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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Extending cell-like maps on manifolds


Authors: B. J. Ball and R. B. Sher
Journal: Trans. Amer. Math. Soc. 186 (1973), 229-246
MSC: Primary 57A60
DOI: https://doi.org/10.1090/S0002-9947-1973-0328950-3
MathSciNet review: 0328950
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Abstract: Let X be a closed subset of a manifold M and ${G_0}$ be a cell-like upper semicontinuous decomposition of X. We consider the problem of extending ${G_0}$ to a cell-like upper semicontinuous decomposition G of M such that $M/G \approx M$. Under fairly weak restrictions (which vanish if $M = {E^n}$ or ${S^n}$ and $n \ne 4$ we show that such a G exists if and only if the trivial extension of ${G_0}$, obtained by adjoining to ${G_0}$ the singletons of $M - X$, has the desired property. In particular, the nondegenerate elements of Bing’s dogbone decomposition of ${E^3}$ are not elements of any cell-like upper semicontinuous decomposition G of ${E^3}$ such that ${E^3}/G \approx {E^3}$. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if $X/G \approx X$ and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in ${E^3}$ are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold $(k,m \ne 4)$. Examples are given of an everywhere wild simply embedded simple closed curve in ${E^3}$ and of a compact absolute retract which embeds in ${E^3}$ yet has no simple embedding in ${E^3}$.


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Keywords: Extension, proper map, cell-like space, cell-like map, monotone map, trivial extension, upper semicontinuous decomposition, dogbone space, tame, locally flat
Article copyright: © Copyright 1973 American Mathematical Society