Talk:Manifold/Archive 4

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Failed Featured Article

Well, manifold is no longer a featured article candidate. It failed with 7 votes in oppositions and no votes in support. Archive of old debate is Wikipedia:Featured article candidates/Manifold/archive1. The objections include

• informal and unencyclopedic tone
• uses first and second persons, diagrams inadequately explained
• a lot of complex mathematical terminology is not even linked to, never mind explained.
• Inline citations

Not all bad though

• The information seems to be excellent, very probably by far the best description available anywhere on the web

so we are making progress, and do have one of the best presentations. Some other critiques, as well from people who have contributed to this Talk page.

I've now created a second archive Manifold/old2 which is an archive of the version of 14:31, 1 February 2006 by Bobblewik which is the last semi-mature version (with dates de wikified) i.e. this version [1].

We now need to reconsider if we wish to work towards another version for featured article status. Its worth reviewing the criteria for featured article:

---

A featured article should have the following attributes:

1. It should exemplify our very best work, representing Wikipedia's unique qualities on the Internet.
2. It should be well written, comprehensive, factually accurate, neutral, and stable. Read Great writing and The perfect article to see how high the standards are set. In this respect:
   • (a) "well written" means that the prose is compelling, even brilliant;
   • (b) "comprehensive" means that an article covers the topic in its entirety, and does not neglect any major facts or details;
   • (c) "factually accurate" includes the supporting of facts with specific evidence and external citations (see Wikipedia:Verifiability); these include a "References" section where the references are set out, enhanced by the appropriate use of inline citations (see Wikipedia:Cite sources);
   • (d) "neutral" means that an article is uncontroversial in its neutrality and factual accuracy (see Wikipedia:Neutral point of view); and
   • (e) "stable" means that an article does not change significantly from day to day (apart from improvements in response to reviewers' comments) and is not the subject of ongoing edit wars;
3. It should comply with the standards set out in the style manual and relevant WikiProjects. These include having:
   • (a) a concise lead section that summarizes the entire topic and prepares the reader for the higher level of detail in the subsequent sections;
   • (b) a proper system of hierarchical headings; and
   • (c) a substantial, but not overwhelmingly large, table of contents (see Wikipedia:Section).
4. It should have images where appropriate, with succinct captions and acceptable copyright status; however, including images is not a prerequisite for a featured article.
5. It should be of appropriate length, staying tightly focused on the main topic without going into unnecessary detail; it should use summary style to cover sub-topics that are treated in greater detail in any "daughter" articles.

---

As a sugestion I think we should look at the other mathematics articles which have featured article status:

Ackermann function, Algorithm, Eigenvalue, eigenvector and eigenspace, Carl Friedrich Gauss, Game theory, Infinite monkey theorem, Margin of error, Monty Hall problem, Blaise Pascal, Prisoner's dilemma, Regular polytope, Marian Rejewski, Trigonometric function.

Personally I quite like the Regular polytope one, good history, good visuals, good applications. --Salix alba (talk) 22:46, 2 February 2006 (UTC)

Comment for Pfafrich

Thanks for catching the indentations; they were accidental, caused by moving material I had put on the talk page several days ago into the article.

I agree that the section on zeroes of a function needed to be rewritten, but I think the concept too important to eliminate entirely. We also need more on analytic continuation. Rick Norwood 23:08, 2 February 2006 (UTC)

• Zeros of functions work well for the sphere, plane, and a few other simple shapes. They are very tricky for non orientable examples, have a hideous eqn for a torus, frequently give rise to non-manifold topology, don't generalise nicely to the more exotic manifold. In general I've found it easier to explain parameterized surfaces (which nicely generalise from graphs of one variable) than algebraic surfaces. I find very little to support this approach. (p.s. just had a nice though for examples: the lie groups and algebras, Special Unitary Group and so on are quite a nice example of how the concept of manifold has application to other fields). --Salix alba (talk) 00:27, 3 February 2006 (UTC)

references

I've added a number of references.

I note that one of the negative votes re: featured article status said that mathematical words were used without references. I'll check the article for that. Rick Norwood 23:23, 2 February 2006 (UTC)

History section

The history section looks good to me. Any idea what more the person wants, who asked for an exansion? I've removed the statement of the Poincarė conjecture as too technical, but left the link and the comment.

I do not understand how the last clause of the last sentence in the history section fits with what the rest of that sentence. The clause in question reads:

"and developed through differential geometry and Lie group theory."

I don't know enough about Lie groups to fix it. I hope someone will tackle that job. Rick Norwood 23:37, 2 February 2006 (UTC)

What's the problem? "The foundational aspects were clarified … and developed." Perhaps you want "further developed"? -- Jitse Niesen (talk) 00:01, 3 February 2006 (UTC)

Here is the sentence:

"The foundational aspects of the subject were clarified during the 1930s by Hassler Whitney and others, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory."

Ah, now I see. I read it as Whitney making intuitions -- what kind of intuitions? -- precise intuitions. When really it says Whitney was making things precise -- making what precise? -- intuitions. Rick Norwood 02:36, 3 February 2006 (UTC)

PS: Good edits, Gauge. Rick Norwood 02:38, 3 February 2006 (UTC)

Well, people did Riemannian geometry without an explicit chart concept; though those guys of course were perfectly adept at changing variables, as a local thing. Lie groups are a more interesting case, because the evidence is that Hadamard didn't really grok the global concept (and he was no slouch at all). This comes up, for example, in thinking about universal covers, where one chart at the origin is not enough. But I think there is a kind of clincher in black hole theory. Where early examples of models of general relativity had singularities, but these were not really noticed initially. Of course, they were noticed, as places at which the functions blew up in some sense. What you need is a concept like lines of longitude at the North Pole, which is one kind of singularity on a perfectly good manifold because the chart (say Mercator projection type) cannot be forced to cover a point; versus real breakdown of manifold structure. So I say around 1920 the chart-style definition was still not formalised enough. Whitney wrote down certain things in order to prove his embedding theorem. Also, around that time, the partition of unity concept was showm to apply to manifolds (paracompact or something). Charles Matthews 10:12, 3 February 2006 (UTC)

Sounds like good material to add to the history section. Rick Norwood 13:31, 3 February 2006 (UTC)

Loom91

You have made your views known. They were not supported. A majority of those who commented on the application for Featured Article status complained that the article was incomprehensible. The introduction you want is even more incomprehensible. You suggest that your intro can be read by someone with an American high school education or the equivalent (an odd choice of prerequisites). I am quite sure your introduction cannot be read by anyone who does not already know what a manifold is. Rick Norwood 14:06, 3 February 2006 (UTC)

I'm quite aware that my introduction is badly worded, but yours is more so. We need something better than either of us has written, and such has not yet been forthcoming. I do not believe the article lead should be a list of examples of different dimensional manifolds, there is much scope for that in the examples section. As for my example being incomprehensible to high school students, I can't see how that can be as I myself understand it quite well without passing high school. Loom91 12:00, 4 February 2006 (UTC)

That may be strongly worded. But I strongly object to any box of prerequisites slapped on the top of the article or anywhere else. Appropriate wikilinks to related topics as the discussion moves along are more than enough to get the reader an idea of the background necessary for this article. Oleg Alexandrov (talk) 17:31, 4 February 2006 (UTC)

Then your objection should be made at the talk page of Wikipedia:Make technical articles accessible and not here. Perhaps people will be happier if the box was called recommended prerequisites? I should also make it clear that the box refers to the article in its currnt state rather than some future article understandable without a knowledge of precalculus, if such a thing is possible. Loom91 12:00, 4 February 2006 (UTC)

I raised the issue at Wikipedia talk:Make technical articles accessible. Oleg Alexandrov (talk) 17:39, 4 February 2006 (UTC)

About the intro being incomprihensible to high school students. Every effort should be taken to make things more acessible. But please do not push that too far. Manifolds are studied seriously only in grad school, and there are good reasons for that. Oleg Alexandrov (talk) 19:18, 3 February 2006 (UTC)

Understanding begins with examples. We learn by proceeding from the particular to the abstract. A paper in a professional mathematics journal, only read by mathematicians, can begin with the abstract because mathematicians are trained to read such papers. But Wikipedia should be useful.

Let me tell you about an experience I had yesterday. As I was walking across campus, a student struck up a conversation, and mentioned that they had flunked professor X's class, but made an A when they retook the same course under professor Y. Her understanding was that "Professor X is just too smart to teach to someone like me." I don't think her analysis of the problem is correct, but my response was, "Well, if you ever need any help, come see me. I'm not nearly as smart as Professor X."

