Bödi, Richard On the embedding of zero-dimensional double loops in locally Euclidean double loops. (English) Zbl 0768.22001 Result. Math. 22, No. 3-4, 657-666 (1992). 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 R. P. Osborne [Mich. Math. J. 13, 57-63 (1966; Zbl 0138.189)] and D. G. Wright [Houston J. Math. 2, 439-447 (1976; Zbl 0332.57002)]. Reviewer: Th.Grundhöfer (Tübingen) Cited in 2 Documents MSC: 22A30 Other topological algebraic systems and their representations 20N05 Loops, quasigroups 57N45 Flatness and tameness of topological manifolds 12K99 Generalizations of fields 57N35 Embeddings and immersions in topological manifolds 57Q35 Embeddings and immersions in PL-topology Keywords:tame embedding; topological double loop; automorphism groups; closed sub- double-loop; positive topological dimension; linear embedding; tameness criteria; Cantor sets Citations:Zbl 0138.189; Zbl 0332.57002 PDFBibTeX XMLCite \textit{R. Bödi}, Result. Math. 22, No. 3--4, 657--666 (1992; Zbl 0768.22001) Full Text: DOI References: [1] R.H. Blng, K. Borsuk, Some remarks concerning topologically homogeneous spaces. Ann. of Math. 81 (1965), p. 100–111 · Zbl 0127.13302 [2] R.D. Edwards, R.C. Kirby, Deformations of spaces of embeddings. Ann. of Math. 93 (1971), p. 63–88 · Zbl 0214.50303 [3] T. Grundhöfer, H. Salzmann, Locally compact double loops and ternary fields. In: O. Chein, H.D. Pflugfelder, J.D.H. Smith (Eds.), Quasigroups and Loops: Theory and Applications, Berlin: Heldermann 1990 · Zbl 0749.51016 [4] V.K.A.M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres, Part I. Proc. London Math. Soc. 3 (1953), p. 29–53 · Zbl 0050.17902 [5] R.P. Osborne, Embedding Cantor sets in manifolds, Part I: Tame Cantor sets in En. Mich. Math. J. 13 (1966), p. 57–63 · Zbl 0138.18902 [6] H. Salzmann, Topological planes. Adv. Math. 2 (1967), p. 1–60 · Zbl 0153.21601 [7] D.G. Wright, Pushing a Cantor set off itself. Houston J. Math. 2(1976), p. 439–447 · Zbl 0332.57002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.