A complete classification of homogeneous plane continua

• 成果类型: Article
• 署名作者: Hoehn, Logan C.; Oversteegen, Lex G.
• 署名单位: Nipissing University; University of Alabama System; University of Alabama Birmingham
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-016-0138-0
• 发表日期: 2016
• 页码: 177-216
• 关键词: pseudo-arc; span; mappings; zero

摘要

We show that the only compact and connected subsets (i.e. continua) X of the plane 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 , 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 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 such that there exists 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.