Continuous maps with the disjoint support property (Q2043724)

It is well known that for each \(n \geq 0\), there is a continuous map \(f : S^n \to \partial \Delta^{n+1}\) with the disjoint support property. Since \(S^n\) and \(\partial \Delta^{n+1}\) are homeomorphic, it is natural to ask whether or not there is a homeomorphism \(h : S^n \to \partial \Delta^{n+1}\) with the disjoint support property. The authors prove that there is no such homeomorphism. Further, they also prove the following. \par Let \(K_n\) be any triangulation of \(S^2\) having \(n\) faces, \(n\geq 4\). They prove that for triangulation \(K_6\) of \(S^2\) having \(6\) faces, there exists a continuous map \(f : S^2 \to \vert K_6\vert \) having the disjoint support property and there is no homeomorphism from \(S^2\) to \(\vert K_6\vert \) having the disjoint support property. For each \(n \geq 8\), they prove that \(S^2\) has at least two non-isomorphic triangulations \(L_n\) and \(K_n\) where the first one admits a continuous map \(f : S^2 \to \vert L_n\vert \) with the disjoint support property and no homeomorphism with the disjoint support property. For \(K_n\), they prove that there is a homeomorphism \(h : S^2 \to \vert K_n\vert \) with the disjoint support property.