A set is a 3 cell if its cartesian product with an arc is a 4 cell

Abstract

We are learning strange things about cartesian products. Shapiro has shown that the cartesian product of a line and a certain example described by Whitehead [16] is topologically E4. The example is a bounded open subset of E3 that is simply connected but which contains a simple closed curve which does not lie in any topological cube in it [3]. Glimm has given a proof of this result in [9] and a different proof is suggested in [5]. Artin and Fox have given [8] an example of a wild arc in S3 whose complement is simply connected though topologically different from E3. Kirkor has shown [i1 ] that the cartesian product of this complement and a line is E4. Bing has described a nonmanifold [2] whose cartesian product with a line is E4 [4; 5]. A modification of this example gives a compact nonmanifold whose cartesian product with a circle is topologically the same as the cartesian product of a 3 sphere and a circle. The modification is a monotone decomposition of S3 whose set of nondegenerate elements is a Cantor set of tame arcs. Rosen has complicated the example of Bing to show [14] that there is a set which is not locally a manifold at any point but whose cartesian product with a line is E4. A proof that the cartesian product of a line and such a decomposition is E4 has also been given by K. W. Kwun. Fox and Artin described [8] an arc in E3 whose complement is not simply connected. Curtis has shown [7] that the cartesian product of a line and the decomposition of E3 whose only nondegenerate element is this arc is E4. He later genralized this result to show that any decomposition of En whose only nondegenerate element is an arc gives En+1 under a cartesian product with a line. One might wonder about any monotone decomposition of E3 each of whose nondegenerate elements is an arc. Poenaru gives [12] an example of a 4 manifold with boundary different from a cell whose cartesian product with an arc is a S cell. One might wonder just how far these unexpected things go.