Topological types of some symmetrical products (Q1385931)

Let \(C(X)\) be a cone and \(S(X)\) be the suspension over \(X\). In [Fundam. Math. 63, 77-88 (1968; Zbl 0172.48003)] \textit{R. M. Schori} extended the results of \textit{K. Borsuk} and \textit{S. Ulam} [Bull. Am. Math. Soc. 37, 875-882 (1931; Zbl 0003.22402)] and of \textit{R. Molski} [Fundam. Math. 44, 165-170 (1957; Zbl 0078.15204)] by proving that \(C(C(X))\approx C(X)\times I\) for any \(X\) \((I= [0,1])\). Using this general relation, he proved that \(I^n(2) \approx C(\mathbb{R}\text{P}^{n-1}) \times I^n\) and \(I(n) \approx C(D^{n-2})\times I\), where \(D^{n-2}\) comprises all elements of \(I^n\) that contain the endpoints 0 and 1 of the segment. In [\textit{R. N. Andersen, M. M. Marjanović} and \textit{R. M. Schori} , Topol. Proc. 18, 7-17 (1993; Zbl 0854.54014)] the spaces \(D^k\) were described as polyhedrons that have the homotopic type of the \(k\)-sphere for even \(k\) and are contractable but non-collapsable for odd \(k\). In this work, the authors prove that \(C(S(X))= S(C(X)) \approx C(X)\times I\) for each space \(X\). They also give a new quotient representation of the iterated suspension \(S^n(X)\) and identify the space of all \(k\)-dimensional balls contained in the \(n\)-dimensional unit ball \(B^n\) with \(C(G_{n,k})\times I^n\) (in particular, for \(k=1\) they give another proof of the equality \(I^n(2) \approx B^n(2) \approx C(\mathbb{R} \text{P}^{n-1})\times I^n)\).