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

MathSciNet review:
0328950

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *X* be a closed subset of a manifold *M* and be a cell-like upper semicontinuous decomposition of *X*. We consider the problem of extending to a cell-like upper semicontinuous decomposition *G* of *M* such that . Under fairly weak restrictions (which vanish if or and we show that such a *G* exists if and only if the trivial extension of , obtained by adjoining to the singletons of , has the desired property. In particular, the nondegenerate elements of Bing's dogbone decomposition of are not elements of any cell-like upper semicontinuous decomposition *G* of such that . Call a cell-like upper semicontinuous decomposition *G* of a metric space *X simple* if 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 are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat *k*-manifold in an *m*-manifold . Examples are given of an everywhere wild simply embedded simple closed curve in and of a compact absolute retract which embeds in yet has no simple embedding in .

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0328950-3

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