The purpose of writing, like the purpose of teaching, is communication. Rick Norwood 14:46, 4 February 2006 (UTC)

Pfafrich's edit

I find Pfafrich's edit an acceptable compromise, modulo a few very minor changes. I hope Loom91 will accept this. Rick Norwood 15:05, 4 February 2006 (UTC)

It seems acceptable to me. I cleaned up the wording a bit to make the lead clearer (I hope), and fixed the weird spacing anomolies caused by the inline comments. In my opinion, making an article "accessible" does not imply a need to display explicit prerequisites. Pfafrich's edit is good in this regard. - Gauge 00:21, 5 February 2006 (UTC)

Intro use of "space"

Once again I must point out that the word "space" is a technical term for mathematicians, leading to misinterpretation by everyone else. But it is clear that all the careful debate and writing of the past is now to be discarded. And so the traditional parents' curse will befall the present editors: may you have children like yourselves. Or in this case, you can watch as the next wave of editors "improves" your version. This is not the way to improve Wikipedia. --KSmrq^T 01:22, 5 February 2006 (UTC)

Excessive subtext

• JA: The number of hidden comment lines in the main article is getting ridiculous. Could editors please try to carry out their inner polylogues on the discussion page? Gratia in futuro, Jon Awbrey 04:06, 5 February 2006 (UTC)

Good point. I'll fix it. Rick Norwood 15:49, 5 February 2006 (UTC)

Precalculus

Of course Precalculus is a part of Calculus, my point was that you don't need to know proper calculus to understand the beginning motivational example section, but you DO need Precalculus. I doubt many of those who studied elementary real analysis in high school retained memories of the wild logical structures after they passed their exams. What I mean to say that if you don't know calculus, don't worry you will get an elementary idea of what we are dealing with. But if you don't know Precalculus (functions, sets etc) then sorry, but you don't stand a chance of getting past the introduction. What're your objections? Loom91 08:29, 6 February 2006 (UTC)

Precalc doesn't comprise a body of knowledge, but rather a level. You say (functions, sets), but functions are studied in analysis, and sets are studied in set theory. Anyone who studied enough of analysis and set theory does not need to study precalc which is not a field of mathematics (I say again). Just because the US educational system spends a year dabbling and names that year "precalc" does not make it a universal requirement. -lethe ^{talk +} 09:41, 6 February 2006 (UTC)

As far as I know, the word "precalculus" is American educational jargon, and has nothing to do with mathematics. It is a way of covering up the failure of the American educational system. Rick Norwood 13:39, 6 February 2006 (UTC)

ha! well put by the previous two comments! But don't you know, precalc is what you study after prealgebra, before you get to prenumbertheory and preriemanniangeometry. you definitely need a little preriemanniangeomerty to understand manifolds. MotherFunctor 07:14, 20 May 2006 (UTC)

Angles

I think we need to tell the reader in the lead what means Euclidean. Nobody except Mathematicians know what it means. The angles maybe the wrong example but we need some pictorial description of what is Euclidean and what is not. The picture must not be 100% mathematical coherent : it must just be clear. A remark can be provided later on to fix the case. Vb 19:50, 18 February 2006 (UTC)

The 180 degrees material is fine, it just doesn't need to be both in the picture caption and in the body of the text. As for "Euclidean", I think most people have at least heard the words "Euclidean geometry", and probably picture points and lines and circles and triangles when they hear it. I think that is as far as we can go in the introduction.

On the other hand, I disagree with "must not be 100% mathematically coherent". I think we can (and have) said things that give the correct impression to the layperson without actually lying.

There is an interesting editorial in the current issue of the NOTICES of the AMS in which a journalist says when news stories are written about mathematics the purpose is not that the public understand the story, but that the public think they understand. I'm offended by that attitude. Rick Norwood 01:15, 19 February 2006 (UTC)

Just to explain my objection. To use angles requires that we have a Reimannian manifold, with a metric tensor. 90% of the article is about topoligical of differential manifolds where the notion of angle is meaningless. Talking about angles early on could confuse the reader: first we talk about angles, then we define something where angles make no sense! That said I feel Reimannian manifold should have a more extensive coverage in the article. --Salix alba (talk) 01:28, 19 February 2006 (UTC)

I would love to see more coverage of Reimannian manifolds. Maybe an introduction here and a link to an article on the subject would be appropriate. Rick Norwood 13:15, 19 February 2006 (UTC)

Well I agree with the corrections you introduced to my contribution. The reference to angles in the caption only is better than what I did. Thanks for the improvement. Vb 11:01, 20 February 2006 (UTC)

Algebra <-->geometry

Here's another pot-shot. According to my understanding of the latest trends and fashions in math (and you know how fashion goes...), the idea of creating a manifold by "gluing together charts from an atlas" is "old-fashioned". The new-fashioned way of doing all this is an appeal to the algebra <--> geometry duality seen, for example, in varieties and ideals, viz, the shape of the space is determined by the coordinate ring (Hilbert's Nullstellensatz and all that). So the new tools are sheaves, and its the geometry of the sheaf that expresses the shape of the space. This is the main idea driving non-commutative geometry. Admittedly, this is all very high-brow and horridly abstract, but I've now seen a half dozen books that make a big sis-boom-bah about this in the preface. Since this is the modern way, shouldn't this article at least mention this? linas 02:46, 8 March 2006 (UTC)

Sounds interesting. Could be a topic for a new article? Do need to take a bit of care with the difference between varieties and manifolds, not all varities are manifolds. Do you know any online references for this? (alas I don't have access to a good library). --Salix alba (talk) 03:00, 8 March 2006 (UTC)

I did see a nice discussion on the algebra-geometry duality online somewhere, but I'm afraid I misplaced it. Certainly under-grad level books on algebraic geometry certainly focus on it in the limited context of "variety"; there seem to be "bigger" parallels which I don't understand. I'm not sure, but I think I was grepping either "quantum group" or "k-theory" when I found it. Possibly look for "commutative C*-algebra" in combination with these. And no matter what, I guarentee that the above searches will turn up a goodly amount of head-scratching stuff making intriguing or even wild claims. linas 04:10, 9 March 2006 (UTC)

The thing about manifolds is that they are everywhere, and there are many different approaches. I think here the best way to deal with sheaves is a brief mention here and a reference to the main article. Rick Norwood 00:01, 9 March 2006 (UTC)

Yeah, I think I meant to say scheme (mathematics) not sheaf. I certainly don't understand the big picture. And also, I do not mean to imply that there's some proof out there that says that the topological construction of atlases and charts is a subset of a grander algebraic approach. Maybe there is some-such theorem but I wouldn't know. I'm an amateur. linas 04:10, 9 March 2006 (UTC)

Page move

• So User:Anthony Appleyard moved Manifold to Manifold (topology). Probably a bit too WP:BOLD for such a well established article, without prior discussion. The names not quite right as manifolds are not just studied in topology Manifold (mathematics) might be better. I've a mind to move it back. Also raises the question of 350 links to the article which now go to a redirect to a disambig page. These will all need fixing. --Salix alba (talk) 15:45, 15 March 2006 (UTC)
  • I have moved it back. I can see the reason for moving it, but if this page is going to be moved, a consensus needs to be established and something needs to be done about all the redirects. Certainly Manifold (topology) is precisely the wrong place to move it to. –Joke 17:17, 15 March 2006 (UTC)
    • Agree with above. The move was the wrong thing to do, and should have been done with discussion anyway. Oleg Alexandrov (talk) 19:14, 15 March 2006 (UTC)
    • Agree. Rick Norwood 19:41, 15 March 2006 (UTC)
• Sorry. Anthony Appleyard 10:57, 16 March 2006 (UTC)
• I have looked through Special:Whatlinkshere/Manifold and similar for pages such as Space shuttle main engine that refer to real manifolds rather than maths. As far as I can manage I have pointed maths manifolds to Manifold and metal manifolds to Manifold (disambiguation). Also wherever I found a pointer to a redirect to manifold I changed it to a pointer to Manifold or Manifold (disambiguation), so that now the only (pointers to redirects to either of these) are user pages and talk pages and lists of redlinks and suchlike. However, manifold still needs to be moved to manifold (mathematics), because most people have not heard of maths manifolds but only of their cars' exhaust manifolds and scuba cylinder manifolds and suchlike. Anthony Appleyard 10:57, 16 March 2006 (UTC)
• I've only heard of math manifolds and not any other type. Loom91 11:06, 16 March 2006 (UTC)
• Then take a look at Manifold (disambiguation). Every car including yours has an exhaust manifold. Anthony Appleyard 17:21, 16 March 2006 (UTC)
• As it happens an exhaust manifold is topologically speaking a Manifold with boundary, I've always found this quite amusing. --Salix alba (talk) 23:43, 16 March 2006 (UTC)
• I don't disagree with moving to manifold (mathematics) although the mathematical usage is the oldest and arguably the most important of the uses on Wikipedia. Seriously, it really is a unifying topic in late 19th and 20th century mathematics and their use has been central in the physics of the past century. So what I'm saying is: just because you haven't heard of it, don't underestimate it's importance. The exhaust manifold page is ancillary to WikiProject Automobiles but in mathematics manifolds have an importance comparable to, say, the four-stroke engine. –Joke 22:06, 16 March 2006 (UTC)
• We should acknowledge Anthony Appleyard's good work in fixing links. Well done, Anthony. Rick Norwood 23:14, 16 March 2006 (UTC)
• I don't have a car, so I don't have an exhaust manifold. I don't think this should be moved to {mathemetics), as this is by far the most important and well-known usage of the term manifold. Loom91 09:14, 17 March 2006 (UTC)
• I still say that far more people have heard of manifolds as parts of car and lorry engines and scuba gear and hydraulic systems etc, than of the abstruse higher maths version. Move it to manifold (mathematics). Anthony Appleyard 11:07, 17 March 2006 (UTC)
• My informal survey of a few non-mathematicians indicates that those who know the word manifold at all know nothing of its use and importance in mathematics. Shocking and tragic but true. Perhaps if we brought this up to featured article quality and then somehow arranged for it to be featured for a week we could change that a little. ;-)     Meanwhile, aside from the inconvenience of fixing all those links (including future ones), it's hard to justify blocking the move. --KSmrq^T 16:12, 17 March 2006 (UTC)

