Talk:Manifold

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Should easily make Good Article ifnominated. Tompw 17:03, 5 October 2006 (UTC)

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old: Talk:manifold/old, Talk:manifold/rewrite/freezer. Archive: Talk:Manifold/Archive1, Talk:Manifold/Archive2, Talk:Manifold/Archive3, Talk:Manifold/Archive4

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Contents 1 Strange visitor from another planet 2 Yet another attempt at an introduction which is both accurate and non-technical. 3 The role of Manifold in the cluster of manifold articles 4 Paragraphs three and four 5 The Circle 6 Conformal manifolds 7 Accessibility concerns 8 Merging from topological manifold 9 Manifold and Topological manifold 9.1 Questions 9.2 Discussion 10 Motivational examples? 11 orientability edits + wikiquette 11.1 Orientability (proposed revision) 12 Help at topological manifold 13 A question for KSmrq 14 On the circle motivational example 15 Gluing 16 Intersecting circles question 17 Impressed with this article 18 Misleading illustration?

[edit] Strange visitor from another planet

fights a neverending battle for truth, justice, and the Wikipedia way. The point being that the battle is indeed neverending.

Over in Mathematics Oleg Alexandrov simply reverts anything that is not clearly a major improvement. But for that to work, the article has to be in very good shape to begin with, which I don't think this article is, now.

Remembering the history of the article, a lot of good work was reverted by a single Wikipedian who found a single source that allowed manifolds to have more than one dimension. That is now reflected in a sentence somewhere in the article, and we are trying to struggle back to the light. Rick Norwood 21:04, 26 April 2006 (UTC)

What the.... are you suggesting that you'd like this article to only (or even mostly) treat 1-dimensional manifolds? 1-dimensional manifolds are pretty boring, lots of stuff that manifold theory is useful for is completely trivial in 1-dimension. I'd like to see you find even a single source that doesn't allow manifolds with more than one dimension. -lethe ^{talk +} 21:10, 26 April 2006 (UTC)

There has been a misunderstanding here. Rick meant that the source allowed a manifold to have charts of different dimensions instead of requiring all charts to be of the same dimension. Loom91 06:40, 27 April 2006 (UTC)

Oh, I see. Thank you for clearing that up. Now that I know what Rick's complain really was, I'll say that I don't see any problem allowing for the fact that we may or may not require the dimension to be constant. It doesn't really get in the way or anything does it? -lethe ^{talk +} 06:58, 27 April 2006 (UTC)

[edit] Yet another attempt at an introduction which is both accurate and non-technical.

I'm going to begin this time by saying what I think we all want to say, and then trying to say it in non-technical languages.

First: a manifold is, in the sense we are talking about, a mathematical object. Second: It is a space (whatever that means) Third: It is Hausdorff and Second Countable (I think we can leave that for later). Finally: Every point has a neighborhood homeomorphic to (looks like) an n-ball.

In other words, I think we all agree on the correct mathematical definition of an n-manifold and that it has nothing to do with sticking pieces together or with close-ups.

Here is my proposed introduction, incorporating the changes suggested by Lethe and Septentrionalis, and now incorporating changes suggested by Loom91:

A manifold is a mathematical space in which every point has a neighborhood which looks like Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. Lines are one-dimensional, planes two-dimensional.

In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and a pair of circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.

Comments? Rick Norwood 22:44, 26 April 2006 (UTC)

You say that we all agree that it has nothing to do with either sticking pieces together or with close-ups. I don't agree, I think that's the essence of the definition. On some level, you must disagree as well, because you use the word "locally" in your description, which is nothing but a technical word for "at small scales" or "close-up". Speaking of which, I think perhaps the word "locally" might be off-putting and could be replaced by "at small scales" or something. I think we can all agree that Hausdorff and countability don't belong in the intro, yes. Finally, homeomorphic to a sphere? Probably you mean ball? I think it's a bit odd to define a manifold as locally homeomorphic to a sphere, since a sphere is topologically nontrivial. A space locally homeomorphic to a sphere would look pretty weird. Maybe the Hawaiian rings would satisfy that definition though? -lethe ^{talk +} 21:54, 26 April 2006 (UTC)

I don't think the minimal set of charts, which is what "sticking things together" usually implies, is the essence of a manifold; rather the maximal set, which I think Rob is thinking of. How about an approach with:

every point has a neighborhood which looks like a Euclidean n-space....

Septentrionalis 22:19, 26 April 2006 (UTC)

(Edit conflict) As I've admonished repeatedly, novices understand the word "space" in a very different sense than the mathematical convention. In fact, manifolds themselves are a natural route to understanding the way mathematicians use the term. The concept of "dimension" is far too advanced and sophisticated to lead with. We can have infinite dimension, and also different dimensions depending on how we measure (hence fractals). A sphere is the boundary of a ball; some neighborhood of every point is homeomorphic to an open ball, not sphere. Omitting the importance of transition maps ignores the huge world of differential manifolds, for which the very definition relies on properties of the transitions — the "stitching", if you will.

And although it does not appear in the lead, let me once again insist that calling Earth's surface "flat" is a bizarre departure from reality. You people need to get outside more and look around (for most parts of the planet). :-D --KSmrq^T 23:46, 26 April 2006 (UTC)

There is a big difference between "locally" and "at small scales". If I spelled out what "locally" means, it means "in a neighborhood of every point". On the other hand I can't imagine what "at small scales" means, given that the open interval (0, 1) is homeomorphic to the entire real line. The absence of scale in topolotical spaces is one of their essential characteristics. Even if we limit ourselves to metric spaces, any constant multiple of a metric is again a metric.

Septentrionalis's suggestion is fine, to replace "locally" by "in a neighborhood", since the technical definition of neighborhood is very close to the non-technical meaning.

And of course I meant n-ball. My bad.

I'll make those two changes above, and this time remember to sign my proposal. Rick Norwood 22:44, 26 April 2006 (UTC)

You're right about the phrase "at small scales". It's too vague to have any meaning. The current solution looks good to me. Now let me say that I'm vaguely uneasy with the heavy reliance on dimension in the introduction, though I'm having trouble putting my finger on exactly why. Instead of "manifolds are usually described by their dimension", how would it be with "Like Euclidean space, manifolds have a notion of dimension." And after the surfaces, some mention that manifolds allow for arbitrary (and let's not address whether "arbitrary" includes infinite) dimension. If the first paragraph mentions only some examples of curves and surfaces, the reader may be left with an impression that manifold is just an inclusive word for curves and surfaces. I dunno.. what do you think? -lethe ^{talk +} 23:35, 26 April 2006 (UTC)

My reasoning on "dimension" is as follows. We need to give some examples, and we need to make clear the difference between a manifold and a CW complex. That difference is, essentially, that all parts of a connected manifold have the same dimension. While people may not know what "dimension" means in general, I think most people know the difference between a line and a plane. If we avoid the word "dimension", how to we get across the idea that a manifold can't start as a plane and then thin out into a line?

