Embeddability of multiple cones (Q929979)

Let us recall, since it is used in the proof, that \textit{W. S. Claytor} [Ann. Math. (2) 35, 809--835 (1934; Zbl 0010.27602)] has proved that a Peano continuum \(X\) is embeddable into \(S^2\) if and only if \(X\) does not contain any of the four Kuratowski curves \(K_i, i= 1, 2, 3, 4\). The paper contains their description. Next we notice that \textit{W. Rosicki} [Colloq. Math. 64, 141--147 (1993; Zbl 0857.54015)] proved the following: If a cone \(C(X)\) of a Peano continuum \(X\) embeds into \(\mathbb R^3\), then \(X\) embeds into \(S^2\). This paper generalizes this by iterating the cone operation of a Peano continuum \(X\), i.e. considering \(C^n(X) = C(\dots(C(X))\dots)\) and its embeddability into \(\mathbb R^{n+2}\). The obtained result says: If for some \(n\) the iterated cone \(C^n(X)\) embeds into \(\mathbb R^{n+2}\), then \(X\) is embeddable into \(S^2\). Proof is by showing that \(C^n(K_i)\), \(i=1,2,3,4\) is not embeddable into \(\mathbb R^{n+2}\). For two of Kuratowski's curves this follows from a result of \textit{R. Cauty} [Proc. Am. Math. Soc. 94, 516--522 (1985; Zbl 0566.54022)], while for the other two it is shown in the paper.