Branched Covers of Open Manifolds

TL;DR

This paper proves that for dimensions 2 and 3, any connected open manifold can be covered by a simple branched cover over Euclidean space, with specific degrees depending on the number of ends, and explores universal base manifolds.

Contribution

It establishes the existence of simple branched covers for open manifolds in dimensions 2 and 3, including bounds on degrees based on the number of ends, and studies universal base manifolds.

Findings

Open 2- and 3-manifolds admit simple branched covers over Euclidean space.

The degree of the cover can be bounded by the number of ends for finite ends.

Existence of countably infinite degree covers for all open manifolds regardless of ends.

Abstract

For $m=2$ and $m=3$ we prove that any connected, oriented, open manifold $M^m$ admits a simple branched covering map over $\mathbb{R}^m$. When $M$ has $k$ ends and $k$ is finite, the degree of the cover can be taken to be $mk$. Regardless of the number of ends, $M$ admits a branched covering map of countably infinite degree over $\mathbb{R}^m$. We also investigate which compact manifolds are universal bases, that is, are branch covered by all compact manifolds in the same dimension.