Simple closed curves in stable covers of surfaces

TL;DR

This paper characterizes when homology classes in stable covers of surfaces can be represented by connected simple closed curves, extending classical results to covers with possibly infinite deck groups.

Contribution

It provides necessary and sufficient conditions for representing homology classes as simple closed curves in stable covers of surfaces, including stabilization techniques.

Findings

Criteria for representing homology classes as simple closed curves

Extension of classical surface topology results to infinite covers

Use of stabilization to achieve desired curve representations

Abstract

Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on $Y$, possibly after passing to a "stabilization", i.e. a $G$-equivariant embedding of covering spaces $X \hookrightarrow X^+$.