A complete classification of homogeneous plane continua

Academic Article

Abstract

  • © 2016, Institut Mittag-Leffler. We show that the only compact and connected subsets (i.e. continua) X of the plane R2 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 S1, 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: C→ X such that f(C) ⊂ g(C) there exists c0∈ C so that f(c0) = g(c0). 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.
  • Published In

  • Acta Mathematica  Journal
  • Digital Object Identifier (doi)

    Author List

  • Hoehn LC; Oversteegen LG
  • Start Page

  • 177
  • End Page

  • 216
  • Volume

  • 216
  • Issue

  • 2