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A complete classification of homogeneous plane continua

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Acta Mathematica

Abstract

We show that the only compact and connected subsets (i.e. continua) X of the plane \({\mathbb{R}^2}\) which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle \({\mathbb{S}^1}\), the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set.

The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps \({f,g\colon C \to X}\) such that \({f(C) \subset g(C)}\) there exists \({c_0 \in C}\) so that f(c 0) = g(c 0). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero.

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Authors and Affiliations

  1. Department of Computer Science & Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, Ontario, P1B 8L7, Canada

    Logan C. Hoehn

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, U.S.A.

    Lex G. Oversteegen

Authors
  1. Logan C. Hoehn
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  2. Lex G. Oversteegen
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Correspondence to Logan C. Hoehn.

Additional information

Dedicated to Andrew Lelek on the occasion of his 80th birthday.

The first named author was partially supported by NSERC grant RGPIN 435518 and by the Mary Ellen Rudin Young Researcher Award.

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Hoehn, L.C., Oversteegen, L.G. A complete classification of homogeneous plane continua. Acta Math 216, 177–216 (2016). https://doi.org/10.1007/s11511-016-0138-0

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  • DOI: https://doi.org/10.1007/s11511-016-0138-0

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