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

DOI:
https://doi.org/10.1090/S0002-9947-1973-0328943-6

MathSciNet review:
0328943

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Abstract | References | Similar Articles | Additional Information

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 $Xβ \subset S$ is said to be a *P*-lift of a set $X \subset S/G$ if $Xβ$ is homeomorphic to *X* and $P(Xβ)$ 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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---,

*Concerning the condition that a disk in*${E^3}/G$

*be the image of a disk in*${E^3}$, Doctoral Dissertation, SUNY/Binghamton, 1971. ---,

*Examples of disks in*${E^3}/G$

*which can not be approximated by P-liftable disks*(to appear).

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Additional Information

Keywords:
Decomposition space,
topology of <IMG WIDTH="31" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${E^3}$">,
lift of a space,
<I>P</I>-lift,
monotone decomposition,
cellular decomposition,
decomposition definable by 3-cells,
tame,
neighborhoods of submanifolds

Article copyright:
© Copyright 1973
American Mathematical Society