Continuous maps with the disjoint support property

Abstract

It is well known that for each $n\ge 0$, there is a continuous map $f :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$ with the disjoint support property. Since $S^{n}$ and $\partial \bigtriangleup ^{n+1}$ are homeomorphic, it is natural to ask whether or not there is a homeomorphism $h :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$ with the disjoint support property. In this paper, we prove that there is no such homeomorphism. Further, we also prove the following. Let $K_{n}$ be any triangulation of $S^{2}$ having n faces, $n\ge 4$. We prove that for triangulation $K_{6}$ of $S^{2}$ having 6 faces, there exists a continuous map $f :S^{2}\rightarrow \vert K_{6}\vert$ 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 $n\ge 8$, we prove that $S^{2}$ has at least two non-isomorphic triangulations $L_{n}$ and $K_{n}$ where the first one admits a continuous map $f :S^{2}\rightarrow \vert L_{n}\vert$ with the disjoint support property and no homeomorphism with the disjoint support property. For $K_{n}$, we prove that there is a homeomorphism $h :S^{2}\rightarrow \vert K_{n}\vert$ with the disjoint support property.