Objections:

1. Neighbourhood of a point does not convey any idea to a layman. We need to use simpler terms, like very small area around the point (ignoring conditions like open).
2. What is an Euclidean Space? As the concept of Euclidean Space is so fundamental to the concept of a manifold, we should at least give the reader an idea that we are talking about your ordinary geometry rather than depend on him learning it from the article on Euclidean Space.
3. "Manifolds are usually described by their dimension." Not true. Dimension is a property of a manifold, specifying the dimension does not specify the manifold.
4. It misses a sentence important for understanding, "but which may have a more complicated structure when viewed as a whole". This may be obvious to the mathematician, but not to the layman.
5. The example part should be in the second para like it is now.
6. The intrinsic view of "gluing together" should be mentioned in the first para, like it is now. We should also refer to non-uniqueness of a coordinate choice, as mentioned by someone along the line.

I also take it that you are proposing to change only the first para and not the entire lead. Incidentally, who first raised an objection to the lead as it was then? Loom91 06:54, 27 April 2006 (UTC)

I can live with the current intro, but please streamline the grammar in the first sentence! I also worry that Euclidean in the first sentence gives a misleading impression, but again, I can live with it. How about one of the following

• In mathematics, a manifold is an abstract mathematical space which has no distinguishing features when viewed up close, but which may have a complicated structure when viewed as a whole.

• In mathematics, a manifold is an abstract mathematical space which, in a close-up view, resembles a space governed by elementary geometry, but which may have a more complicated structure when viewed as a whole.

• User:MarSch's patchwork metaphor, mentioned above, is also a good first sentence.

Rmilson 13:24, 27 April 2006 (UTC)

As discussed above, I think the non-mathematical definition of neighborhood and the mathematical definition are close enough to convey understanding, whereas the idea of a "small area around a point" conveys the wrong impression, that is, that neighborhoods must be small.

Euclidean space is linked. I have no objection to explaining it here if (and this is why I didn't try) you can do so in no more than a sentence or two. If a person really has no idea what Euclidean space means, they can follow the link.

To say that manifolds are usually described by their dimension is not the same as suggesting that they are specified by their dimension. But any discussion of manifolds in mathematics usually begins by specifying the dimension. See, for example, 3-Manifolds by Hemple or Topology of 4-Manifolds by Freedman and Quinn.

I'll restore the mention of more complicated structure.

I'll put the examples in the second paragraph.

While some manifolds are constructed by "gluing", many are not.

I'm only working on the very beginning of the introduction because an attempt to do more was instantly reverted. However, I note that while I'm here trying to work with the other people interested in editing the article, many people have made major rewrites of the article without discussing them.

The problems with "close up" are discussed above. Objections to the "patchwork" idea are what started the current bout of revision. Rick Norwood 14:52, 27 April 2006 (UTC)

This discussion has rambled through so many suggestions and replies I'll just pick a spot and indentation to comment. First, this is explicitly a survey article. It must begin by speaking to a broad, general, untrained audience. It must include, not exclude, topological manifolds, differentiable manifolds, Banach manifolds, and so forth and so on. It must leave the details and implications of the specialities to the specialized articles. Such an article is surprisingly hard to write well, much harder than the specialty articles. The lead and introduction attract a disproportionate amount of attention, and thrashing rather than concensus. Every Johnny-come-lately editor (and I was once one) sees a problem and knows just how to fix it; and so does the next, and the next, and ….

Mathematicians see many definitions; precise postulates are our bread and butter. Yet as writers for the general public, it is clear that few mathematical editors appreciate the larger use of definitions, as covered by the Wikipedia definition article. Folks, it ain't easy! We would like someone who encounters the term "manifold" in who-knows-what context to be able to read the first paragraph and get the general idea. We must not depend on unfamiliar mathematical language, but rather on familiar common experience. We need not try to pack all the possibilities into a single sentence; in fact, that approach is doomed.

What should we mention about manifolds in the lead? I'd suggest we say something to indicate that our focus is not number theory or abstract algebra, but topology and/or geometry. We need to point out local uniformity (as a rule), global flexibility, and overlap consistency. This methodology is echoed in other constructs as well, including fiber bundles and schemes, but has important differences from the gluing of a CW complex.

And we need to realize that no matter how brilliantly we write, someone — now or later — will be dissatisfied. --KSmrq^T 21:14, 27 April 2006 (UTC)

I like your point of view about this article a lot. I'd like to see how that point of view translates into article space. Have you written a version of the intro that we can consider for inclusion? -lethe ^{talk +} 21:57, 27 April 2006 (UTC)

I'll wait for a few more comments, and then put the paragraph above in place of the one we've got now. Not everybody likes this one, but not everybody likes the old one, and while I try to establish a consensus, the article is rapidly changing.

On the subject of definition -- just how different the views of various people are on that subject was brought home to me when my son, a college student, reported the following exchange in an English class on critical thinking. He said something to the effect that a definition was arbitrary, and any definition would do, as long as everyone in the discussion agreed to observe it. The teacher was of a totally different view -- definitions are right or wrong, and if everyone agrees to a wrong definition, it is still wrong. I think you can guess which side of the argument I'm on. Rick Norwood 21:38, 27 April 2006 (UTC)

I think that prof's view is pretty much totally untenable. -lethe ^{talk +} 21:51, 27 April 2006 (UTC)

Several hours have gone by without an objection, so I'm going to replace the current intro. If you want changes, please move forward rather than back. Rick Norwood 00:28, 28 April 2006 (UTC)

I made one change in the intro above. I replaced "looks like" with "resembles" because it seemed to improve the flow. Rick Norwood 00:40, 28 April 2006 (UTC)

[edit] The role of Manifold in the cluster of manifold articles

At the moment there are articles Manifold, Topological manifold, Complex manifold, Differential manifold Symplectic manifold, Riemannian manifold, Pseudo-Riemannian manifold, Kähler manifold and Calabi-Yau manifold. Clearly the last eight are about precise mathematical concepts and this one is a more general article. Shouldn't the main focus here be (1) an intuitive overview and history (well-addressed at present) and (2) a comparison of the different concepts and their interrelationship. If so, is there an argument for rearranging material (and avoiding sections dealing with exactly the same things)? For example, this article defines a manifold using the topological manifold definition, which is both doing something that is done in the more specialised article and providing only part of the truth for the other types of manifold (like the relevance of the topology of the plane to Euclidean geometry). The general description of atlases is much more appropriate though, but could do with more emphasis on the fact that for each class of manifolds we demand different compatibility conditions. Any views? Elroch 15:55, 27 April 2006 (UTC)

