A complete classification of homogeneous plane continua

TL;DR

This paper classifies all homogeneous plane continua, showing they are either a circle, pseudo-arc, or circle of pseudo-arcs, and characterizes the pseudo-arc via hereditarily indecomposable and span zero properties.

Contribution

It provides a complete classification of homogeneous plane continua and introduces a new characterization of the pseudo-arc.

Findings

Every non-degenerate homogeneous plane continuum is homeomorphic to a circle, pseudo-arc, or circle of pseudo-arcs.

Any planar homogeneous compactum is a product of a point, one of the three continua, with a finite set or Cantor set.

A continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable and has span zero.

Abstract

We show that every non-degenerate homogeneous plane continuum is homeomorphic to either the unit circle, the pseudo-arc, or the circle of pseudo-arcs. It follows that any planar homogenous compactum has the form $X \times Z$, where $X$ is a either a point or one of these three homogeneous plane continua, and $Z$ is a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc: a non-degenerate continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable and has span zero.