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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conditions under which disks are $ P$-liftable

Author: Edythe P. Woodruff
Journal: Trans. Amer. Math. Soc. 186 (1973), 403-418
MSC: Primary 57A10; Secondary 54B15
MathSciNet review: 0328943
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Abstract: A generalization of the concept of lifting of an n-cell is studied. In the usual upper semicontinuous decomposition terminology, let S be a space, $ S/G$ be the decomposition space, and the projection mapping be $ P:S \to S/G$ . A set $ X' \subset S$ is said to be a P-lift of a set $ X \subset S/G$ if $ X'$ is homeomorphic to X and $ P(X')$ is X.

Examples are given in which the union of two P-liftable sets does not P-lift. We prove that if compact 2-manifolds A and B each P-lift, their union is a disk in $ {E^3}/G$, their intersection misses the singular points of the projection mapping, and the intersection of the singular points with the union of A and B is a 0-dimensional set, then the union of A and B does P-lift.

Even if a disk D does not P-lift, it is proven that for $ \epsilon > 0$ there is a P-liftable disk $ \epsilon $-homeomorphic to D, provided that the given decomposition of $ {E^3}$ is either definable by 3-cells, or the set of nondegenerate elements is countable and $ {E^3}/G$ is homeomorphic to $ {E^3}$. With further restrictions on the decomposition, this P-liftable disk can be chosen in such a manner that it agrees with D on the singular points of D.

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Keywords: Decomposition space, topology of $ {E^3}$, lift of a space, P-lift, monotone decomposition, cellular decomposition, decomposition definable by 3-cells, tame, neighborhoods of submanifolds
Article copyright: © Copyright 1973 American Mathematical Society