Branched Covers of Open Manifolds

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.

arXiv commentMinor changes and clarifications. Corollary 2.2 is also strengthened to include even dimensions. This version includes 21 pages, 12 images, 1 footnote, 1 graphic