An example of a wild $(n-1)$-sphere in $S^{n}$ in which each $2$-complex is tame

Abstract:

The main purpose of this note is to give an example promised in the title (for $n \geqq 5$). The example is the k-fold suspension $(k \geqq 2)$ of Bing’s 2-sphere in ${S^3}$ in which each closed, nowhere dense subset is tame. Our efforts were motivated by recent results of Seebeck and Sher concerning tame cells in wild cells and spheres. In fact, we are able to strengthen one of Seebeck’s results in order to prove that every embedding of an m-dimensional polyhedron in our wild $(n - 1)$-sphere $S(n - m \geqq 3)$ can be approximated in S by an embedding that is tame in ${S^n}$.