Conditions under which disks are $P$-liftable

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.