# Which manifolds have a circle as their boundary?

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

Which other manifolds share that property?

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Doesn't the boundary of the Möbius strip have two components? – Mercy Jul 4 '12 at 17:56
@Mercy : No. Just one. – Michael Hardy Jul 4 '12 at 17:57
Would it just be every surface without a boundary, since you could just puncture it? – Michael Hardy Jul 4 '12 at 17:58
Yes I see, but it seems it isn't really a circle, meaning that it's not contained in a plane. – Mercy Jul 4 '12 at 18:01
@Mercy : It's topologically a circle. That's what I had in mind. I.e. it's homeomorphic to the ordinary circle in the plane. – Michael Hardy Jul 4 '12 at 18:03
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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.
In this case, the different ways to attach a disk ($2$-handle) to a surface with boundary $S^1$ are in one-to-one correspondence with elements of $\pi_1(\mathrm{O}(0)) = 0$. See the following answer regarding the relevant part of Gompf and Stipsicz on this topic: math.stackexchange.com/a/131947/24934 – Henry T. Horton Jul 4 '12 at 23:58