid: 24 December 2017

zbMATH: 1385.57022
MathSciNet: MR3743708
Digital Object Identifier: 10.1216/RMJ-2017-47-7-2133

Subjects:
Primary: 57N15
Secondary: 57N16, 57N45, 57N50

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 7 • 2017
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