What you propose sounds good to me. I wasn't aware of the article topological manifold. A quick look suggests that this article and that article should be merged. Rick Norwood 17:35, 27 April 2006 (UTC)

I may be exposing my biases here, but it's my opinion that if one of those articles were to be merged with this, it should be differential manifold, not topological manifold. -lethe ^{talk +} 20:04, 27 April 2006 (UTC)

No, a merger is wrong. This article was explicitly created and designed as an overview, while an article on topological manifolds needs to focus on features specific to topology, ignoring all the variant structures of, say, Milnor spheres. Similar remarks apply to each of the specializations. --KSmrq^T 20:13, 27 April 2006 (UTC)

Point taken. Still, there is a lot of repetition. Given that all those articles exist, we ought to use them for the more technical aspects of this long article. Rick Norwood 21:40, 27 April 2006 (UTC)

Yeah, no merger. We'd have to have a big fight to see whether topological manifolds are "the most important" class of manifolds or differential manifolds are. And what fun would that be? And of course, this article would have to become far more technical, whichever choice we made. On a related note, do you guys consider differential structures to be "geometric" or "topological"? I lean towards the latter, and consequently feel comfortable calling the the exterior derivative a concept of differential topology (rather than differential geometry), but I'd like to hear other people's thoughts (over at talk:exterior derivative, where rmilson brought it up). -lethe ^{talk +} 21:50, 27 April 2006 (UTC)

Between differentiable manifolds and topological manifolds, it is not a question of which is more important but which is simpler. As for whether differentiable structures are geometric or topological, I would say, neither. I'd go with analytic. This is how I divy up pure math. It all starts with numbers and shapes. Out of that comes algebra and geometry. Combine the two to get analytic geometry. Throw limits into the mix to get calculus. Extend algebra to abstract algebra, extend geometry to topology, extend calculus to real and complex analysis. Combine abstract algebra and topology to get algebraic topology. Combine abstract algebra and analysis to get Frechet spaces, Minkowski spaces, Hilbert spaces and the like. Combine topology and analysis to get differentiable manifolds. Put into a large bowl, toss well, and out comes category theory. Look at them through a microscope and find set theory. Logic is the dressing that keeps it all from falling apart. Rick Norwood 22:53, 27 April 2006 (UTC)

I disagree with a lot of what you say.

1. What the article manifold is about is dictated by our naming conventions. What is the most common/important usage of the word, and what is the least ambiguous. Those are the two rules for naming articles, which are sometimes in contradiction with each other. The first rule says this should be about differential manifolds (if you think that's the most common/important class of manifold, as I do). The second convention says that this article should be about all manifolds (since that's the least ambiguous interpretation of the word manifold). We have no naming convention which says that our article should be about the mathematical object with the smallest structure.
2. I think limits give you topology, not calculus. For calculus, you need derivatives or integrals (that is, you need a differential structure). Lots of places have limits which I would not call calculus
3. Extend geometry to topology? You think topology is an extension of geometry? Even though topological spaces have smaller structures than (many) geometric spaces? This seems to be in contradiction with, for example, your position on the naming. And in any case, I can't support viewing topology as an extension of geometry, just because you take the course in topology after you take the course in geometry.
4. Combining algebra and topology ought to give you topological groups, topological rings, etc. Algebraic topology is not really a combination of the two, even though you need material from both your topology course and algebra course.
5. Do you have a different definition of Minkowski space than I do? For me, Minkowski space is a finite dim vector space with a non-definite inner product. A decidedly geometric place, for doing relativity theory.
6. About Frechet space and Hilbert space, I divide my TVSes into abstract TVSes and concrete functional spaces. The functional spaces are what are studied in analysis (which is, after all, the study of real or complex numbers and functions). Lebesgue space, Hardy space, Sobolev space, etc
7. Therefore I view abstract TVSes like Hilbert space, Banach space, Frechet space as objects in the coincidence of algebra and topology (which, as I said, is not algebraic topology). They are topological groups. Of course, there is necessity to know about these guys in order to do analysis in an abstract way.

on the other hand, I might agree with some extended version of your description of category theory. I might also agree with your description of set theory if you had said "some choice of microscope". There are other choices. -lethe ^{talk +} 23:22, 27 April 2006 (UTC)

The sort of picture I had was to introduce some definitions which have a higher level of abstraction (eg an abstract notion of compatibility of charts which would, hopefully, be applicable to all the types of manifolds which a specific choice of the compatibility condition). Also, it would be nice to describe the tree of types of manifolds with increasing amounts of structure in a unified way, perhaps with a nice diagram. Another issue might be to cover the ways in which different types of manifold can be given more structure in ways which may, or may not, be unique to within isomorphism. This is rather a lot, but I am sure there are several who know parts of this picture very well. With regard to the difference between differential geometry and differential topology, perhaps it doesn't matter, as long as one agrees on the axioms. :-) Elroch 23:04, 27 April 2006 (UTC)

I was just riffing on the idea of structure, in response to your question of whether differentiable manifolds were geometric or topological. It's fun to talk about, but not relevent to this article. So, in the spirit of fun, I'll respond to your comments, after an

I agree that this is fun. A lot more fun that actually trying to write the article, that's hard work! This talk page is swamped with miles of comments which are mostly not helping the article and are pissing off KSmrq, so I feel a little bit bad about prompting this series of digressions. Oh well. -lethe ^{talk +} 00:58, 28 April 2006 (UTC)

((off topic)) warning to people with better things to do with their time. Historically, the application of limits to calculus and the application of calculus to limits came long before the invention of topology. Essentially, calculus is the study of three linear operators, the limit, the derivative, and the integral. Those operators were invented in the reverse of that order, but the limit comes first logically. Topology is an extension of geometry in the direction of abstraction. Both deal with sets of points, and many of the examples from topology arise from geometry, just as abstract algebra is an extension of algebra in the direction of (what else) abstraction. And many of the examples of abstract algebra rise from the real numbers. And, of course, out of the combination of topology and abstract algebra, you get both topological algebra and algebraic topology. Funny but true story: when I was doing the paperwork for my Ph.D. an administrator in the registrar's office didn't want to give me credit for both of those two course because to him they sounded the same.

