Induced mappings on symmetric products, some answers (Q1637163)

Let \(X\) be a nondegenerate metrizable continuum. Certain hyperspaces of \(X\) (with the Vietoris topology) are explored, in particular, \(F_n(X)\), the collection of nonempty subsets of \(X\) of cardinality \(\leq n\). The quotient space \(SF_n(X)=F_n(X)/F_1(X)\), \(n\geq 2\), comes into play in this study. In case \(f:X\to Y\) is a map of such continua, there are induced maps \(f_n:F_n(X)\to F_n(Y)\) and \(Sf_n:SF_n(X)\to SF_n(Y)\). The authors are interested in certain classes \(\mathcal{M}\) of maps and the relations among the statements \(f\in\mathcal{M}\), \(f_n\in\mathcal{M}\), and \(Sf_n\in\mathcal{M}\). Four classes \(\mathcal{M}\) of maps are considered: almost monotone, atriodic, freely decomposable, and joining. To give some flavor to the results, recall that a surjective map \(f:X\to Y\) of continua is called almost monotone if for every subcontinuum \(B\) of \(Y\) that has nonempty interior in \(Y\), \(f^{-1}(B)\) is connected. Consider the statements: \((1)\) \(f\) is almost monotone. \((2)\) \(f_n\) is almost monotone. \((3)\) \(Sf_n\) is almost monotone. The authors point out that \((2)\Rightarrow(1)\) and \((2)\Rightarrow(3)\) have previously been established. In Section 2, they show that \((1)\) does not imply \((3)\), but that in case \(n=2\), \((3)\Rightarrow(1)\). Let us just mention that the last three sections deal respectively with atriodic, freely decomposable, and joining maps.