On planarity of compact, locally connected, metric spaces

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 K5 or K3,3. 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 K5, K3,3, or any generalized thumbtack, or the disjoint union of a sphere and a point.

• Type: Journal
• Year: 2011
• Where: COMBINATORICA
• Authors: R. Bruce Richter, Brendan Rooney, Carsten Thomassen
• Keywords: Combinatorica, COMBINATORICA 2011, Disjoint Union, Metric Space, Optimization