We classify topological 4 -manifolds with boundary and fundamental group \mathbb{Z} , under some assumptions on the boundary. We apply this to classify surfaces in simply-connected 4 -manifolds with S^{3} boundary, where the fundamental group of the surface complement is \mathbb{Z} . We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every 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} . For surfaces we have a similar result, and in particular we show that every 2 -handlebody with S^{3} boundary contains a pair of exotic discs.
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