((back on topic)) Anyway, the important point is this: the reason for starting with a simple structure is that this seems to be a portal page, and portals should be widely accessable. As for which kinds of manifolds are "most important", it seems clear to me that the answer is that everyone thinks the kind of manifolds he or she works with are most important. I'm in knot theory, work almost entirely with 3-manifolds, and almost never use differentiable structure. But I am aware of the importance of other kinds of manifolds. Rick Norwood 00:23, 28 April 2006 (UTC)

Well, as a portal page, I might be able to get on board your argument. Give some prominence to the (layman's description of the) topological manifold. But I shouldn't like to see other classes of manifolds fall too far behind in their consideration in this article. -lethe ^{talk +} 00:58, 28 April 2006 (UTC)

RN: Thanks to Elroch for putting back the links, which evidently "cut and paste" removes. And I agree with Lethe that differentiable manifolds, at least, need to be high in the article. Rick Norwood 12:28, 28 April 2006 (UTC)

[edit] Paragraphs three and four

Paragraph three seems fine to me as is.

I would like to rephrase paragraph four in paragraph form rather than in bullet point form. The Wiki style sheet does not forbid bullet points, but frowns on them. Here is how paragraph four reads now:

"Additional structures are often defined on manifolds. Examples of manifolds with additional structure include:-

• differentiable manifolds on which one can do calculus
• Riemannian manifolds on which distances and angles can be defined
• symplectic manifolds which serve as the phase space in classical mechanics
• the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity."

I suggest:

"Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity."

No change in the words (except the addition of "and") only in the punctuation.

Comments? Rick Norwood 00:45, 28 April 2006 (UTC)

Oh definitely. This should be not controversial, just go ahead and do it. I would probably support some cleaning up of the language too, what do you think? Perhaps conflating pseudo-Riemannian structures with Riemannian, and mentioning GR along with Riemann? -lethe ^{talk +} 00:48, 28 April 2006 (UTC)

I'm going to take your advice and do it, and you can fix the Riemannian stuff -- for me, Riemannian manifolds, Bozhe moi! . Rick Norwood 00:52, 28 April 2006 (UTC)

The wikipedia guidance against bullet points seems excessively pedantic. The justification for this guidance is that it follows normal practice in paper encyclopedias, where a bulleted list uses more paper. On a web page it has no such disadvantage and makes the separation between items in the list much clearer to the reader. I am strongly in favour of the use of bulleted lists rather than free format text, simply on the basis that they are definitely more effective at getting the information across to the reader. Elroch 11:51, 28 April 2006 (UTC)

Hmm.. I guess I was wrong about this being non-controversial. Well, anyway, I didn't like the list in this instance, although I'm not opposed to using bulleted lists in general (check out the article Locally convex topological vector space, which I wrote, where I use itemized lists throughout the entire article. I think I went overboard.) -lethe ^{talk +} 12:09, 28 April 2006 (UTC)

I agree that there is a place for bullet points, but that they are not needed in this paragraph. Rick Norwood 12:29, 28 April 2006 (UTC)

[edit] The Circle

I've been reading the next section, The Circle, in light of the following reviewer's comment:

"a lot of complex mathematical terminology is not even linked to, never mind explained."

It seems to me that, at least in this section, the links have been provided now, and the pictures are pretty, but I have to wonder who the section is intended to address. If the audicence has, say, a working knowledge of analytic geometry, then the exposition is elementary, but what will "projection onto the first co-ordinate" mean to a majority of readers?

not much; and it should be possible to explain the idea without the vocabulary. Septentrionalis 22:12, 28 April 2006 (UTC)

The article has to become technical at some point, but is this too soon?

Comments? Rick Norwood 01:14, 28 April 2006 (UTC)

• I don't think it is possible to treat Manifolds without assuming at least Analytic Geometry after the lead. Analytic Geometry is high-school math, so we should be fine unless we are facing primary school students or adults who have managed to completely forget their high-school math, and I don't think we need to consider those readerships beyond the lead. Loom91 08:17, 28 April 2006 (UTC)

Many well known people have embraced the quote, "There is no such thing as algebra after high school," including people who should no better. Rick Norwood 12:31, 28 April 2006 (UTC)

• The arcs of the circle are described as semicircles, but not drawn as such. shouldn't they each be 180 degrees, or am I missing something? Ojcit 19:13, 29 September 2006 (UTC)

Yes, you're right, the picture doesn't match the text. The yellow arc is certainly not the entire top half of the circle. Michael Kinyon 19:18, 29 September 2006 (UTC)

No, you're both wrong, though the mistake is natural and we've discussed the relationship between language and figure before. While it is true that the yellow arc itself is not a semicircle, its horizontal extent is exactly half the circle, as can be seen from the projection. That is, the yellow arc covers the top half of the circle, but obviously is not even part of the circle. If the yellow arc were actually drawn as a full semicircle, it would be huge, cumbersome, and less helpful. How do we know? We tried.

I would support rewording "(the yellow part in Figure 1)" to read "(indicated by the yellow arc in Figure 1)". On the other hand, from a topological point of view it is completely irrelevant whether we use the full top half or just enough to have an open set in common with its neighboring charts. Therefore we could refer to "arcs" rather than "semicircles".

Personally, I trust readers to be able to absorb the concept discussed and illustrated in this introductory section without the encumbrance of revised language. So far, my trust has been rewarded. --KSmrq^T 00:12, 30 September 2006 (UTC)

Well, in my case, you're preaching to the choir. I know what is and isn't relevant topologically, and I know that the yellow arc is not intended to be part of the circle. All I said is that the figure doesn't match the text, and indeed, it doesn't. It says that the yellow part is the upper half. Instead of "indicated" how about "(covered by the yellow arc in Figure 1)"? It's actually closer to what is meant while keeping the informal tone of the section.

Regarding the accuracy of the figure (which I concede is not as important as its pedagogical utility), it actually took me a few seconds of staring to see that the yellow and red arcs do, in fact, "line up" correctly with the circle. The reason I did not realize it at first is that it is immediate to the eye that the blue and green arcs do not line up correctly; they and their corresponding vertical line segments are too short. I would suspect that lengthening them slightly would be too difficult at this stage of the game.

Your trust, incidentally, is quite heartening. Michael Kinyon 12:34, 30 September 2006 (UTC)

I'm a little uncomfortable with "covers", partly because it has a technical meaning in topology (as in “covering map”), and partly because it works less well for the other charts.

As for trust, in the end we have no choice. Until we develop a technology that can transfer understanding directly from one brain to another, learning will remain an active process for the reader. We cannot do all their thinking for them, and it is counterproductive to try. Good writing does not just instruct readers, it engages them.

