Homogeneity degree of some symmetric products (Q2405080)

For a metric continuum \(X\), let \(F_{n}(X)\) denote the hyperspace of nonempty subsets of \(X\) with at most \(n\) points, and let \(hd(X)\) denote the \textit{homogeneity degree}, i.e. the cardinality of the set of orbits for the action of the group of homeomorphisms of \(X\) onto itself. In \textit{P. Pellicer-Covarrubias} [Topology Appl. 155, No. 15, 1650--1660 (2008; Zbl 1151.54026)], continua \(X\) for which \(hd(F_{2}(X))=2\) were studied. In general, the computation of \(hd(F_{n}(X))\) is a difficult task. In [\textit{I. Calderón} et al., Topology Appl. 221, 440--448 (2017; Zbl 1387.54008)], it was proved that if \(P\) denotes the pseudo-arc, then \(hd(F_{2}(P))=3\). In the paper under review, the authors completely determine \(F_{n}(X)\) when \(X\) is either a manifold without boundary or the unit interval \([0,1]\). They prove the following. THEOREM. If \(X\) is an \(m\)-manifold without boundary, then: (a) if \(m \geq 3\), then \(hd(F_{n}(X))=n\), (b) if \(m=2\) and \(n \neq 2\), then \(hd(F_{n}(X))=n\), (c) if \(m=2\) and \(n=2\), then \(hd(F_{2}(X))=1\), (d) if \(m=1\) and \(n \neq 3\), then \(hd(F_{n}(X))=n\), (e) if \(m=1\) and \(n=3\), then \(hd(F_{n}(X))=1\). THEOREM. For the unit interval we have: (a) if \(n \notin \{2,3\}\), then \(hd(F_{n}([0,1]))=2n\), (b) if \(n \in \{2,3\}\), then \(hd(F_{n}([0,1]))=2\).