(1)(2)-Homogeneous continua with cut points

Abstract

A space is said to be _21-homogeneous provided that there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. It is shown that if X is a 1/2-homogeneous continuum with at least one cut point, then X has either uncountably many cut points or only one cut point c. In the former case, X is 1/2-homogeneous if and only if X is an arc or X is a compactification of the reals R-1 whose remainder is the union of two disjoint, nondegenerate, homeomorphic homogeneous continua and the ends of X are mutually homeomorphic and 1/3-homogeneous. In the latter case, the closures of the components of X - {c} are mutually homeomorphic and 2-homogeneous at c, and ord(c)(X) >= 4

furthermore, if ord(c)(X) <= omega, X is a locally connected bouquet of simple closed curves. Conversely, the two conditions about the components of X - {c} are shown to imply X is 1/2-homogeneous under an additional assumption, which is shown by examples to be both required and restrictive. (C) 2007 Elsevier B.V. All rights reserved.