A complete classification of homogeneous plane continua

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 $f,g:C\to X$ such that $f(C)\subset g(C)$ there exists $c_0\in C$ 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.