A hierarchy for closed $n$-cell complements

Abstract

Let $C$ and $D$ be a pair of crumpled $n$-cubes and $h$ a homeo\-morphism of Bd$C$ to Bd$D$ for which there exists a map $f_h: C\to D$ such that $f_h\mid$ Bd$C=h$ and $f_{h}^{-1}$Bd$D$=Bd$C $. In our view, the presence of such a triple $(C,D,h)$ suggests that $C$ is ``at least as wild as" $D$. The collection $\mathscr {W}_n$ of all such triples is the subject of this paper. If $(C,D,h)\in \mathscr {W}_n$, but there is no homeomorphism such that $D$ is at least as wild as $C$, we say that $C$ is ``strictly wilder than" $D$. The latter concept imposes a partial order on the collection of crumpled $n$-cubes. Here, we study features of these wildness comparisons, and we present certain attributes of crumpled cubes that are preserved by the maps arising when $(C,D,h) \in \mathscr {W}_n$. This effort may be viewed as an initial way of classifying the wildness of crumpled cubes.