Good, so is a consensus emerging then? –Joke 16:59, 17 March 2006 (UTC)

I have no objection to the move Anthony proposes, if he is willing to take on the huge task of fixing all those links. The negative reaction at first was, I think, because the move was sudden, unannounced, and unexpected. Rick Norwood 21:35, 17 March 2006 (UTC)

Please slow down a bit, Anthony

I appreciate your efforts to improve the article, but ...

I think "resembles" is more encyclopedic than "looks like". I'm told that bullet points are unencyclopedic and paragraph form is prefered.

When I first came to wiki, I rewrote extensively. Now, I've learned to be more cautious -- in most cases, unless an article is in a state of severe neglect, I limit my rewrite in any one article to a single section and then wait a day for comments. A lot of people are very active in working on this article, and many points have been discussed in detail. Why not try just a few changes, and see what happens. Rick Norwood 21:56, 17 March 2006 (UTC)

• Yes please. Anthony, you are the last in a long line of people very eager to improve this article without much prior consensus (Rick has been there too :) Take it easy, and talk a lot, if you want to have any chance at your changes sticking in the final version. :) Oleg Alexandrov (talk) 02:06, 18 March 2006 (UTC)

• In a list of information, I still think that a plain list with bullet points is clearer than essay style. In "essay style", the rule "vary the expression" makes it good for literary effect but confusing to read and hard on the brain when searching for a particular desired bit of information.

I know no such rule. See elegant variation. –Joke 18:43, 18 March 2006 (UTC)

What does encyclopaedic mean? To me it means "what is clearest in an encyclopaedia for people looking through for information". For that purpose I prefer a plain list to tangling everything together in so-called "best essay style", and avoiding long-winded Latinisms if there is good plain English that means the same.

Anthony Appleyard 08:03, 18 March 2006 (UTC)

I would have preferred an answer of the form: "Thanks guys, you have a good point. I will try to work more with others.". :) Oleg Alexandrov (talk) 08:17, 18 March 2006 (UTC)

Well guys, I think Anthony has a reasonable point. Loom91 08:54, 18 March 2006 (UTC)

Perhaps its best if new contributors make proposals for changes first. As to tangling everything together really that is the point of this article, i.e. a general overview of all aspects of the topic. More technical details go in the more specific articles topological manifold and differentiable manifold. Pesonally I like bullet points, this might have something to do with my learning style, I think there are good psycological reasons why bullet points often work better on the web, a very defferent medium to print. --Salix alba (talk) 10:29, 18 March 2006 (UTC)

I feel like this page needs some kind of bold shake up in order to become an example of Wikipedia's best. Unfortunately, whenever anyone tries to do anything, the response they get is "slow down" and "discuss on talk". That's fine, I guess, but I never see any real discussion on talk except to tell people that they really ought not make any changes, lest people become offended. This is causing article to ossify and making this talk page sound a bit like the European parliament. –Joke 18:53, 18 March 2006 (UTC)

I have been following this page for a year now. It gets "bold shakeups" all the time, and the next "smart ass editor" just overwrites the previous one; with null result in the end. That's not the way to go. Oleg Alexandrov (talk) 19:55, 18 March 2006 (UTC)

• Yes the intro does seem to get rewritten on a weekly basis, personaly I tend to refrain from editting intros. In the couple of months I've been watching this article is has got a lot better, the history is more extensive. Greater treatment of various different types of manifold, and there was a lot more improvments before that. Don't be dishartend Oleg!

My main gripe with the article at the moment is the orientability and examples. Personally I'd have a top level section on properties of manifolds, with orientability as a sub section of that, other properties worthy of note include: connectedness, curvature, Euler characteristic (leading on to very brief note on homology/homotopy). I'd also have a seperate examples section, including the current examples and also some more exotic examples, R^n, space-time, and orthogonal group. --Salix alba (talk) 20:54, 18 March 2006 (UTC)

As for boldness, the article has been completely rewritten more than once, including a major collaboration lasting for many weeks. Noticeable improvements now seem to come from more tightly targeted boldness. We have pondered the importance of orientability before, and the existence of a section devoted to it is a happy outcome. The question remains of when and how to first mention it. We sneak in compactness (but not the term) early, but non-orientability cannot occur without at least a 2-manifold. And part of the discipline of writing for Wikipedia is that the earlier the material occurs in an article, the broader the audience it must address. For example, anyone who understands English should be able to read the first sentence and decide if the article is of interest. So as a general matter of encyclopedic style, we need to give a feel for what manifolds are before we launch into a catalog of their properties. Also, since this is an overview article, properties like curvature (which depend on extra structure) don't really belong here, and not everyone is interested in Betti numbers and Euler characteristic; thus we banish those to more specialized articles. --KSmrq^T 21:25, 18 March 2006 (UTC)

Topological vs. Differential

I found some inconsistencies in the text. The definition uses the topological point of view and does not mention charts or atlases. Yet most of the rest of the article takes the differential point of view and explains at length what a chart and an atlas is. I believe that a novice reader would ask themself: Why do I need to know what a chart or an atlas is if they are not used in the definition? Of course, the answer is that they are used in the definition of a differential manifold and differential manifolds are very, very important, but, while possible, it is hard to understand that from the article.

I think that if we added an explicit definition of a differential manifold the text would become clearer. After a precise definition is given, it should be much easier to follow the explanations in the section "Charts, atlases and transition maps". Or we could define topological manifold to be a topological space equipped with an atlas of homeomorphisms, which is sightly incorrect, but would make the rest of the text much easier to understand. What do you think? I have some difficulties with English, so I prefer not to make the changes myself.

By the way, is there a topological manifold, which cannot be equipped with a differential structure?--tzanko 21:28, 28 March 2006 (UTC)

Differentiable manifolds are mentioned in the introduction, and there is a link there to the article on differentiable manifolds. In the section just before the section on charts, atlases, and transition maps, this statement appears: "One of the most important is the differentiable manifold, which has a structure that permits the application of calculus." Every topological manifold can be equipped with a differentiable structure, in higher dimensions more than one. But there are many areas of topology that use manifolds without reference to a differentiable structure, knot theory for example. Rick Norwood 21:37, 28 March 2006 (UTC)

What you said "every topological manifold can be equipped..." seems to contradict what I said. -lethe ^{talk +} 21:45, 28 March 2006 (UTC)

I stand corrected. Thanks. Rick Norwood 21:47, 28 March 2006 (UTC)

See Donaldson's theorem, which shows that there exist topological manifolds which do not admit smooth structures in dimension four. I believe that all topological manifolds of dimension less than four admit one (and only one) smooth structure. Now you didn't specify smooth structure. Maybe you can do better if you allow weaker differentiability, I don't know. -lethe ^{talk +} 21:44, 28 March 2006 (UTC)

This is a worthwhile item that has come up in the archives: Bishop and Goldberg's book Tensor Analysis on Manifolds (ISBN 0486640396) mentions a difficult theorem by Whitney, that any C¹ manifold can be massaged into a real-analytic, and hence C^∞ (smooth), manifold. --KSmrq^T 00:45, 29 March 2006 (UTC)

Sounds like a manifold version of Hilbert's fifth problem. I'll check out that book. A section about this topic would look nice, but maybe it would go better in differential structure than here. -lethe ^{talk +} 07:02, 29 March 2006 (UTC)

