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

The image of the branch set of a piecewise linear (PL)‐branched cover between PL 𝑛n‐manifolds is a simplicial (𝑛−2)(n−2)‐complex. We demonstrate that the reverse implication also holds: an open and discrete map 𝑓:𝕊𝑛→𝕊𝑛f:Sn→Sn with the image of the branch set contained in a simplicial (𝑛−2)(n−2)‐complex is equivalent up to homeomorphism to a PL‐branched cover. peerReviewed