  We study locally flat, compact, oriented surfaces in $4$-manifolds whose
exteriors have infinite cyclic fundamental group. We give algebraic topological
criteria for two such surfaces, with the same genus $g$, to be related by an
ambient homeomorphism, and further criteria that imply they are ambiently
isotopic. Along the way, we prove that certain pairs of topological
$4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries,
and equivalent equivariant intersection forms, are homeomorphic.
