A hierarchy for closed n-cell-complements

Abstract

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 ${\mathcal{W}}_n$ of all such triples is the subject of this paper. If $(C,D,h)\in {\mathcal{W}}_n$ 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 $(C,D,h)\in {\mathcal{W}}_n$. The effort can be viewed as an initial way of classifying the wildness of crumpled cubes.

21 pages. Small updates. Theorem 6.1 in old version has been replaced by Theorem 6.3. A new theorem 6.1 has been added. To appear in Rocky. MT. J. Math

References in corpus (1)

• On the Shrinkable U.S.C. Decomposition Spaces of Spheres