Extending cell-like maps on manifolds

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 ${E^3}$.