Talk:Manifold

other old discussions: Talk:manifold/old, Talk:manifold/rewrite/freezer.

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Proposed changes

To start, I believe the four paragraphs of Motivational should be elsewhere. I propose the following:

The first should be worked into section 5, regarding construction.

The second should go in section 3.

The third may be addressed in either the intro, or some sort of prerequisite mathematics entry;

The fourth should either be in the introduction, when the concept's importance is mentioned, or in a new section towards the end, in an "applications" entry;

I will begin drafting these changes. If there are objections or better suggestions, let's discuss here. Horsesizedduck (talk) 22:46, 11 June 2021 (UTC)

Please be more accurate. Which are the "Motivational" paragraphs you are talking of? The paragraphs of the lead, or the subsections of the section "Motivating example"? In any case, the paragraphs of the lead must be keep, even if some could be (and must be) shortened and made less technical. In particular, "embedded", "immersed", and "compactness" do not belong to the lead and the example of the sphere should be either replaced by that of the circle, and/or the proof of non-homeomorphism must be skipped. D.Lazard (talk) 07:31, 12 June 2021 (UTC)

I apologize, I meant the "motivational examples" section, right before the circle is introduced. Again, I'm sorry. I agree that the lead's second paragraph should not use the terms "immersed", "compactness"; the sphere should also not be appearing in the motivational examples. Horsesizedduck (talk) 11:36, 12 June 2021 (UTC)

I'll leave a link to what (I think) was a better version of this page. https://wiki.riteme.site/w/index.php?title=Manifold&oldid=111890033 This might be useful for comparison. Horsesizedduck (talk) 11:40, 12 June 2021 (UTC)

Please explain/correct/delete claim that the non-negative x-axis, in 2D, "special" under polar coordinates?

The assertion is made that the origin and positive x-axis are special under polar coordinates.

This is bizarre and needs explanation or correction (perhaps via deletion). Is the word "minus" being used in some unclear fashion?

Charts, atlases, and transition maps

. . .

Charts

. . .

In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R 2 {\displaystyle \mathbb {R} ^{2}} \R^2 minus the positive x-axis and the origin. Another example of a chart is the map χtop mentioned above, a chart for the circle.

2601:1C1:C100:F420:1CAC:3F44:5927:BCE (talk) 02:17, 23 July 2021 (UTC) Just another drive by.

Wiki-link to Boundary (topology)

In the section "Manifold with boundary", there is an (as i think) misleading link: It reads "(See also Boundary (topology))." In that Article, the set-theoretic topology notion of "boundary" is explained, i.e. for a subset S of a topological space X, the set-theoretic boundary of S is the intersection of the closure of S and the closure of X\S. This is totally misleading here, at most we could say "(Do not confuse with Boundary (topology))." Any comments? Thanks in advance. --Himbeerbläuling (talk) 12:54, 26 March 2022 (UTC)

Lead picture?

The lead picture of the projective plane seems uniquely bad to me. The self-intersections in the picture give the wrong impression- the thing in the picture, interpreted at a glance, is not a manifold. Reading the caption we see that OK, yes, it really is a manifold, but it happens to be one for which no picture in R³ can be quite accurate. So why is this the lead-off example? The concept of "manifold" is not intuitively complicated, so we don't need a picture that makes it seem wild.

I think any of the pictures currently in Surface (topology) could work. Maybe there are others that would be better. Staecker (talk) 18:13, 10 April 2023 (UTC)

I see that User:D.Lazard swapped out the immersed projective plane for an immersed Klein bottle. That misses the point. It is the immersion that makes these examples confusing. Why not a nice double torus or something like that, for which an embedding is possible? That way it would be more obvious from the image that all points have disk neighborhoods. In the immersed images, it looks like the points of self-intersection have non-disk neighborhoods — they do have non-disk neighborhoods in the image of the immersion, and the preimage, where their neighborhoods are disks, is not actually shown. —David Eppstein (talk) 20:03, 10 April 2023 (UTC)

I agree. Moreover, only expert can easily understand the relationship between the image and the projective plane. So, I have exchanged this image with that of the Klein bottle. D.Lazard (talk) D.Lazard (talk) 20:05, 10 April 2023 (UTC)

See above comment. I think your change made zero difference to the actual problem identified in the original comment. —David Eppstein (talk) 20:07, 10 April 2023 (UTC)

Sure, but I did not have the time to search in Commons. My edit was only an understandability improvement before a better choice resulting from a consensus. Also, I have not removed the image of the Boy's surface, although I think it is not a good choice for illustrating the projective plane D.Lazard (talk) 07:11, 11 April 2023 (UTC)

Merge proposal

@David Eppstein, D.Lazard, and Horsesizedduck: I propose merging Topological manifold into Manifold. These two articles appear to be about exactly the same mathematical concept. Mathwriter2718 (talk) 18:41, 28 June 2024 (UTC)

They are not about the same concept, any more than, say, natural number, integer, rational number, and real number are about the same concept.

The general concept is manifold. It has several widely-studied and more-specific variations, just like the general concept of a number has variations such as the ones listed above. These include topological manifold, differentiable manifold, analytic manifold (not the same as differentiable despite the current merge tags there), piecewise linear manifold, and Riemannian manifold. Those are all distinct topics and we should neither confuse readers by mixing them up into one article nor pick a winner by saying that one of them (topological manifold, say) is somehow the main concept that others are subsidiary to.

Many of the others can be interpreted, by ignoring some of their structure, as special cases of topological manifolds, but that is no different in principle than saying that integers are special cases of real numbers; it does not mean that real numbers are the main concept and it does not mean that number should be merged with real number. —David Eppstein (talk) 18:49, 28 June 2024 (UTC)

I agree with you, I should have read this page more closely before putting in this proposal. Closing. Mathwriter2718 (talk) 18:59, 28 June 2024 (UTC)

Looking at the given formal definitions, a manifold is a topological manifold, and a topological manifold is a manifold if and only if it is second countable. None of the two articles discusses non-second-countable manifolds, and Topological manifold does not link to Non-Hausdorff manifold.

So, I strongly support the merge. At first glance, for realizing this merger, it suffices to redirect Topological manifold to Manifold, and to add to the latter a sentence saying that "topological manifold" is often used instead of "manifold" for emphasizing that no further structure is considered. D.Lazard (talk) 10:50, 29 June 2024 (UTC)