Talk:Cross-cap

message: :There is virtually nothing in the article (other than the lovely graphics) that is not false.  Starting with the misguided attempt to roughly define a cross-cap in the first sentence.
* First of all, a cross-cap is never topologically equivalent to a Moebius strip.  It is a continuous image by a certain type of map of the (closed) Moebius strip into 3-space, a map that has an open interval's worth of double-points.
* As a topological subspace of 3-space, it is the space obtained by starting with a closed disk D2,  choosing an interval in the disk's interior -- say the image of [-1,1] via an embedding h: [-1,1] → int(D2) -- and then identifying the points h(x) and h(-x) with each other for all x in (0,1].
* A cross-cap may have a boundary that is a round (perfect) circle, but is required only to have a boundary that is an unknotted simple closed curve.
* Further, there are continuous deformations of the usual picture of a Moebius band to a picture that is still topologically a Moebius band embedded in 3-space (i.e., with no self-intersections), such that its boundary is a perfect circle.  So this perfect-circle-boundary property in no way characterizes a cross-cap.
* The word cross-cap has been erroneously used to mean a cross-cap with a disk glued on to its boundary (making a continuous image of a projective plane).   But this is an error, and it should not be perpetuated in this article. Rather, the article should warn people to avoid this misuse of the word.Daqu 00:53, 4 December 2006 (UTC)
replies: text: Better late than never, I've now corrected many errors in the first paragraph.  No further errors caught my attention at this time.Daqu (talk) 06:19, 19 January 2013 (UTC)
thread_title: Accuracy of the article

message: The stated parametric equations do not give the shown image. The fraction 2/3 should be removed in the formula for Z[u,v].  —Preceding unsigned comment added by 81.210.243.30 (talk) 14:05, 24 June 2008 (UTC)
thread_title: Parametric Equations

message: Immersion means that something is locally an embedding. The pictures shown fail to be an immersion at the singular points (the image of any small neighborhood around these points in the domain will self-intersect). The cross-caps I usually saw in math books never were immersions. Furthermore, the picture shows a cross-cap (which is a map from the disk to the 3-sphere) with a disk glued on, so the label is misleading.
99.38.248.12 (talk) 23:59, 5 August 2010 (UTC)
thread_title: Cross-Cap not (necessarily) an Immersion

message: Now that the nature of the cross-capped disk has been exposed as a model of the projective plane, I think that section no longer belongs in this article.
For example the same surface may be obtained by joining the edge of a Möbius strip to itself. We all know that the projective plane may be divided into a Möbius strip and a disc. If we shrink the Möbius strip to a line, we get the model shown of a self-intersecting disc. But we can alternatively shrink the disc to a point (via the Jordan curve theorem), which is topologically equivalent to zipping the edge of the Möbius strip to itself.
All this has precious little to do with cross-caps, but much to do with the topology of the projective plane.
— Cheers, Steelpillow (Talk) 09:33, 21 April 2012 (UTC)
thread_title:  What is the projective plane doing here? 

message: So, what should this article discuss instead? Some suggestions:
On any non-orientable surface in R^3, a pair of cross-caps may be transformed into a handle, and vice versa. The Klein bottle is, or used to be, a minor exception, I'm not sure if it still is?
The cross-cap is a good example of a surface which is neither an embedding nor an immersion. I don't know if there is an appropriate term to use, beyond very general words like "injection" or "mapping".
While the cross-cap and Möbius strip are not homeomorphic, they have the same intrinsic topology (e.g. Euler characteristic and number of boundaries). Can the correct terminology for this be added?
[edit] although the Whitney umbrella is linked to, its relationship to the cross-cap could be made explicit.
— Cheers, Steelpillow (Talk) 09:44, 21 April 2012 (UTC)
thread_title:  What else to say? 

message: There is only one illustration with the article, and it is inexplicably not of a crosscap. Instead it is the image of a mapping of the projective plane into R3 that includes a crosscap.
A much better idea would be to include a picture of a crosscap, not of something else.65.157.43.50 (talk) 00:56, 15 August 2020 (UTC)
thread_title:  If there is only one illustration accompany this article, it should not be of something else!