Continuously homogeneous continua and their arc components

Certain Homogeneous Unicoherent Indecomposable Continua

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [l, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has give...

SOME THEOREMS ON n-HOMOGENEOUS CONTINUA

A Complete Classification of Homogeneous Plane Continua

We show that the only compact and connected subsets (i.e. continua) X of the plane R2 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 S1, 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 th...

Homologically arc-homogeneous ENRs

The so-called Modified Bing–Borsuk Conjecture, which grew out of a question in [1], asserts that a homogeneous Euclidean neighborhood retract is a homology manifold. At this mini-workshop on exotic homology manifolds, Frank Quinn asked whether a space that satisfies a similar property, which he calls homological arc-homogeneity, is a homology manifold. The purpose of this note is to show that t...

Use of a New Technique in Homogeneous Continua

Recently Ungar has employed a theorem due to Effros [3] in the study of homogeneous continua. It is the purpose of this paper to use the same theorem to clarify and simplify Bing's proof that the only plane homogeneous continuum containing an arc is a simple closed curve [ 1 ]. Preliminary definitions and lemmas. Let M denote a continuum (= compact, connected, metric space). Then M is said to b...