In fact, when I give a lecture I often like to leave an open question or two, to engage the audience. The kind of people who like to learn and do research find it irresistible. Likewise, a wise author ends a paper with suggestions for further research, partly because that will lead to more interest and more citations! (A famous example is Hilbert's problems.) --KSmrq^T 20:49, 30 September 2006 (UTC)

Good point about "covers". It's informally correct, but could be confused with its technical meaning. Maybe your suggested rewording is the way to go.

I don't think anyone would disagree with your platitude about what constitutes good writing. (Yes, I given open questions in talks and papers, too.) It's simply that in the case of the sentence over which we're splitting hairs, we have writing which is neither instructive nor engaging. The rest of the section is fine.

By the way, the hairs I'm splitting here are grey, not yellow, red, green, or blue. :-) Michael Kinyon 21:46, 30 September 2006 (UTC)

[edit] Conformal manifolds

Looking at the introduction again, it struck me that there is an obvious class of manifolds which isn't mentioned, namely those where angles are meaningful but distances aren't, which I would call "conformal manifolds" (it appears others agree, but this seems to be given only limited coverage on wikipedia, in the article Conformal geometry). In the 2-dimensional real case, these would be the Riemann surfaces, with holomorphic transition maps, but in higher real dimensions the transition maps would need to be higher dimensional conformal maps (just Mõbius maps, I think). Elroch 12:32, 28 April 2006 (UTC)

I'm familiar with conformal transformations on Riemannian manifolds, but what exactly is a conformal manifold? Can the notion of conformal be defined without a metric? Edit: looking at the article, I see that it's an equivalence class of metrics. Makes sense. -lethe ^{talk +} 12:55, 28 April 2006 (UTC)

I believe a conformal manifold can be defined directly by saying it is a manifold where the transition maps are conformal maps between open subsets of Rⁿ, as this will ensure angles are well-defined. The extra structure of tangent bundles on Riemannian manifolds is not really necessary, and is a bit like defining a topological manfold starting from a differentiable manifold.

But you need an (equivalence class of) inner product on Rⁿ to define conformal maps there, which means a (n equivalence class of) metrics on the manifold. -lethe ^{talk +} 14:06, 28 April 2006 (UTC)

On a little further reflection, the issue is whether the very limited conformal maps available for n>2 mean that there is some way of getting a unique metric structure on a conformal manifold, which would justify not treating them as a special class (except in dimension 2) Elroch 13:16, 28 April 2006 (UTC)

Yes, maybe that's the right way to think of it. A Riemannian manifold is a manifold with a reduction of the structure group of the frame bundle from GL(n) to O(n) while the conformal manifold ought to be a reduction to the conformal group. -lethe ^{talk +} 14:18, 28 April 2006 (UTC)

There's a couple of other ways to describe conformal structure. A common approach is to say that a conformal structure is an equivalence class of Riemannian structures: two metric tensors are judged equivalent if they differ by scaling factor. One can also employ a Cartan connection. Conformal structure is a well developed topic in differetnial geometry and is of considerable interest in GR. Kobayashi's Transformation Groups in Differential Geometry is a reference. Rmilson 14:27, 28 April 2006 (UTC)

Thanks, Rmilson. So basically, the metric (to within a scaling factor) comes "free" with a general conformal manifold, just like in the case of Riemann surfaces, despite the much smaller conformal group? Might be worth noting somewhere. Elroch 15:34, 28 April 2006 (UTC)

Of course the smaller group makes it easier, not harder, and the key thing is that all conformal maps are very nearly orthogonal maps near any point. Elroch 15:38, 28 April 2006 (UTC)

The comment about the metric coming 'almost for free' is spot on. The issue of the group is trickier, and I don't think the comments to date get it quite right. First of all, the idea of getting conformal structure by reduction of structure of the frame bundle doesn't work. The problem is that the group of conformal transformations of 'n-dimensional space' is larger than GL_n, not a subgroup of GL_n. Likewise conformal n-space is not just R^n, but rather R^n + one point at infinity --- topologically S^n. Usually the way one gets CO(n) (the conformal group) acting on conformal space is to have the proper Lorentz group, the connected component of SO(n+1,1), act on the projectivized light cone, PK^n. To be precise, we start with SO(n+1,1) acting on n+2 dimensional Minkowski space, then we restrict to the action on the n+1 dimensional lightcone K, and then we projectivize to get the action of SO(n+1,1) on the space of null lines, thats PK^n. For example, CO(2) is a subgroup of SO(3,1), and 2-dimensional conformal space is S^2, the celestial sphere of past (or future or both) directions. So one can't do conformal geometry by using G-structures. One really needs a Cartan connection, which is a slightly different gadget.

Ah. Well, not larger than GL(n) (there are linear maps which are not conformal), but not contained in GL(n) either (since there are conformal maps which are not linear). So we shouldn't expect to get a vector bundle at all. Thank you, that cleared up some of these things a bit for me. -lethe ^{talk +} 22:13, 28 April 2006 (UTC)

A final remark about a potential pitfall in all this. I think there was some mention of the Liouville theorem to the effect that conformal transformations (when one considers such things locally or infinitesimally) of n-dimensional space for n>2 form a finite-dimensional Lie group. However the relationship between this group and a general (curved) conformal structure is rather subtle. It's analogous to the relationship between Euclidean transformations and Riemannian geometry. Sharpe's book on Cartan geometries is a reference. Cartan's own writing is also a great resource. The notation is archaic, but he had a certain eloquence and was good at communicating geometric ideas and motivation. A useful resource in this regard is here [1] Rmilson 17:10, 28 April 2006 (UTC)

It's probably worth noting that to define a conformal manifold in the direct way I mentioned, the transition maps are between open subsets of Rⁿ, so there's no need to deal with the 1-point compactification. It's my understanding that just like in differential geometry, one can construct the manifold using an atlas, from which all the bundles arise naturally, as a convenient alternative to defining the manifold using bundles. (I still have Graham Allen's notes on this somewhere around). With regard to Lethe's comment about the metric coming from the inner product on the open subsets of Rⁿ, this isn't quite trivial, as conformal maps do not preserve distances. Very close to a point, however, they almost do, which allows the extension of a concept of distance from near a point to the whole space, intuitively by tiling it with very small tiles on which distances are well-defined. Elroch 20:04, 28 April 2006 (UTC)

P.S. it's not trivial that this is well defined when you get to a point using two different routes, but it can be proved for Riemann surfaces: anyone know this result for a higher dimension conformal manifolds? Elroch 20:08, 28 April 2006 (UTC)

I believe that the construction you describe will yield the trivial conformal geometry. In 2D, the group of local conformal transformation is much richer (holomorphic functions) than the group of global conformal transformations (Mobius xforms). In 3d and higher, all local conformal transformations are restrictions of global ones -- that's Liouville's theorem. Rmilson 20:45, 28 April 2006 (UTC)

Well that's what comes of extrapolating what I half recall about Riemann surfaces and differential geometry with a bit of guesswork. I should defnitely brush up on this stuff. Elroch 22:02, 28 April 2006 (UTC)

[edit] Accessibility concerns

There has for long been a tug-of-war over this article concerning accessibility to lay person versus technical accuracy. Part of the reason is that manifolds are one of the most important concepts in mathematics and it is quite conceivable that laypersons may wish to find out about it. It is worthwhile to note that three other such technical subjects that are of great interest to laypersons have come up with a novel solution.

