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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A characterization of manifold decompositions satisfying the disjoint triples property


Author: Dennis J. Garity
Journal: Proc. Amer. Math. Soc. 83 (1981), 833-838
MSC: Primary 54B15; Secondary 57N15
MathSciNet review: 630031
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Abstract: A metric space $ X$ satisfies the Disjoint Triples Property $ ({\text{D}}{{\text{D}}_{\text{3}}})$ if maps $ {f_1}$, $ {f_2}$ and $ {f_3}$ from $ {B^2}$ into $ X$ are approximable by maps $ {\tilde f_1}$, $ {\tilde f_2}$ and $ {\tilde f_3}$ with $ \cap _{i = 1}^3{\tilde f_i}({B^2}) = \emptyset $. Those CE decompositions of manifolds satisfying $ {\text{D}}{{\text{D}}_3}$ and yielding finite-dimensional nonmanifold decomposition spaces are shown to be precisely those intrinsically 0-dimensional decompositions which yield nonshrinkable null cellular decompositions under amalgamation. This characterization results in another proof of the fact that $ {E^n}/G \times {E^1}$ is secretly 0-dimensional where $ G$ is a CE usc decomposition of $ {E^n}$, $ n \geqslant 4$, with $ {E^n}/G$ finite dimensional.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0630031-6
PII: S 0002-9939(1981)0630031-6
Keywords: Cell-like decomposition, cellular set, disjoint disks property, disjoint triples property, intrinsic dimension, null sequence, secret dimension, shrinkable
Article copyright: © Copyright 1981 American Mathematical Society