Self-universal crumpled cubes and a dogbone space

Abstract

The question of whether each self-universal crumpled cube is universal is answered negatively by presenting an example of a dogbone space which is not topologically E3 but which can be expressed as a sewing of two crumpled cubes, one of which is selfuniversal. C. D. Bass and R. J. Davernian [2] presented a brief paper indicating that the solid Alexander horned sphere is an example of a crumpled cube which is self-universal but not universal, thus answering negatively the question asked by C. E. Burgess and J. W. Cannon in [4] of whether each self-universal crumpled cube is universal. The validity of the example presented by Bass and Daverman depends on a claim that a certain upper semicontinuous decomposition of S3 into points and tame arcs, described in [2], is not topologically S3, which in turn depends on the validity of four lemmas which are stated in [2, §2]. The proofs of these four lemmas appear to entail nontrivial arguments which are not included in [2]. In this note, we present an example of a dogbone space, an upper semicontinuous decomposition of S3 into points and tame arcs whose nondegenerate elements can be expressed as the intersection of a tower of solid double tori, which is not topologically S3. The dogbone space can be described as the result of a sewing of two crumpled cubes, one of which is self-universal. Thus, the question asked by Burgess and Cannon in [4] is answered negatively. The argument will be based essentially upon work by Casier [5] and the author [1]. Some recent work by Eaton [6] includes a different proof that the solid Alexander horned sphere H, used in the example presented by Bass and Daverman, is not universal. This was done by sewing H to the crumpled cube F described by Stallings [7] so that the wild points of Bd //are sewn to the Cantor set of nonpiercing points of F. Other methods developed by Eaton, in papers cited in [6], should offer alternative ways Presented to the Society, October 30, 1971 ; received by the editors August 18, 1971. AMS 1970 subject classifications. Primary 57A10.