# On proper branched coverings and a question of Vuorinen

## TL;DR

This paper investigates when proper branched coverings in Euclidean space are homeomorphisms, confirming Vuorinen's question for specific cases involving the branch set and dimension.

## Contribution

It proves that proper branched coverings with compact branch sets are homeomorphisms in three dimensions or when the branch set is empty, advancing understanding of their injectivity.

## Key findings

- Mappings are homeomorphisms in 3D when branch set is empty.
- Mappings are homeomorphisms in all cases when the dimension is 3.
- Confirmed Vuorinen's question for these specific cases.

## Abstract

We study global injectivity of proper branched coverings defined on the Euclidean $n$-ball in the case when the branch set is compact. In particular we show that such mappings are homeomorphisms when $n=3$ or when the branch set is empty. This proves the corresponding cases of a question of Vuorinen from [Vuo79].