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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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    Do you mean cross-cap as a subspace of an abstract 2-manifold (a Möbius band), or do you mean cross-cap as a geometric feature of a self-intersecting surface in 3-space, or do you mean something else?2012-05-04
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    As a subspace of a Mobius band I think. It states in my notes that $RP^2$ minus a disc is homeomorphic to a cross cap, and I was looking for an explicit definition (perhaps that is it...?)!2012-05-04
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    Then "cross-cap" is just a synonym for "Möbius band". Both are homeomorphic to $RP^2$ minus a disk.2012-05-04
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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.

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    Is there a topological, rather than geometrical construction that one can do... I can't get my head around the geometrical one! Thanks!2012-05-04
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    No, sorry. Look at the variety of images on 2012-05-05