\(\frac 1 2\)-homogeneous suspensions
A metric continuum \(X\) is said to be \(\frac 1 n\)-homogeneous provided that there are exactly \(n\) orbits for the action of the group of homeomorphisms of the space onto itself. An \(n\)-theta is the suspension of the discrete space with \(n\) points. The authors study metric continua \(X\) for which the suspension \(Sus(X)\) is \(\frac 1 2\)-homogeneous. The main results are: Theorem. Let \(X\) be either a local dendrite or a continuum containing a free arc. Then \(Sus(X)\) is \(\frac 1 2\)-homogeneous if and only if \(X\) is an \(n\)-theta for some positive integer \(n\), different from 2. Theorem. The only dendroid \(X\) for which \(Sus(X)\) is \(\frac 1 2\)-homogeneous is the arc. The authors include the following question. Question. Let \(X\) be a compact metric space such that \(Sus(X)\) is \(\frac 1 2\)-homogeneous. Must \(X\) be either homogeneous or \(\frac 1 2\)-homogeneous? Following the line of the paper, it would be interesting to solve the following problem. Problem. Determine those finite graphs (dendrites, local dendrites, dendroids) \(X\) for which \(Sus(X)\) is \(\frac 1 3\)-homogeneous.
scientific article; zbMATH DE number 5647967
scientific article