A hierarchy for closed n-cell-complements

Let $C$ and $D$ be a pair of crumpled $n$-cubes and $h$ a homeomorphism of $\text{Bd }C$ to $\text{Bd }D$ for which there exists a map $f_h: C\to D$ such that $f_h|\text{Bd }C =h$ and $f_{h}^{-1}(\text{Bd }D)=\text{Bd }C$. 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 $C$ 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 $(C,D,h) \in \mathscr{W}_n$. The effort can be viewed as an initial way of classifying the wildness of crumpled cubes.