Simple closed curves in stable covers of surfaces

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 $X\hookrightarrow X^{+}$.

27 pages, 5 figures. Comments welcome!