On the shrinkability of decompositions of -manifolds

Author:
William L. Voxman

Journal:
Trans. Amer. Math. Soc. **150** (1970), 27-39

MSC:
Primary 54.78

DOI:
https://doi.org/10.1090/S0002-9947-1970-0261577-8

MathSciNet review:
0261577

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Abstract: An upper semicontinuous decomposition of a metric space is said to be *shrinkable* in case for each covering of the union of the nondegenerate elements, for each , and for an arbitrary homeomorphism from onto , there exists a homeomorphism from onto itself such that

(1) if , then ,

(2) for each , (a) and (b) there exists such that .

Our main result is that if is a cellular decomposition of a -manifold , then if and only if is shrinkable. We also define concepts of local and weak shrinkability, and we show the equivalence of the various types of shrinkability for certain cellular decompositions. Some applications of these notions are given, and extensions of theorems of Bing and Price are proved.

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DOI:
https://doi.org/10.1090/S0002-9947-1970-0261577-8

Keywords:
Cellular decompositions of -manifolds,
shrinkability,
0-dimensional decompositions,
weakly shrinkable,
locally shrinkable

Article copyright:
© Copyright 1970
American Mathematical Society