So obviously dimension 4 is a special case. In dimension 4, R^4 admits uncountably many smooth structures, S^4 is unknown, and there exist manifolds which admit none. In d<4, it's as nice as it could be: every manifold admits one, and only one. How about more than 4? The R^n and spheres admit finitely many structures, but I don't know whether there are any which admit none. Donaldson's theorem doesn't tell us. I would like to guess that all manifolds of d>4 admit at least one, but it's just my guess. -lethe ^{talk +} 07:35, 29 March 2006 (UTC)

There are higher dimensional manifolds not admitting differentiable structure. For example, I know of one in eight dimensions (Tamura), and there are more. In addition, the existence of a triangulation in dimensions greater than four is not known and is a big open question; any smooth manifold can be given a smooth triangulation (by Munkres, Whitehead, et al). BTW, I should mention there are non-triangulable 4-manifolds. So the question of how special dimension four is, is really still open. --C S (Talk) 17:21, 13 April 2006 (UTC)

Ah neat. So it's open whether higher D manifolds are triangulable? I remember asking my algebraic prof that, after he invoked without justification a theorem of Rado that all surfaces are, and he muttered some stuff that made it sound like he didn't know. Why do you bring up triangulation here though? Is existence of a triangulation related to the existence of a smooth structure? Or just another structure whose existence becomes unclear in high dim? -lethe ^{talk +} 17:29, 13 April 2006 (UTC)

Well, that theorem of Rado is pretty simple to prove, relatively speaking, compared to dimension 3, so let's give your prof a break :-) I'm a little surprised he didn't know though, but I would guess this is more known among older topologists harking back to the golden age of topology, rather than the younger ones. Triangulations are very closely related to smooth structures. The theorem I mentioned shows that every smooth structure arises from smoothing a triangulation. Now, there are different ways to smooth, so different smooth structures can arise. However, in dimensions up to four, it's known you can always smooth a triangulation. But I believe that's unknown for higher dimensions (not sure). One way to show uniqueness of smooth structure for dimension 2 and 3 is to study smoothings of triangulations. --C S (Talk) 02:17, 14 April 2006 (UTC)

Rationale for reversion

Anthony Appleyard has been rewording many paragraphs throughout the article, and doing some reformatting. In some cases there is a good idea behind the edits, such as substituting a plain word for a fancy one. But in most cases the result is a longer, more complex sentence structure. Studies show that shorter, simpler sentences are easier to read. They may seem a little choppier, but that's a sacrifice worth making. Also, please do not use underlining; it is a wretched holdover from the typewriter. Eliminating single blank lines after section headings does not affect the page layout, but makes editing less pleasant. --KSmrq^T 09:31, 29 March 2006 (UTC)

I wish Wikipedia had a "line item" revert. Rick Norwood 21:39, 29 March 2006 (UTC)

Since the technology already gives a paragraph-by-paragraph compare, it seems like it should be possible. Ah well, I'm not holding my breath. --KSmrq^T 23:10, 29 March 2006 (UTC)

OK, I can understand the benefit of using simpler words, but changing "constructed" or "obtained" to "made"? Come on, those are not advanced vocabulary. Take it to simple:manifold at the simple English wikipedia. -lethe ^{talk +} 13:09, 30 March 2006 (UTC)

I liked the most recent edit. For one thing "made ... made" uses the same word for the same concept, an advantage over "constructed ... obtained". Rick Norwood 15:11, 30 March 2006 (UTC)

And you like "met" for "encountered"; "but" for "however", "part" for "portion"? I think it sounds far too colloquial. -lethe ^{talk +} 15:26, 30 March 2006 (UTC)

• As Mark Twain said, the difference between almost the right word and the right word is the difference between a lightening bug and lightening. If only we could get Mark to write the article for us, we would be in good shape. All I can say is that the edit read well to me. I like long words when they serve a purpose. Right now I'm reading Avram Davidson's "The Scralet Fig", and a typical sentence reads, "Only the lictor's leather and legal face, the vetal's marmoreal countenance, did not change." But in writing mathematics, I almost choose the shortest word that will serve, because most people find mathematics hard to read. Of the three examples you give above, I see no advantage of the longer word over the shorter, and in the case of "part" for "portion", I think "part" expresses the idea better. Rick Norwood 17:33, 30 March 2006 (UTC)
• Ditto. I also see no need for Latin where there is an English word, or unnecessary extra expressiveness until the more-expressive word becomes a cliché and loses its expressiveness. Anthony Appleyard 07:07, 1 April 2006 (UTC)

Perhaps you meant to say cliché? --KSmrq^T 07:57, 1 April 2006 (UTC)

Good edit, Anthony. And, KSmrq, while there is every reason to insist on correct spelling in the article, there is no need to play "gotcha" on the talk page. Rick Norwood 13:51, 1 April 2006 (UTC)

Not guilty. You'll notice that chiché is not a redlink, so it's possible you Anthony used it intentionally. --KSmrq^T 20:03, 1 April 2006 (UTC)

I too am confused by Anthony's reference to chiché. He took the time to wikify it, so maybe that's what he really meant. And I still think that any program of replacing words with a Romance-based etymology with words with a German-based etymology is strange. Again I tell you, if you want articles that are written in rudimentary vocabulary, we have a place for it: simple.wikipedia.org. Take it over there. -lethe ^{talk +} 20:15, 1 April 2006 (UTC)

So, KSmrq, you are seriously suggesting that Anthony intended to say, "...until the more expressive word become a municipality in Guatemala and looses its epressiveness."

Sorry. "Cliché". Typo. Sorry. I have now corrected the original typo. Anthony Appleyard 08:29, 3 April 2006 (UTC)

Nobody is suggesting limiting our writing to the basic -- what is it? -- 8000 words? -- that are as much as -- what? -- 60% ? -- of the people understand. The point is that there is no advantage to using a longer word just because it is longer. I have not liked all of Anthony's edits, but I liked his most recent edit, because I thought it made the concepts clearer. Rick Norwood 14:51, 2 April 2006 (UTC)

I didn't say I'd followed the link. ;-)

Incidentally, we needn't look to German-based versus Romance-based; one of my pet peeves is "utilize" versus "use". --KSmrq^T 21:14, 2 April 2006 (UTC)

• "Anthony's reference to chiché. He took the time to wikify it,": It does not take me much time or thought to type double square brackets round a word as I type it. Anthony Appleyard 17:19, 3 April 2006 (UTC)

Basic English

(Off topic)

"1:1 At the first God made the heaven and the earth. 1:2 And the earth was waste and without form; and it was dark on the face of the deep: and the Spirit of God was moving on the face of the waters. 1:3 And God said, Let there be light: and there was light. 1:4 And God, looking on the light, saw that it was good: and God made a division between the light and the dark, 1:5 Naming the light, Day, and the dark, Night. And there was evening and there was morning, the first day."

The above is from the Basic English Bible; a version of the Bible written using only the 800 Basic English words. I got curious, after my previous post, about how close my esitmate of 8000 basic English words was. I still think most people know 8000 words, but it is interesting how much you can say with just 800. Rick Norwood 15:05, 2 April 2006 (UTC)

off the charts

I may be mistaken, but I have not heard the phrases "chart" and "atlas" used except in the setting of a differentiable manifold, with the requirement that the transition maps be diffeomorphisms. In the setting where I do most of my work, PL-topology, covering maps and quotient topologies are used instead of charts and atlases. Does anyone else know of a case where "chart" and "atlas" are used without the requirement of differentiablilty? Rick Norwood 15:48, 2 April 2006 (UTC)

• "I have not heard the phrases "chart" and "atlas" used": presumably that means "Within mathematics I have not heard ...". Anthony Appleyard 08:29, 3 April 2006 (UTC)

• Now who's playing "gotcha"? Rick Norwood 13:18, 3 April 2006 (UTC)

I don't know what PL-topology is, but where do you get the local structure (local homeomorphisms to Euklidian balls) from if not from charts. Even for differentiable and complex manifolds, the altas business is used in the definition and in some how further implicitly, but I never saw a mathematical proof on manifolds using the word atlas once you understood the definition. Besides, formally the definition I know uses maximality of the atlas.

PL-topology studies objects constructed from unions of simplices, which are the convex hulls of sets of points, said sets having the property that no point is contained in the convex hull of the other points. The PL stands for "piecewise linear".

Clearly, you are arriving at manifolds through differential topology. For a different approach, I suggest Munkres "Topology". I can provide other references if you are interested. Rick Norwood 14:12, 14 April 2006 (UTC)

The language of charts and atlases is often used as it unified many different viewpoints, including that of PL manifolds. One definition of PL manifold is to have an atlas so that transition maps between charts is PL. --C S (Talk) 15:35, 19 April 2006 (UTC)

Pointless paragraph in intro

• I removed this paragraph from the intro earlier, because I didn't think it was necessary.

