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The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together.

When defining the connected sum of surfaces with boundary, is the boundary of each surface allowed to touch its removed disk?

By my intuition it seems like a bad idea to allow that, because that might lead to "not nice" behaviours. But I can't really think of anything in the formal definition that could forbid it - removing an open disk doesn't prevent the disk from touching the surface boundary.

Herng Yi
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    Either the disks have to be included in the boundary, or not touching it. Have a look, for example, at the glossary at the beginning of "lectures on the topology of 3-manifolds", N. Saveliev. – Léo May 03 '14 at 13:47
  • The glossary says that the disk is to be removed from the interior for the usual connected sum (not boundary connected sum). But are the removed disks closed or open? If they are closed then I can understand how removing them from the interior would prevent the disk from touching the boundary, but the definitions I've seen use the removal of open balls (a disk is a 2-dimensional ball). – Herng Yi May 03 '14 at 14:13
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    I don't have the reference in front of me, but what it should say is that the open disk to be removed is the interior of a closed disc which is disjoint from the boundary. – Lee Mosher May 03 '14 at 15:03
  • @LeeMosher ah yes that would clear everything up. It would be great if you could find the reference though. – Herng Yi May 04 '14 at 01:24

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