Covers of surfaces

TL;DR

This paper investigates the types of covers of orientable surfaces, revealing universal properties, classifying certain covers, and identifying possible homeomorphism types of characteristic covers, especially for infinite-type surfaces.

Contribution

It provides a comprehensive classification of covers of orientable surfaces, including universal, abelian, and characteristic covers, with new results on their homeomorphism types.

Findings

Every non-abelian surface covers every noncompact surface.

Identified universal abelian and $ ext{Z}/n ext{Z}$-homology covers.

Non-locally finite characteristic covers have four homeomorphism types.

Abstract

We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.