A technical mathematical definition of a manifold is given below. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and a working knowledge of calculus and topology would be helpful.

• Rick Norwood put it back, but I'm afraid it is still superfluous.

Ben Standeven 01:43, 11 April 2006 (UTC)

Yeah, looks superfluous to me. -lethe ^{talk +} 01:50, 11 April 2006 (UTC)

• I thought we finished this debate. The paragraph is necessary and not superfluous. Loom91 09:16, 11 April 2006 (UTC)

• • I can't see any debate in the history, except about whether it should be a template or a prose paragraph. Why is this paragraph necessary? Regular polytope doesn't have a disclaimer telling people they need to know group theory to understand the math behind polytopes. Ben Standeven 22:02, 11 April 2006 (UTC)

• • • Well, in that case it should. Informing readers of what they should be familiar with if they expect to understand the whole article does good, or at the very least does no harm, so I don't see why you want to remove it. Eventually all articles in Wikipedia on specialised topics should contain information about prerequisites for understanding. Loom91 08:20, 12 April 2006 (UTC)

If you start reading the article, either you will understand it or you won't. Scaring people off with vague dependencies is pointless. Announcing what is given below likewise adds nothing. --MarSch 17:03, 12 April 2006 (UTC)

My Additional Objections

• I don't like the following text from the circle example:

Manifolds need not be connected (all in "one piece"): a pair of separate circles is also a topological manifold.
Manifolds need not be closed: a line segment without its ends is a manifold.
Manifolds need not have finite extent: a parabola is a topological manifold.
Other topological manifold examples include a hyperbola and the locus of points on the cubic curve y² - x³ + x = 0, which are neither connected, nor closed, nor finite.

• I think it is too choppy, and reads like a bulletted list without the bullets. But I don't see any good way to fix this. Ben Standeven 03:09, 11 April 2006 (UTC)

This was a recent change. Perhaps you'd like to look back in the history and see if you like its predecessor better. I'm not thrilled with this version myself, but I get tired of holding back the tides of darkness. It's a rare editor who is a master of both the mathematics and the English language. Ah well; this is "the free encyclopedia that anyone can edit." --KSmrq^T 08:55, 11 April 2006 (UTC)

• From the history section we have "Abelian varieties were already implicitly known in Riemann's time as complex manifolds". How can this be, if Riemann invented the term "manifold"? Ben Standeven 03:51, 11 April 2006 (UTC)

There are a bunch of strange examples that should be removed. Since the line, the parabola, and a cubic curve and for that matter the graph of any polynomial are all homemomrphic to the real line, it is pointless to multiply examples. Also, I find the example of a complex manifold a little further down confused and confusing. Rick Norwood 20:24, 12 April 2006 (UTC)

The examples are not strange; the text that accompanied the image has been progressively modified until the reason for their existence is clouded. I've restored the language introduced with the image. Sadly, the history of this article (and many others) is, like evolution of species, not a monotonically improving march forward. --KSmrq^T 09:28, 13 April 2006 (UTC)

Why is the section on the mathematical definition not only the shortest section in the article, but also incorrect (since it doesn't mention Hausdorff or secound countability)? This article kinda sucks. Seems like a massive amount of effort went in to making the discussion seem accessible and elementary, while correctness, completeness and conciseness were badly neglected. What is to be done though? I don't think I have the energy for a rewrite at the moment. -lethe ^{talk +} 14:24, 13 April 2006 (UTC)

KSmrq, I still do not understand why you think we need more than one example of the graph of a polynomial. Also, I find the following unintelligable:

"(We may take a different view in algebraic geometry, where we consider complex points on the quartic curve ((x−1)²+y²−1)((x+1)²+y²−1)=0, whose real points alone form a pair of circles touching at the origin.)"

Why introduce complex manifolds in this section and without any groundwork? How does this example provide any useful information?

Lethe, I agree with your comments, and will try to do something to fix the problems you bring up. Rick Norwood 14:42, 13 April 2006 (UTC)

"Graph of a polynomial"? First, that's factually inaccurate, for the usual meaning of graph; the illustration depicts real algebraic curves. Second, each example explicitly states what it's about, and none of those statements fits your (rather perverse) characterization. It is merely convenient, but totally incidental that the examples use algebraic curves. If I must belabor the obvious, the examples are about the freedom manifolds have: to be connected or not, to be compact or not (without using the term just yet), to be finite or infinite, and to have multiple components that are not all alike. I find it bizarre that, despite the text being as explicit as it is, you miss the point.

As for complex manifolds: The specific curve given is a pair of unit circles touching at the origin as a real curve; thus not a manifold. (I hope that's obvious to a mathematician, accustomed to x²+y²−1 = 0 as the implicit form of a unit circle centered at the origin, and knowing that the product of two polynomials gives the union of their curves.) However, algebraic geometry habitually works in projective space and with complex curves. As a complex curve, this is a manifold. Anyone who is familiar with that discipline will reflexively point out the rich connection between manifolds and properties of complex curves. I'm guessing the increasing abstractness of modern algebraic geometry means most non-specialist mathematicians today know almost nothing about it, not even the much more accessible classical material. A pity. But perhaps the temptation to mention the difference between a real curve and a complex curve should be resisted at this point. After all, this is meant to be an elementary and introductory section. Still, the remark is parenthetical, which is a hint that it can be ignored if it seems too advanced.

I won't fight the death of the algebraic geometry remark, but I will vigorously oppose excision of any of the examples.

As for Hausdorff and second countability: We once talked about how restrictive a definition to use. That included the idea of an n-manifold versus a more general concept (Jitse cited Marsden) allowing components of different dimension. My view is that we should mention common restrictions but acknowledge the existence of broader variants. Apparently the restrictions were lost (or never added). Bottom line, I strongly support mention of restrictions, but not so soon we scare the novices. --KSmrq^T 17:58, 13 April 2006 (UTC)

I have to agree with Rick on the algebraic variety. --MarSch 15:25, 13 April 2006 (UTC)

Whether or not the comment is necessary, I don't know, but certainly it's intelligible: it simply says that algebraic varieties include things that are not manifolds, like two tangent circles. -lethe ^{talk +} 15:29, 13 April 2006 (UTC)

KSmrq, how about you keep all of your examples except the one about algebraic geometry. In fact, keep that one, but put it in the article on algebraic geometry. Deal? Rick Norwood 19:03, 13 April 2006 (UTC)

Works for me. In fact, I think that's what I just said. :-D

I made a small but important change to your needed addition of Hausdorff and second countability, to accommodate the fact that not everyone requires them. --KSmrq^T 23:51, 13 April 2006 (UTC)

In my universe, spaces which are locally homeomorphic to Euclidean space are very sensibly called locally Euclidean spaces. But well, I guess if there are sickos who want to pretend that non-Hausdorff spaces are manifolds, so be it. -lethe ^{talk +} 00:04, 14 April 2006 (UTC)

I guess we should mention the long line and the line with two origins to clarify the purpose of the conditions for a manifold? -lethe ^{talk +} 00:09, 14 April 2006 (UTC)

I'm not happy at all with the addition of all these technical conditions. The purpose of this article is to explain to a broad audience what the concept of a manifold is. It is not about any specific manifold such as topological manifolds or differentiable manifolds. Therefore talking about things like Hausdorffness and second-countability is not constructive. Nor is using algebraic curves. --MarSch 10:42, 14 April 2006 (UTC)

The article can and should have both. Above the ToC, it should have an introduction that anyone can read. But it should also have a correct, mathematical definition, clearly labled as such. Rick Norwood 13:55, 14 April 2006 (UTC)

I couldn't disagree with MarSch more. If we don't include those conditions, technical though they be, then the article fails to have a correct definition, which is a very bad sin. Putting readability before correctness is something I just can't abide. It's halfway down the article, buried in a section called "mathematical definition", so it won't detract from the readability too much. -lethe ^{talk +} 16:25, 14 April 2006 (UTC)

That def is not for manifold but for topological manifold or something. There IS no definition of manifold, except if you take it as a synonym for a particular kind of manifold. Otherwise you'd best start merging in both top.man. and differentiable manifold. Sigh. --MarSch 09:24, 19 April 2006 (UTC)

Likewise, I'm unhappy with the definitions introduced by Invisible Capybara and others recently. In initiating and designing and discussing this present form of the Manifold article we reached agreement among many parties with many interests that we would be careful to not settle on one definition to the exclusion of others. And more recently I've emphasized that aspect in my comments here. Generally speaking, a manifold typically begins as a topological n-manifold; but it need not be Hausdorff etc., it may not have fixed n, it may demand other structure, or it may be locally something other than an open ball in Euclidean space. Just because an editor has never seen anything other than a very basic manifold, the temptation is to impose individual limitations on a group topic. Please don't.

