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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Author: Michael Starbird
Journal: Trans. Amer. Math. Soc. 249 (1979), 203-215
MSC: Primary 57N12
MathSciNet review: 526318
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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\vert{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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0526318-3
PII: S 0002-9947(1979)0526318-3
Keywords: Cell-like, decompositions
Article copyright: © Copyright 1979 American Mathematical Society