A complete classification of homogeneous plane continua

作者：Hoehn Logan C^*; Oversteegen Lex G

来源：Acta Mathematica, 2016, 216(2): 177-216.

DOI：10.1007/s11511-016-0138-0

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.

• 出版日期2016-6