Never give a specific definition as if there are no others; always be inclusive. It is unwieldy and unhelpful to give an umbrella definition that covers all possible variants. (Though this is explicitly an umbrella article.) It is also unhelpful to define a manifold as a topological manifold, ignoring the folks who never use any manifold that isn't at least differentiable, if not smooth. --KSmrq^T 17:37, 19 April 2006 (UTC)

And, yes, I know it's a pain in the ass to write this way. When we speak in broad generalities, nobody knows what we're talking about. When we luxuriate in closely delimited definitions, we may be perfectly clear and correct, but we disenfranchise all those whose manifolds don't fit. The reward for our effort is that when we succeed in balancing the extremes, we communicate broadly and powerfully, and that can be deeply gratifying.

Two useful approaches: (1) Give a liberal definition and then point out the typical restrictions. Taken to extremes, this is Bourbaki style, I suppose. (2) Give a tight definition and then expand with generalizations. Often this is the easier way to learn, and the usual way a topic evolves historically.

I think it's a matter of taste and judgement which of these options, or something else, works better in a given circumstance. What doesn't work (here) is giving just the liberal definition or just the tight definition, without the followup. --KSmrq^T 00:45, 20 April 2006 (UTC)

I think the style preferred on wikipedia is (2): use a restricted definition and generalize later in the article. I've noticed that Oleg frequently prefers this approach, and I occasionally clash with him on that. I often prefer the most inclusive definition. But in this case, I fall on the side of more restrictive definition. It may just be an indication of my limited experience, but I feel like only in some marginal areas would one want to allow non-Hausdorff non-second countable manifolds (anywhere in physics, plus symplectic or Riemannian geometry, which I would argue are the main area that we have to cover, do not consider such cases, for example). Anyway, however we treat it, whether showcased in the main definition, or shunted to a later section, those restrictions have to be mentioned. If we don't mention them, we've got a wrong article. -lethe ^{talk +} 03:11, 20 April 2006 (UTC)

Yes I feel to throw the terms Hausdorf and second countable will be a discouragment to the Lay reader. We do also mention them below in the differential manifold section.

Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable.

this latter presentation I feel is better, technically correct, but worded in a non frightening way.

I also feel that there are problems with the article, in the order of the sections. It feels more natural to place the mathematical definition section closer to the discussion of the various sub-types, after all there is no mathematical object called simply a manifold, its always a topological-M or differential-M etc. What do people think the ordering of sections should be? --Salix alba (talk) 18:18, 19 April 2006 (UTC)

Actually, there is a mathematical object called simply a manifold, although usually the dimension is appended. See, for example, book titles Calculus on Manifolds by Spivak or 3-Manifolds by Hempel. According to Munkres Topology, page 225, "An m-manifold is a Hausdofff space X with a countable basis such that each point x of X has a neighborhood that is homeomorphic with an open subset of R^m." Rick Norwood 21:07, 19 April 2006 (UTC)

In this case they are refering to what we refer to as topological manifolds. --Salix alba (talk) 21:52, 19 April 2006 (UTC)

Yes, that is the most general kind of manifold. All other kinds of mathematical manifolds are topological manifolds plus some additional structure. My point is that some authors do not use the adjective "topological". Rick Norwood 23:42, 19 April 2006 (UTC)

We should make this clear in the article. --Salix alba (talk) 00:01, 20 April 2006 (UTC)

diff. Man. and especially smooth man. are also usually called just manifold. They are not a subclass of top. man. since you may have more than one diff. structure on a top. man. See exotic sphere. And there are some top. man. which allow no diff. structure. So Rick Norwoods comment that top. man. are the most general is simply not true. --MarSch 12:10, 20 April 2006 (UTC)

Rick's use of "most general" has the sense of "least restrictive". Rings are more general than fields, topological manifolds are more general than differential manifolds. In both examples, we impose additional restrictions on the more specialized object.

My concern is that in the past we discussed manifolds of a kind unfamiliar to me, that might not be merely restrictions of n-manifolds. I am not qualified to represent their point of view, but feel I should mention it now. For example, the article mentions infinite-dimensional Fréchet manifolds. --KSmrq^T 14:28, 20 April 2006 (UTC)

I don't think you've taken MarSch's point. The analogy with rings versus fields illustrates the point perfectly: a field is a ring which satisfies additional axioms. A smooth manifold is not a topological manifold with more axioms, it is actually a topological manifold with more structures, with more sets attached to it. In the language of category theory, this difference goes like this: the forgetful functor from Fld to Rng is fully faithful, while the functor from Smooth to TopMan is merely faithful. In the latter case, when removing structures, truncating the signature, it makes little sense to try to compare generality of the structures. It's like saying that a set is a generalized kind of group. True of monoids, makes little sense for sets. It's true that all groups have underlying sets, but mostly irrelevant. -lethe ^{talk +} 16:53, 20 April 2006 (UTC)

I'm not convinced. We can define a topological manifold as having continuous transition maps, a differentiable manifold as further stipulating differentiable transitions, and smooth as infinitely differentiable. In fact, we speak of C⁰, C¹, and so on, which tends to support a hierarchical view. And we also have the theorem of Whitney that any C¹ manifold can be made real analytic. (Not so for C⁰.)

I do agree that we introduce, not only restrictions, but also choices. (Hence we have multiple differential structures on higher spheres.) I just don't find that a compelling argument against topological manifolds being more general.

In any case, so long as we find language and a sequence of statements that covers enough variations, I'm happy. --KSmrq^T 19:17, 20 April 2006 (UTC)

Hmm... I think despite my fancy category talk and Rick's endorsement, KS has got me. A manifold is a space with an atlas, and smoothness is an axiom that the atlas may satisfy. Atlases are additional structures, so the comments are good for manifolds versus topological spaces, but not for smooth manifolds versus topological manifolds. So now I think I agree with Rick's description of topological manifolds as the most general class, but not with his description about extra structure, nor with MarSch that smooth manifolds are not a subclass of top. manifolds. -lethe ^{talk +} 20:53, 20 April 2006 (UTC)

I think Lethe expresses my point well. Every manifold is a topological manifold with additional structure, just as every group is a set with additional structure. Of course, it all depends on what the meaning of "is" is. Rick Norwood 20:27, 20 April 2006 (UTC)

I think lethe's first explanation using category theory is correct, although I'm not too fluent in faithfulness. I'm can understand that the concept of additional structure seems convincing but in view of the evidence that it is not restrictive this so-called additional structure argument is false. A topological manifold is not just a topological space with an atlas. It is an equivalence class of those things. Atlases which are differentiably compatible are automatically also topologically compatible. But the converse is not true. A topologically equivalent class of topological spaces with atlas may split into several differentiably equivalent classes of topological space with atlas. So a topological manifold is more basic than a differentiable manifold but it is not true that a diff. mani. is more general than a top. mani. because some top. man. do not have a diff. structure. Using the set/group analogy: sets are more basic than groups because each group has an underlying set, and groups would be more general than sets, if each set had at least one group structure. I don't think this is so, but I don't know. But what about Banach manifold. Do they have an underlying top. mani.? No they are a strict extension. Such as ordinals/cardinals are a strict extension of natural numbers. And what about non-Hausdorff top. manifolds? They are also a strict extension. It's like saying the empty set is the most general set. All other sets are just the empty set with some more structure. Well yes, but NO. --MarSch 11:10, 24 April 2006 (UTC)

reply to lethe edit summary

It's a little awkward responding to a question raised in an edit summary! In response to my "typo, fine tuning" summary, lethe's summary says "This is a weird typo, Kmsrq (also, isn't mathbf preferred in math formulas?))". So first, that should be "KSmrq". The typo is "give"→"given". Font selection for TeX has evolved, and I'm not sure what's now preferred. Looks like mathbf, the historic choice, is the common Wikipedia choice, so I shouldn't have changed it. --KSmrq^T 11:39, 14 April 2006 (UTC)

The wierd typo I was referring to was changing B to Bold as the name of the unit ball. I assume that was a typo? You don't actually want to name your unit ball "Bold"? You have to admit, that is kinda weird. As for tex, I don't care that much. I was just fixing the typo and so changed that as well. -lethe ^{talk +} 16:22, 14 April 2006 (UTC)

It was a very logical typo, just forgetful. Typically mathematical italics are letterspaced farther apart than ordinary italics, and I was trying to see if there was also any difference with the boldface. (Didn't see any.) You see, "B" for ball and "B" for bold; obvious, eh? Sorry to leave the debris from my experiment; thanks for catching it. --KSmrq^T 16:33, 14 April 2006 (UTC)

By the way, I have been having persistent problems getting your name right ever since I met you. Does it stand for something or is it short for something? That might help me remember. -lethe ^{talk +} 18:08, 14 April 2006 (UTC)

Yes, it stands for something; but if I tell you what, then I lose my anonymity. I'd like to switch to just KS. --KSmrq^T 22:49, 14 April 2006 (UTC)

KS, I can do that. From now on in my head I shall call you Kansas. -lethe ^{talk +} 22:56, 14 April 2006 (UTC)

I've always assumed KSmrq is a dragon, a relative of the one Bilbo met, Smaug, greatest and dreadfullest of all calamities : ) Rick Norwood 21:02, 14 April 2006 (UTC)

Now try saying KSmrq out loud three times fast. -lethe ^{talk +} 21:51, 14 April 2006 (UTC)

What is not a manifold?

