Which manifolds have a circle as their boundary?

Question

The boundary of a disk or of a Möbius band is a circle.

Which other manifolds share that property?

Answer 1

For the compact case, I believe the answer is, as you said in the comments above, any closed surface with a single puncture, i.e. a disk removed. I claim that this completely classifies compact surfaces with boundary $S^1$. This is because you can glue a disk to the surface along its boundary to obtain a closed surface, and there is a unique way to do this (see, for example, Example 4.1.4(c) in Gompf and Stipsicz's 4-Manifolds and Kirby Calculus). So by the classification of surfaces, there should then be a unique surface with boundary $S^1$ corresponding to each closed surface.