## Extending cell-like maps on manifolds

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- by B. J. Ball and R. B. Sher PDF
- Trans. Amer. Math. Soc.
**186**(1973), 229-246 Request permission

## 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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## Additional Information

- © Copyright 1973 American Mathematical Society
- 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