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

TL;DR

This paper proves that open, discrete maps with branch set images contained in a simplicial complex are equivalent to PL branched covers, establishing a converse to a known property of PL branched covers.

Contribution

It demonstrates that the reverse implication of the branch set image property characterizes certain open, discrete maps as PL branched covers.

Findings

Open and discrete maps with branch set images in simplicial complexes are PL branched covers.

The result provides a converse to the known characterization of PL branched covers.

Establishes a topological equivalence under homeomorphism for these 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 $f \colon \mathbb{S}^n \to \mathbb{S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent up to homeomorphism to a PL branched cover.