Quantum mechanics, Special relativity and General relativity all provide a non-technical introduction to the subject in the form of a separate companion article, often called a trampoline article. This way, the article can stay focused on providing a formal encyclopedic approach to the matter and outsource accessibility issues.

Now I know that not all agree with having such articles on Wikipedia, but those who do may consider writing such a companion article to Manifold that will provide an elementary approach based on high-school math and geared towards getting the point across rather than provide technical definitions. Such a move will hopefully relax some of the drawn-out edit wars over this article. For an idea of how such an article might be presented, visits to Introduction to quantum mechanics and Introduction to special relativity may prove fruitful.

Currently all such articles belong to physics, it won't be bad for the other great branch of Sciences, Mathematics to have a few under its belt also. And what better place to begin than the frightfully complicated yet phenomenally important concept of manifold? Loom91 11:31, 1 May 2006 (UTC)

A while ago Manifold was split up exactly with the intent of making it a very accessible article. Split off were for example topological manifold and differentiable manifold. Colloqually those are also called manifold. Now that Rick Norwood has also finally discovered the existence of topological manifold perhaps we can all agree once again that this article is not about a specific mathematically well-defined object, but about the underlying idea or the red line running through all different kinds of manifold. --MarSch 14:17, 1 May 2006 (UTC)

Excellent suggestion, in my opinion. This should stop the see-saw of accessibility versus mathematical precision. The slightly painful question is how much of the current article is inappropriate to an informal introduction to manifolds? Also, I think it is quite important that a more mathematically sophisticate reader who wanders in here finds good directions to the more mathematical articles (of which there are quite a large number - I found some more on specialised types of manifold after I listed the ones I already knew of somewhere above). Elroch 16:37, 1 May 2006 (UTC)

This is the trampoline article. Also, MarSch and I continue to point out difficulties with any single definition of manifold that attempts to encompass all the variations simultaneously. It is impossible; nor is this unusual in definitions, as any dictionary will reveal. We want to convey the general idea, some suggestive examples, a few specific definitions, and lots of links. I think I'm repeating myself. --KSmrq^T 19:27, 1 May 2006 (UTC)

For the record, I do not support the splitting into introduction to manifolds. I would also excise those physics articles, if I had my way. -lethe ^{talk +} 16:50, 1 May 2006 (UTC)

I agree with MarSch, this is in effect the trampoline article, and there is no need for a new introduction to manifolds article. There could be a case for putting a statement at the top saying

this is a non-technical overview of the concept of a manifold, for technical descriptions see the articles on the sub types: topological manifold, differential manifold.

I had a quick look at Introduction to quantum mechanics and in many ways it is superior to quantum mechanics. This is reflected in the poor state of the more technical manifold articles, which could do with some attention. --Salix alba (talk) 19:57, 1 May 2006 (UTC)

Actually, from a mathematical point of view, I don't think the idea of defining a "general" manifold is impossible, and I am sure someone must have done this some time. A general definition could be based on an abstract set of continuous maps S from open sets of a vector space to open sets of a vector space (S is going to be the transition maps) with some appropriate properties (for example: the restriction of a map in S to a smaller open set is in S; all the maps are invertible, and the inverse of every map in S is in S). However, although such a definition seems highly appropriate for an article which is about all types of manifold, and could be made to cover most types of manifold, it is rather abstract for a non-mathematical readership. Elroch 23:00, 1 May 2006 (UTC)

I've been thinking about defining a general manifold too :) and I'm pretty sure it should involve category theory ;)

Let C be a category, let {A, B} ⊂ C objects and m a morphism from A to B. A gluing is a tuple (A, m, B) where m is an isomorphism. A C-manifold is an (equivalence class) of a set of gluings. Of course the difficulty is in defining the equivalence relation, but then this def. easily encompasses top. and diff. manifold, by taking the category to be that of finite dimensional vector spaces with continuous or diff. morphisms. --MarSch 08:28, 2 May 2006 (UTC)

Of course the isomorphism m should be between subobjects of A and B otherwise A and B would be isomorphic.--MarSch 10:35, 2 May 2006 (UTC)

I was thinking of the categorical viewpoint as well. The type of category called a groupoid could well have been designed expressly for this purpose. So, we say a class of manifolds is defined by atlases with transition maps in a particular groupoid G of maps between open sets of a topological vector space (might as well be as general as reasonable). It might be handy to add on conditions like the restriction of a map to a smaller open set is also in G. What other conditions would be useful? Elroch 18:25, 2 May 2006 (UTC)

The article states that "All manifolds are topological manifolds", so it seems to me a definition of a topological manifold will define all manifolds. Are there any manifolds that do not satisfy the definition of topological manifolds? And if it is indeed true that all manifolds are topological manifolds, what is topological manifold doing in an article of its own? Loom91 11:04, 2 May 2006 (UTC)

If your definition of manifold includes the fact that it be a topological manifold, then all manifolds are topological manifolds. By definition. It's got an article of its own because this article is trying to be less technical and more intuitive, while that article deals with the strictly topological properties of manifolds on a technical level. -lethe ^{talk +} 11:56, 2 May 2006 (UTC)

In that case we have a serious misnaming issue here! If topological manifolds are the same thing as manifolds, then the article on toplogical manifold with its technical details should be moved to Manifold (naming policy suggesting the more common name to be adopted when two names refer to the same thing) and any attempts to present an intutive approach in a separate article should go at Introduction to manifold. If you don't like introduction forks, I don't see why you like misnamed introduction forks. It seems to me to be seriously misleading readers to have two separate articles on the same topic without explaining the fact. Loom91 12:35, 2 May 2006 (UTC)

Topological manifolds are not the same thing as manifolds, vector spaces are not the same thing as modules, and groups are not the same thing as monoids. Nevertheless, it happens to be the case that manifolds are also topological manifolds, vector spaces are also modules, and groups are also monoids. -lethe ^{talk +} 13:37, 2 May 2006 (UTC)