There's lots of examples of what a manifold is, but no examples of what isn't a manifold. Maybe a simple counter example would go a long way to helping people (me) understand? 81.103.232.63 23:21, 20 April 2006 (UTC)

The figure 8 counterexample is in the article now. An idealized umbrella with a one dimensional handle and a two dimensional top would not be a manifold. Should we add more examples? Rick Norwood 23:26, 20 April 2006 (UTC)

I thought perhaps the non-examples of the long line and the line with two origins should be in here, so show why we like paracompactness and Hausdorffness. For another nonexample, no indiscrete space can be a manifold, and discrete spaces can only be 0-dimensional manifolds if they're countable. -lethe ^{talk +} 01:05, 21 April 2006 (UTC)

Sounds like good material for the article to me. Rick Norwood 12:36, 21 April 2006 (UTC)

I support adding good counterexamples. A link to physics' string theory might be a fun way to show a practical way being a manifold is nice. How about a T-junction of (infinitely thin) walls? Or the cubic real algebraic curve y² = (x+1)(x²−1), which has an isolated point, thus is not an n-manifold? Or y² = x³, to remind us that a curve with a cusp can be, surprise, a smooth manifold? Some time back the elementary part of the article mentioned a string dangling from a sphere; it's a natural. --KSmrq^T 18:06, 21 April 2006 (UTC)

Contact Request: Manifold Page Curator

I am seeking to contact the Manifold page curator. Thanks.

Sincerely, gmjaet@sbcglobal.net GeMiJa 16:35, 21 April 2006 (UTC)

Yes? We're listening... What do you want? Talk pages are how you contact the authors of articles at wikipedia, and other interested parties, and you've arrived here. So what can we help you with? -lethe ^{talk +} 16:52, 21 April 2006 (UTC)

I want something proved.

Introduction is misleading. An essential concept is missing.

A key idea in manifold theory, something that goes all the way back to Ricci's absolute differential calculus, is covariance with respect to arbitrary changes of coordinates. I would put it like this:

• a manifold can be described locally by a system of coordinates
• however no choice of coordinate system can be preferred over any other choice

So the sentence that a manifold "in close up view resembles Euclidean space" is misleading. This sentence is valid for describing a Riemannian manifold (indeed that is how Cartan thought about Riemannian manifolds), but it is not right for general manifolds.

I understand that the goal is to find wording that is comprehensible to an interested layperson, but we are not there yet. The difficulty is that manifold is an abstraction, an axiomatic framework where we forget as much geometric information as possible.

In a way, the examples from elementary geometry like curves and surfaces are misleading, because they all suggest a notion of distance and angle. It might be worth remembering that the first person to write down manifold-type axioms was H. Weyl in his book "The idea of a Riemann surface". So the example of a complex algebraic curve, a 2-dimensional entity that lives in 4-dimensional space is more enlightening. At least there is less structure (conformal rather than Riemannian).

Bottom line: the present introduction does a good job of motivating the idea thata manifold also has a global structure. However, the freedom in the choice of local structure is not mentioned. This makes the current entry inadequate, both for the expert and for the layperson.

Please add new sections at bottom of talk page, not top, per Wikipedia convention. Fixed. --KSmrq^T 03:47, 24 April 2006 (UTC)

I read the phrase "in close up view resembles Euclidean space" as a layman accessible stand-in for something like "some neighborhood of every point of an n-manifold is homeomorphic to Bⁿ." We have tried being totally precise and complete in the introduction, and found that for a survey article like this it just doesn't work. Certainly at some point we wish to point out that we have multiple choices for an atlas, but the intro is not the place to do it. --KSmrq^T 04:01, 24 April 2006 (UTC)

My apologies for the misplacement of comment. Thanks for the reformat and the comment. I am not advocating being totally precise in the intro. This would be, as you say, a mistake. What worries me is that, currently, the intro conveys an incorrect impression. We should get the word Euclidean out of the intro. The idea is that every neighborhood is homeomorphic to R^n is the same as saying that there exist local systems of coordinates. However, right now, the inexperienced reader will think that distance is a meaningful concept in a manifold. A general manifold is a very stretchy and flabby kind of space which is suitable for a very limited kind of geometrical theory. The introduction has the ambitions of conveying the idea of manifold without giving the precise definition. The present wording is a flawed attempt in this direction. Can we do better? Rmilson 11:00, 24 April 2006 (UTC)

In the case of topological manifolds, which this article is not about (is about), what we want is the underlying topological space of Euclidean space. In the case of diff. manifolds which this article is also not about (or is also about) we want the underlying topological space plus differentiable structure. Mathematically a good way to capture the structure you want is to use a sheaf of functions, called the structure sheaf. Unfortuantely this is not so simple to state simply. --MarSch 11:20, 24 April 2006 (UTC)

Frequently a manifold is defined as the pair (M,A) where M is a topological space and A the atlas of charts. In this definition you are restricted in the local coordinate systems, i.e. it must correspond to one of the charts in the atlas. In certain cases there is only one chart in the Atlas and hence only one coordinate system. Perhaps

• a manifold can be described locally by a system of coordinates (not necessarily unique)

would do. --Salix alba (talk) 12:32, 24 April 2006 (UTC)

• Saying that the space is locally homeomorphic to an Euclidean Space does not necessarily remove covariance. We need to make the distinction between the affine Euclidean space and the Euclidean vector space. If we talk about the affine space, then no choice of co-ordinates is implied. However I can't see how we can get this point across in the introduction. Loom91 07:00, 25 April 2006 (UTC)

draft revision of first paragraph

In mathematics, a manifold is an abstract multi-dimensional space where one cannot meaningfully make a choice of a coordinate system. In contrast to Euclidean space , one cannot distinguish between a straight line and a curve, nor define the distance between two points. Another difference is that many manifolds possess a complicated spatial structure when viewed as a whole. There are finitary manifolds that are nonetheless unbounded ; other manifolds possess holes, regions where a circle cannot be continuously shrunk to a point.

The surface of Earth is an example of a manifold; locally it seems to be flat, but viewed as a whole it is round. However, since manifold geometry aims to be a minimalist theory, the surface of a sphere and the surface of an ellipsoid cannot be distinguished as manifolds. A manifold can be constructed by "gluing" separate Euclidean spaces together; for example, a world map can be made by gluing many maps of local regions together, and accounting for the resulting distortions. Alternatively, a manifold can be defined by a system of equations. However, as a geometric entity in and of itself, a manifold exists independently of any ambient space.

Rmilson 13:07, 24 April 2006 (UTC)

comments on proposed revision

• The terminology multi-dimensional space is an informal way of saying that a manifold is locally like R^n. After all, what is dimensionality of space other than a local system of address, aka a coordinate system, aka a local homeomorphism with a piece of R^n Rmilson 13:07, 24 April 2006 (UTC)

• we have to make it clear to the reader that a manifold is a minimalist setting to do geometry. topological manifold (modulo the non-existence of differential structure pathologies) is indeed the most minimal setting, but it is debatable whether one can do 'geometry' in a topological manifold. So, assuming that we want to give the intro a geometrical flavour, we should aim for an informal definition of differential manifold in the intro, and discuss topological manifolds a few paragraphs down where we deal with the various common structures. Rmilson 13:07, 24 April 2006 (UTC)

If you scroll back up the talk page, you will discover long discussions over whether to mention topological manifolds first or to mention differentiable manifolds first. As best I can tell, everybody wanted to mention the manifolds they work with personally first. I do think that tolological manifolds are simpler, and should come first, but then I work with topolotical manifolds, don't I. Rick Norwood 20:02, 24 April 2006 (UTC)

I don't like

• where one cannot meaningfully make a choice of a coordinate system

there are plenty of manifolds where one can meaninfully make a choice of coordinate system, eg Rⁿ. Likewise there are manifolds (Reinmanian) where you can distinguish between lines and curves. We should focus on what the core of what a manifold (of all types) is, rather than emphesise chateristics which some don't have. These can be moved later.

