A HIERARCHY FOR CLOSED n-CELL COMPLEMENTS

Let C and D be a pair of crumpled n-cubes and h a homeomorphism of Bd C to Bd D for which there exists a map f(h) : C -> D such that f(h) | 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 W-n of all such triples is the subject of this paper. If (C,D,h) is an element of 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) is an element of W-n. The effort can be viewed as an initial way of classifying the wildness of crumpled cubes.