Simple closed curves in stable covers of surfaces

Abstract

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