On the embedding of zero-dimensional double loops in locally Euclidean double loops
Let \(D\) be a topological double loop on the topological space \(\mathbb{R}^ n\). Then \(n\in \{1,2,4,8\}\). Sub-double-loops arise naturally in the study of automorphism groups of \(D\). The conjecture that every closed sub-double-loop has positive topological dimension is known to be true for \(n \leq 2\). The author proves that a closed sub-double-loop of dimension zero is always tamely embedded in \(D\) (i.e. the embedding is equivalent to a piecewise linear embedding), and that its complement in \(D\) is simply connected. To prove this, the author establishes three tameness criteria for Cantor sets in \(n\)-spheres. These criteria are variations of results of \textit{R. P. Osborne} [Mich. Math. J. 13, 57-63 (1966; Zbl 0138.189)] and \textit{D. G. Wright} [Houston J. Math. 2, 439-447 (1976; Zbl 0332.57002)].
scientific article; zbMATH DE number 147375