In this talk I will discuss work in progress in which we classify topological 4-manifolds with boundary and fundamental group $\mathbb{Z}$, under some mild assumptions on the boundary. We apply this classification to provide an algebraic classification of surfaces in simply-connected 4-manifolds with $S^3$ boundary, where the fundamental group on the surface complement is $\mathbb{Z}$. We also compare these homeomorphism classifications with the smooth setting, showing for example that ever Hermitian form over $\mathbb{Z}[t^{\pm 1}]$ arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group $\mathbb{Z}$. This work is joint with Anthony Conway and Mark Powell.