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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Conditions under which disks are $P$-liftable
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by Edythe P. Woodruff PDF
Trans. Amer. Math. Soc. 186 (1973), 403-418 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 186 (1973), 403-418
  • MSC: Primary 57A10; Secondary 54B15
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0328943-6
  • MathSciNet review: 0328943