More on induced maps on \(n\)-fold symmetric product suspensions (Q2787001)

The present paper is a continuation of the paper [the first author, Topology Appl. 158, No. 10, 1192--1205 (2011; Zbl 1223.54012)]. Let \(n \geq 2\) and let \(X\) be a continuum. The \textit{\(n\)-fold symmetric product suspension} of \(X\), \({\mathcal S}{\mathcal F}_n (X)\), is the quotient space \({\mathcal F}_n (X) / {\mathcal F}_1 (X)\) with the quotient topology. The \textit{quotient map} is denoted by \(q_X^n : {\mathcal F}_n (X) \twoheadrightarrow {\mathcal S}{\mathcal F}_n (X)\) and \(q_X^n({\mathcal F}_1 (X))\) is denoted by \(F_X ^n\). Let \(n \geq 2\) and let \(f: X \to Y\) be a function between two continua \(X\) and \(Y\). The functions \({\mathcal F}_n (f)\) and \({\mathcal S}{\mathcal F}_n (f)\) are defined as follows. {\parindent=0.6cm\begin{itemize}\item[--] \({\mathcal F}_n (f) : {\mathcal F}_n(X) \to {\mathcal F}_n(Y)\), \({\mathcal F}_n (f) (A) = f(A)\) for each \(A \in {\mathcal F}_n(X)\); this function is called the induced map of \(f\) on \(n\)-fold symmetric products of \(X\) and \(Y\). \item[--] \({\mathcal S}{\mathcal F}_n (f): {\mathcal S}{\mathcal F}_n (X) \to {\mathcal S}{\mathcal F}_n (Y)\), NEWLINE\[NEWLINE {\mathcal S}{\mathcal F}_n (f)(\chi)= \begin{cases} q_Y^n({\mathcal F}_n(f))(( q_X^n )^{-1}(\chi)), & \text{ if } \chi \neq F_X ^n \\ F_y^n, & \text{ if } \chi = F_X ^n \end{cases}NEWLINE\]NEWLINE NEWLINENEWLINE\end{itemize}} Both functions \({\mathcal F}_n (f)\) and \({\mathcal S}{\mathcal F}_n (f)\) are continuous (see [\textit{S. Macías}, Topics on continua. Boca Raton, FL: Chapman \& Hall/CRC (2005; Zbl 1081.54002)] and [\textit{J. Dugundji}, Topology. 8th printing. Boston: Allyn and Bacon, Inc (1973; Zbl 0397.54003)], respectively). Let \({\mathcal M}\) be one of the following classes of maps between continua: almost monotone, atriodic, freely decomposable, joining, monotonically refinable, refinable, semi-confluent, semi-open, simple and strongly freely decomposable maps. The authors examine the relationship between the following statements: {\parindent=0.6cm\begin{itemize}\item[--] \(f \in {\mathcal M}\); \item[--] \({\mathcal F}_n (f) \in {\mathcal M}\); \item[--] \({\mathcal S}{\mathcal F}_n (f) \in {\mathcal M}\). NEWLINENEWLINE\end{itemize}} Note that all these classes of maps are not included in Barragán's paper and all their definitions can be found in the present paper. Also, in the present paper the reader can find many open questions and in this aspect the article could be very interesting for further research.