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I have looked this up in several places but can only find heuristic descriptions. What I'm after is a definition like

If X has certain properties, the cross cap on (of?) X is the quotient of Y with Z...

or something similarly precise.

Many thanks in advance.

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    Thanks - I understand now!2012-05-12

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Generally, a cross-cap refers to the embedding of a Möbius band into 4-space created via a movie with the birth of a circle, a loop in the circle made by a type I move, a handle attached to sort-of turn the figure-8 inside out, and then removed the loop. This can be capped-off by a disk to embed the projective plane into 4-space. It can be drawn in a snap.