Spheres, symmetric products, and quotient of hyperspaces of continua (Q407374)

For a continuum (i.e., a nonempty compact connected metric space) \(X\) and a positive integer \(n\), let \(F_n (X)\) denote the \(n\)-th symmetric product of \(X\), that is, the hyperspace of all nonempty subsets of \(X\) with at most \(n\) points, endowed with the Vietoris topology. \textit{R. Bott} [Fundam.\ Math.\ 39, 264--268 (1953; Zbl 0050.17801)] proved the third symmetric product \(F_3 (S^1)\) of the circle \(S^1\) is homeomorphic to the 3-sphere \(S^3\). In this paper, the authors prove that, for a continuum \(X\), \(F_n(X)\) is homeomorphic to the \(m\)-sphere for some \(m\leq n\) if and only if either \(n=3\) or \(n=1\), and \(X =S^1\). The authors also study continua \(X\) such that \(F_n(X)/F_1(X)\) is homeomorphic to \(C_1(X)/F_1(X)\), where \(C_1(X)\) is the hyperspace of subcontinua of \(X\) with the Vietoris topology, and \(F_n(X)/F_1(X)\) and \(C_1(X)/F_1(X)\) are quotient spaces obtained by shrinking \(F_1(X)\) to a point in \(F_n(X)\) and \(C_1(X)\), respectively. They prove the following two theorems: For a finite-dimensional arcwise connected continuum \(X\), \(F_2(X)/F_1(X)\) is homeomorphic to \(C_1(X)/F_1(X)\) if and only if \(X\) is homeomorphic to the unit interval \([0,1]\). If \(Y\) is an arcwise connected continuum and \(n \geq 3\), then \(C_1(Y)/F_1(Y)\) is not homeomorphic to \(F_n(X)/F_1(X)\) for every finite-dimensional continuum \(X\).