• I took a look at the topological manifold article and this article, and it seems that this article mentiones numerous times directly or indirectly that all manifolds are topological (which also seems intuitive to me) and this is reinforced by the topological manifold article, which seems to be an inferior version of this article. I think we have a large-scale double article scandal here and we either need to do an immediate merge or clarify that it is possible for some manifolds not to be topological. I can't see how the latter can be done, as manifold and topological manifold seems to have the exact same definition. If the concern here is non-technical versus technical approach, then we have to make it clear thrugh proper naming and use of templates ({{introduction|Manifold}} and {{seeintro}}). Whatever it is, manifold is a high-quality article and topological manifold is a resonably large one, this pathological situation must be taken care of at once. Loom91 12:48, 2 May 2006 (UTC)

I went through the talk page of topological manifold and it seemed to have been created explicitly as a fork to contain technical details. This is something new, the forking is usually done the other way. If that is the wish of the editors, the whole situation is still in a royal mess. It's the article on Manifold that should contain any gory technical details as par encyclopaedia convention, it's the educationalist approach that needs to be moved. Not to mention the whole thing needs to be made clear to the reader at the beginning as is done in other article/trampoline pairs such as Quantum mechanics/Introduction to quantum mechanics. There appears to have been some serious communication gaps here and I think it's time for damage control.Loom91 13:08, 2 May 2006 (UTC)

All manifolds are topological. That is, all manifold definitions mention open sets, and open sets are elements of some topology. Also, the word manifold is well and carefully defined in numerous books, and I have cited several book titles that use "manifold" without any adjective except in some cases a dimension, e.g. Hempel's 3-Manifolds and Spivak's "Calculus on Manifolds". The topological manifold article was created in 2002 by Toby Bartels. There were no further edits until June 2005, when MarSch copied a large chunk of this article over there.

If there is a place for an article topological manifolds, it should contain such subjects as connectedness, compactness, and why manifolds should be first countable and Haussdorff, material too technical for inclusion here. Rick Norwood 13:27, 2 May 2006 (UTC)

I think the currect division is good, and do not agree with Loom. -lethe ^{talk +} 13:46, 2 May 2006 (UTC)

How do you see the role of the two articles, Lethe? Which should be broad and introductory, which more specialized and technical? Rick Norwood 15:05, 2 May 2006 (UTC)

Manifold as the less technical article, topological manifold and differential manifold and other specialized classes can be more technical. -lethe ^{talk +} 15:19, 2 May 2006 (UTC)

Sounds good to me. Rick Norwood 21:43, 2 May 2006 (UTC)

[edit] Merging from topological manifold

First a note. Loom91, when you post a merge tag you should explain on talk why the article should be merged, I mean you should have some reasons for that, right?

I disagree with a merger. The manifold article is well-written, the product of a lot of good work, and doing a merger will just force us to reshuffle everything. Also, we should have a separte article on topological manifolds, since the subject is important enough, and besides putting that topological stuff over here will make this article more complicated.

I suggest that a lot of the energy used on arguing at this article be better used in improving topological manifold and differentiable manifold. Oleg Alexandrov (talk) 15:29, 2 May 2006 (UTC)

Object to merger. I'm pleased to agree with Oleg. I'm especially pleased that this manifold article, after a long evolution, is seen as a good article. After much wrangling over the intro, the body finally got some of the attention it needed. Now, as Oleg says, we need to turn our attention to the specialized articles and bring them up to a higher standard.

When we discuss a topological manifold, we have topological concerns. When we discuss a Banach manifold, we have very different concerns. And the folks who spend their careers working with differentiable manifolds tend to see topological manifolds as an inferior mutant lacking essential features. Insisting that manifold means topological manifold risks the uproar of an angry crowd. It's just not so. --KSmrq^T 21:07, 2 May 2006 (UTC)

Yes, Loom91, I think KSmrq has got it exactly right with his last sentence. Just because all manifolds are examples of topological manifolds does not mean that the phrase "toplogical manifold" and "manifold" are synonymous. Manifolds can be examples of topological manifolds with also many other interesting things happening. Therefore the merger is inappropriate. -lethe ^{talk +} 23:55, 2 May 2006 (UTC)

[edit] Manifold and Topological manifold

Let's structure this so we can make better progress.

[edit] Questions

• Are all manifolds topological manifolds and all topological manifolds manifolds?

If yes, should there be two different articles on the same topic under different names?

If no, should the distinction be made clear in the articles instead of saying/implying manifolds are the same things as topological manifolds?

[edit] Discussion

I don't have the technical qualifications to answer the first question, but I will answer no and yes to the second and third questions (there should not be two articles on the same topic, which is obvious, and if a distinction exists it should be underlined). If the goal is to create one non-technical and one technical article, then the naming is HIGHLY misleading. The naming should reflect the nature of these articles in that case. Either all manifolds are topological manifolds and vice-versa, or there exists a distinction between the two. Either way the current state of affairs can not be satisfactory. What if Science (journal) decides to survey the quality of Wikipedia's math articles and comes upon this situation? Loom91 07:17, 3 May 2006 (UTC)

It's true that all topological manifolds are manifolds and all manifolds are topological manifolds. It's simply the definition of manifold. Nevertheless, the two phrases do not connote the same topic. The point of the article topological manifold is to focus on those properties of manifolds which do not depend on other kinds of manifolds (which are topological manifolds but also more). I think we're all decided that there is a need for two separate articles. Now we just need someone to fix the article topological manifold to make it clear what its purview is. -lethe ^{talk +} 07:57, 3 May 2006 (UTC)

It's true. The intro makes it seem like it is exactly the same topic as this. I rewrote the intro to make it clear that a more technical and specific topic is intended. Now we just need to find someone to rewrite the rest of the article to agree with the intro. -lethe ^{talk +} 08:44, 3 May 2006 (UTC)

Some Banach manifolds are _NOT_ topological manifolds. Specifically the infinite dimensional ones. Differentiable manifolds may have an underlying topological manifold structure, but that does not mean they _ARE_ topological manifolds. There is no one-to-one correspondence between them. In the category of topological manifolds it is not possible to identify the subcategory of differentiable manifolds. This article is not an introduction to topological manifold and/or differentiable manifold, it is an overview article for _ALL_ different manifold types.--MarSch 11:32, 3 May 2006 (UTC)

I want to disagree with your comment here, but before I do, let me say that I think we're all in agreement on this point: this article is an overview of all objects that can be called manifolds of some kind. Also, while I don't think we agree with Loom about changing to a dismbig or merging, I do agree with him that the present state of the article topological manifold leaves the matter in a rather unacceptable state (but see my recent edit.

