Abstract
Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5 or K 3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5, K 3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.
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An erratum to this article is available at http://dx.doi.org/10.1007/s00493-014-2967-9.
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Richter, R.B., Rooney, B. & Thomassen, C. On planarity of compact, locally connected, metric spaces. Combinatorica 31, 365–376 (2011). https://doi.org/10.1007/s00493-011-2563-1
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DOI: https://doi.org/10.1007/s00493-011-2563-1