On embeddings with locally nice cross-sections

Abstract:

A $k$-dimensional compactum ${X^k}$ in euclidean space ${E^n}(n - k \geqq 3)$ is said to be locally nice in ${E^n}$ if ${E^n} - {X^k}$ is $1$-ULC. In this paper we prove a general theorem which implies, in particular, that ${X^k}$ is locally nice in ${E^n}$ if the intersection of ${X^k}$ with each horizontal hyperplane of ${E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $k$-dimensional polyhedron $P$ in ${E^n}$ ($n - k \geqq 3$ and $n \geqq 5$) is tame in ${E^n}$ if each $({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $1$-ULC. However, by strengthening our general theorem in the case $n = 4$, we are able to prove this result for $n = 4$ as well. For example, an arc $A$ in ${E^4}$ is tame if each horizontal cross-section of $A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).