Some results on crumpled cubes

Abstract

Introduction. This paper deals with some problems on crumpled cubes. In [6] R. H. Bing raised the following questions: (1) Is the union of a crumpled cube and a 3-cell topologically S3? (2) Under what conditions will the union of two crumpled cubes sewn together along their boundaries be S3? (3) Under what conditions will the union of two copies of the same crumpled cube sewn together along their boundaries by the identity homeomorphism be S3? In [2] Bing proves that the union of two copies of the Solid Alexander Horned Sphere sewn together along their boundaries by the identity homeomorphism is S3. In [1] B. J. Ball proves that there exists a crumpled cube C such that two copies of C sewn together along their boundaries (not by the identity homeomorphism) is not topologically S3. In [7] Casler proves that the union of two copies of the Solid Alexander Horned Sphere sewn together along their boundaries is S3. In this paper it is proved in ?2 that question (1) has an affirmative answer. This result has recently been obtained independently by Norman Hosay [9]. Using this result, it is shown in ?3 that if C is a crumpled cube and B is a unit ball in E3, then there exists an upper semi-continuous decomposition G of B such that (1) B/G is homeomorphic to C, and (2) if g is a nondegenerate element of G, then g is point-like and the intersection of g with the boundary of B is a onepoint set. Corollary 2 of ?3 shows that if C and D are crumpled cubes, S is a tame 2-sphere in S3, and C and D are sewn together along their boundaries, then there exists an upper semi-continuous decomposition G of S3 such that the intersection of each nondegenerate element of G with S is a one-point set and S3/ G is homeomorphic to the union of C and D. In ?4 it is shown that if two copies of a particular bad crumpled cube are sewn together by the identity homeomorphism along their boundaries the result iS S3, and it is shown that if two copies of a particular nice crumpled cube are sewn together by the identity homeomorphism along their boundaries the result is not S3.