Cell-like $0$-dimensional decompositions of $E^{3}$

Abstract:

Let G be a cell-like, 0-dimensional upper semicontinuous decomposition of ${E^3}$. It is shown that if $\Gamma$ is a tame 1-complex which is a relatively closed subset of a saturated open set U whose boundary misses the nondegenerate elements of G, then there is a homeomorphism $h:{E^3} \to {E^3}$ so that $h|{E^3} - U = {\text {id}}$ and $h(\Gamma )$ misses the nondegenerate elements of G. This theorem implies a disjoint disk type criterion for shrinkability of G. This criterion in turn provides a direct proof of the recent result of Starbird and Woodruff that if G is an u.s.c. decomposition of ${E^3}$ into points and countably many cellular, tamely embedded polyhedra, then ${E^3}/G$ is homeomorphic to ${E^3}$.