If we take only the manifold structure on Rⁿ, i.e. the class of manifolds diffeomorphic to Rⁿ, then we forget so much information, that we can no longer recover the usual Cartesian coordinates. The intuition behind general manifolds is somewhat different from the intuition behind Riemannian manifolds.Rmilson 20:50, 24 April 2006 (UTC)

don't like

• There are finitary manifolds

never heard the term finitary before. I do like references to fact that manifolds can be globally complex, a double torus is a nice example of a locally simple but globally complex manifold. There is I feel a bit to much emphesis on very simple manifolds, circle and sphere, and not enough examples illustrating the diversity of manifolds. --Salix alba (talk) 20:41, 24 April 2006 (UTC)

Yes, I understand your point. One wants to say compact, but in an informal way. Maybe one should just say finite. For example, there are cosmological theories in which space is a compact manifold, but they say space is finite.

• Far too technical. The average reader will scan it once and leave promptly. Also, the absence of a preffered coordinate system is emphasised so much as to make it seem the defining property of a manifold. At most an one line mention in the intro will suffice. Further delibaration can be made later into the article. I.m doing this right now.Loom91 07:03, 25 April 2006 (UTC)
  • I will argue that the absence of a preferred coordinate system is the core concept underlying the definition of topological and differential manifolds. This was stated informally in the old-style absolute differential calculus, and formalized by H. Weyl in his book, The concept of a Riemannian surface for a 2-dimensional setting. Hassler Whitney gave the general axioms in the 30s in terms of charts and atlases. A chart is the inverse of a system of coordinates. In the first case, we have a mapping from Rⁿ to the manifold. In the other case, from the manifold to Rⁿ. Rmilson 10:35, 25 April 2006 (UTC)

A good first sentence

The following versions of introductory sentences have been used and reverted in recent days:

1. In mathematics, a manifold is an abstract mathematical space which, in a close-up view, resembles the spaces described by Euclidean geometry met in elementary math courses, but which may have a more complicated structure when viewed as a whole.
2. In mathematics, a manifold is an abstract mathematical space which, in a close-up view, resembles elementary Euclidean space, but which may have a more complicated structure when viewed as a whole.
3. In mathematics, a manifold is an abstract mathematical space which is made up of simple parts. Those simple parts are Euclidean spaces, for example parts of a line or of a plane or of three dimensional space.

Version 1 is has an awkward sentence structure. What do elementary math courses have to do with all this? I believe that is why revisions 2 and 3 were offered. Surely there is way to smooth out that sentence without changing its essential meaning? Rmilson 10:47, 25 April 2006 (UTC)

The big problem with 1 and 2 is "close up view". I see what the author is trying to say, but I don't think "close up view" really captures the idea of a manifold being made up of simple parts, that is, being locally Euclidean. Rick Norwood 15:38, 25 April 2006 (UTC)

Of course you realize, that your informal description of a manifold makes it sound like some kind of a cell-complex? Rmilson 16:09, 25 April 2006 (UTC)

Yes. Which is why the example of a cell complex which is not a manifold (the figure 8) is given below. Can you think of an elementary way to phrase the idea of homeomorphic neighborhoods? I think the introduction needs that. I've been trying to think of something, but haven't thought of anything yet. Rick Norwood 18:39, 25 April 2006 (UTC)

No choice of phrasing can satisfy everyone. Here are some possibilities:

1. In mathematics, a manifold is an abstract multi-dimensional space.

   The notion of dimension is based on assigning coordinates, and one can't assign coordinates near the 'crotch' of the figure 8.

   Precicely, thats why the figure 8 is not a manifold.

2. In mathematics, a manifold is an abstract space that can be described by one or more system of coordinates (not necessarily unique).

   This was suggested by User:salix alba and is quite good, but might be deemed to be too technical

3. In mathematics, a manifold is an abstract space that accomodates multi-dimensional coordinate grids

   This is something along the lines above, but using less technical language.

4. In mathematics, a manifold is an abstract multi-dimensional space that can be broken up into simple parts. Whatever the decomposition, each part can be described by one or more system of coordinates; however, it may not be possible to meaningfully choose amongst the different possibilities.

   The above is synthesis approach. Is it too 'technical'? Too wordy?

5. A manifold is a mathematical space which is constructed, like a patchwork, by bending and gluing together copies of simple spaces.

   I found this in the history. The wording was introduced by MarSch on 23 October 2005[2] to address peer review comments. I find the wording to be a quite effective metaphor. It was a very stable opening sentence, surviving to February 2, 2006[3]

Rmilson 01:35, 26 April 2006 (UTC)

I hate to say it, but I think both "multi-dimensional" and "system of coordinates" are going to fail to communicate. I've been thinking about it for a few hours, and I'm going to take another shot at it. Wish me luck. Rick Norwood 19:36, 25 April 2006 (UTC)

Good luck. Also, take a look at what MarSch wrote late last year. He went through a peer review, and tried many variations based on the reviewers' comments. May I also suggest that you wait a day, so that other can make comments? Rmilson 19:52, 25 April 2006 (UTC)

I like number 5 above by MarSch better than the current first sentence, which might even be wrong (a manifold is locally homeomorphic to simple parts, which is not the same thing as made up of simple parts). -lethe ^{talk +} 20:57, 25 April 2006 (UTC)

I've given it my best shot. But I won't make any other changes in this article today. Rick Norwood 20:58, 25 April 2006 (UTC)

To Lethe -- yeah, but you want to explain local homeomorphism to a non-mathematician. Made up of simple parts is true. The way in which it is "made up" is explained more fully below. Rick Norwood 21:00, 25 April 2006 (UTC)

In my opinion, "made up of simple parts" is false and should therefore be excised. A manifold is made up of points along with a topology and an atlas. Which one of those is supposed to be a simple space? As for explaining homeomorphism to a non-mathematician, what's wrong with the phrase "looks like"? -lethe ^{talk +} 21:02, 25 April 2006 (UTC)

To Lethe -- "looks like" might work. How about if we take out the "simple parts" (but that is echoed throughout the article, so it will require a major rewrite) and said every point on a one-manifold is in set that looks like part of a line, etc. Rick Norwood 21:11, 25 April 2006 (UTC)

To quote Santayana, "Those who cannot remember the past are condemned to repeat it." This entire discussion is madness; repetitive, unhelpful madness. The archives are packed with endless discussions, diverse viewpoints, valiant attempts. Many of those older attempts are far better than any of the current attempts. They capture the mathematics better. They communicate the essentials better. They fit the article better. This is not a wise pursuit. --KSmrq^T 23:53, 25 April 2006 (UTC)

I quite agree here. Maybe we should compile a list of all past intros, put it at the top of the talk page as a reference to all further discussions. That said heres another go, exploiting a sewing metaphore,

• A manifold is a mathematical generalisation of the idea of a patchwork quilt cover. Lots of individual patches are sewn together to form a curve, surface or higher dimensional object.

--Salix alba (talk) 00:32, 26 April 2006 (UTC)

You raise a valid point. Having perused the history I too suspect that there were past versions of the introduction that are more palatable than the recent attempts. I note your longstanding involvment in this endeavor. Is there a constructive way to deal with this difficulty? A silent revert war? Elect an executive editor? Review past efforts and choose the best available? Or declare 'manifold' to be a lost cause and move on? Rmilson 00:40, 26 April 2006 (UTC)

And what, pray tell, is an executive editor? -lethe ^{talk +} 02:20, 26 April 2006 (UTC)

I think a better question will be what is a non-executive editor? Loom91 08:03, 26 April 2006 (UTC)

An executive editor is a different approach to writing the intro. Something that may (or may not) help overcome the instability caused by the non-convergence of opinions. The elected individual

1. creates a sandbox with a proposed intro
2. receives comments and revises for a specified period of time - say a week
3. no further changes to the intro are permitted w/o prior discussion

Rmilson 10:25, 26 April 2006 (UTC)

Unnecessary, unacceptable, unwiki. If you wish to propose Wikipedia:mediation, I would think it premature.

• The sentence made up of simple parts is bad because it conveys 0 information. The next sentence says that the simple parts are euclidean spaces, and that is the information conveyed by the two sentences and therefore should be in the first sentence. The phrase simple parts conveys no relevant picture of the concept to a non-mathematician.

As for the elementary math part, I'm not particularly happy with it but added it to give an idea of what an Euclidean space was. Those who studied Euclidean space in high-school don't necessarily know that they are looking at an Euclidean space. Coordinate systems are not introduced before Analytic Geometry, and even then often in a limited way so relying on that exclusively in the intro makes the article inaccessible.

Let's consider this perspective: write for the potential FA reviewers, who are not all math graduates. Throwing unintuitive things like non-unique coordinate patches in their face will likely deter them from reading the rest. Loom91 08:03, 26 April 2006 (UTC)

Better would be to write for the readers; especially the reader who does not yet know what a manifold is. But then my opinion of the articles FA likes is usually low. Septentrionalis 21:39, 26 April 2006 (UTC)