Open and Discrete Maps with Piecewise Linear Branch Set Images are Piecewise Linear Maps

Abstract

The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds: an open and discrete map $f:{\mathbb{S}}^n\to {\mathbb{S}}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent up to homeomorphism to a PL branched cover.