Monodromy representations of completed coverings

TL;DR

This paper studies completed branched coverings between PL manifolds, characterizing when monodromy representations exist and their properties, including discreteness and the nature of the branch set.

Contribution

It provides a characterization of the existence of monodromy representations for completed coverings and links this to the discreteness of completed normal coverings.

Findings

Monodromy representations exist under certain conditions.

Completed coverings with monodromy are discrete.

The image of the branch set is closed.

Abstract

In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterization for the discreteness of a completed normal covering. We also show that completed coverings admitting a monodromy representations are discrete and that the image of the branch set is closed.