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I understand how the mobius band and the real projective plane are different (the first is a manifold with boundary, for example, while the second is a compact, closed surface and a true manifold).

1. Is a cross-cap the name for spaces which are homeomorphic to the mobius band and which are sewn into holes cut from surfaces in order to increase their non-orientable genus? (The orientable counterpart being handle attachment -- the mobius band, in contrast, has a "twist".)

2. Or is the cross-cap what results from the connected sum of a disk and a mobius band, i.e. the real projective plane?

Core Question:

I.e. (up to homeomorphism) is a cross-cap 1. a mobius band or 2. a real projective plane? Or something else entirely?

The terminology does not seem to be consistent between sources, for instance Wikipedia https://en.wikipedia.org/wiki/Cross-cap, or the various mentions of the term "cross-cap" in Gross, Tucker, Topological Graph Theory.

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You can search on the web for images of the cross-cap; it is a model in 3 dimenions, with self intersections, of the projective plane.

The projective plane can also be seen as a Mobius band with a disc glued onto the boundary; but this cannot be dome in 3 dimensions. A representation of this gluing and its relation to rotations in 3 dimensions is shown in part of this presentation Out of Line.

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  • $\begingroup$ So when people refer to doing surgery with cross caps, do they mean removing a disk from them and then attaching the resulting mobius band? $\endgroup$ Commented Jul 31, 2016 at 16:05

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