Each crumpled 4-cube is a closed 4-cell-complement
In studying the wildness of an (n-1)-sphere \(\Sigma\) topologically embedded in the n-sphere \(S^ n\), it can be useful to tackle the pathology one side at a time. Feasability is ensured by the following fairly general theorem: for each crumpled n-cube C in \(S^ n\) (i.e., C is the closure of a component of \(S^ n-\Sigma)\) and each \(\epsilon >0\) there exists an embedding h of C in \(S^ n\) moving points less than \(\epsilon\) for which the closure of \(S^ n-h(C)\) is an n-cell. This was proved for \(n=3\) by \textit{N. Hosay} [Notices Am. Math. Soc. 9, 666 (1963)] and \textit{L. L. Lininger} [Trans. Am. Math. Soc. 118, 534-549 (1965; Zbl 0134.430)], for \(n\geq 5\) by the author [Mich. Math. J. 24, 225-241 (1977; Zbl 0376.57007)], and for \(n=4\) in the paper under review. Instrumental in each of these arguments is the locally flat approximation theorem for embeddings of (n-1)-manifolds in n-manifolds, the 4- dimensional version of which was established recently by \textit{F. D. Ancel} [Topology Appl. 24, 13-40 (1986; Zbl 0607.57011)]. Ancel's paper also surveys the history of these results as well as their (possible) extensions to embeddings in n-manifolds of generalized n-manifolds with boundary X, where \(\partial X\times (-1,1)\) is required to be an n- manifold.
wildness of an (n-1)-sphere topologically embedded in the n-sphere
locally flat approximation theorem for embeddings of (n-1)-manifolds in n-manifolds
scientific article; zbMATH DE number 4001002