On planarity of compact, locally connected, metric spaces

R. Bruce Richter, Brendan Rooney, Carsten Thomassen

    Research output: Contribution to journalJournal articlepeer-review

    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.
    Original languageEnglish
    JournalCombinatorica
    Volume31
    Issue number3
    Pages (from-to)365-376
    ISSN0209-9683
    DOIs
    Publication statusPublished - 2011

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