I will concede the point that Banach manifolds are not manifolds according to the standard definition, but I would put forth the suggestion that this is more a matter of taste than anything else. We could easily adopt a convention for the word "manifold" that allows for Banach manifolds or manifolds modeled on any TVS, and I think (?) I've seen authors that do this.

That said, I would like to return to a subject we debated a few days ago: are differential manifolds special examples of topological manifolds, or are they topological manifolds with more structure. Do differential manifolds constitute a subcategory of of topological manifolds? I don't understand your point about one-to-one correspondence. There isn't a one-to-one corrrespondence between groups and monoids. I consider every group to be a monoid. Do you not? Or am I missing your point? -lethe ^{talk +} 12:17, 3 May 2006 (UTC)

The difference between the case of groups and monoids seems rather important: a group is simply a monoid that satisfies another axiom, but a topological manifold defined by a maximal atlas, or equivalence class of altases (which seems the best definition), if it has one differentiable structure, has an infinite number of incompatible differentiable structures which give isomorphic differentiable manifolds, (and in some cases many non-isomorphic structures). The only way to argue that differentiable manifolds are special types of topological manifolds is to say two topological manifolds are only the same if they have exactly the same (non-maximal) atlas, and then say if the transition maps happen to have the right differentiable properties, it is a differentiable manifold. Of course, one can talk about the class of topological manifolds that have some differentiable structure, but there is no natural identification between the topological manifold and a differentiable manifolds, and in some instances the differentiable structure isn't even unique to within isomorphism.

The category of topological manifolds each with a maximal atlas (or equivalence class of compatible atlases) does not have a subcategory of differentiable manifolds. The category of topological manifolds with some specific (non-maximal) atlas does have a subcategory of differentiable manifolds using a similar definition. Elroch 16:56, 3 May 2006 (UTC)

On reflection, despite my prejudice towards the "equivalence class of atlases" definition, the obvious conclusion is that the "specific atlas" definition of a manifold is preferable, and makes the relationship of topological and differentiable manifolds very similar to that of monoids and groups. Elroch 21:36, 3 May 2006 (UTC)

I guess it's pretty easy to see that if the definition of manifold were "space with atlas", then differentiability would be just another axiom for an atlas to satisfy. I made that precise point earlier in this thread. Now I guess I need to figure out what happens when you do a maximal atlas or equivalence class of atlases. So like the real line has one chart given by the identity and another chart given by x³. As topological manifolds, the two charts are equivalent. As differential manifolds, they are not. Nevertheless, there is a differential structure for (containing) the latter chart (under which it is isomorphic to R). -lethe ^{talk +} 02:23, 4 May 2006 (UTC)

The criterion for whether a differential manifold is a special class of topological manifolds is that the functor from Smooth --> TopMan be fully faithful. I think the example I gave above (standard real line versus real line with x³) shows that this functor is not full; no morphism between the two differential manifolds gets mapped to the identity morphism on the underlying topological manifold. It's also not an embedding, since different differential manifolds may have different underlying topological manifolds (a functor that's not injective on objects cannot be injective on morphisms). I think the functor is faithful, as we expect of forgetful functors, but that's not enough. For the image of a functor to be a subcategory, it must be either full or injective on objects. This one is neither. So SmoothMan cannot be regarded as a subcategory, nor can it be regarded as top. mans with extra axioms (this was why I switched sides in the first place. The confusion stems from the mistake of thinking in terms of a single atlas, instead of an equivalence class of them). OK, I think I'm on board. I'm switching sides again. Now I agree with MarSch, Rick and Elroch. How about you, KSmrq, are you convinced? -lethe ^{talk +} 03:53, 4 May 2006 (UTC)

It might be worth mentioning that if the objects in the "manifold with a specific atlas" category are specific sets with specific atlases, then if the same set appears with two compatible atlases then there is a natural isomorphism between these two manifolds deriving from the identity map on the sets. Thus each equivalence class of manifolds with compatible atlases has a set of natural isomorphisms between each two of them which can be used to form a well-defined quotient of this category called "category of manifolds with an equivalence class of atlases" which may be easily identified with the category of manifolds with a maximal atlas. Elroch 00:14, 6 May 2006 (UTC)

It would be nice if your statement could be translated as: the category of manifolds is a quotient category of the category of spaces with specific atlas mod equivalence of atlas". But I think the standard definition of quotient category allows to identify morphisms within hom-sets, not objects. -lethe ^{talk +} 01:02, 6 May 2006 (UTC)

Thanks for pointing out my loose use of the word quotient. What I meant was there is a surjective functor (which is analogous to a quotient map from a group) between the category of manifolds with a specific atlas and the category of manifolds with an atlas defined to within equivalence (or to the category of manifolds with a maximal atlas). In a sense, this functor loses nothing of importance because the set of natural isomorphisms between manifolds with equivalent atlases makes these classes act like a single object. (I think to justify the word "natural" here, one uses the forgetful functor to the category of sets). If you have a category where there is a class of objects each two of which is related by a specific isomorphism, it seems obvious that only trivial structure is lost by collapsing that class of objects down to a single object in the obvious way. Elroch 15:11, 6 May 2006 (UTC)

P.S. I think its necessary for the composition of any two isomorphisms to be one of the other isomorphisms for this to make sense (which is of course true here). Elroch 23:04, 6 May 2006 (UTC)

[edit] Motivational examples?

To me, it seems that some of the "motivational examples" present a somewhat unusual viewpoint. For almost all purposes (including as a Riemannian manifold) a hyperbola and a parabola are identical 1-manifolds. To me the text suggests they are two distinct examples of manifolds, which is only true as algebraic varieties or as embedded manifolds, or some equivalent structure that provides some information about how things look from "outside" the object. Can anyone explain to me why this approach has been chosen, rather than emphasising intrinsic properties of manifolds, with lines, circles, spheres and tori (non-orientable manifolds are dealt with in another section). This has got to be a very important section to a general article aimed largely at non-mathematicians. Elroch 16:30, 3 May 2006 (UTC)

This objection has been raised before. A parabolic graph is connected, a hyperbolic graph is not connected. Rick Norwood 16:36, 3 May 2006 (UTC)

ok, it does make sense. I am going to add some text to emphasise what structure is important to a manifold, rather than an algebraic variety. Elroch 17:03, 3 May 2006 (UTC)

[edit] orientability edits + wikiquette

I have made some changes to the orientability section of the entry. Perhaps I owe the other editors a note of explanation, so at to avoid needless revert conflicts.

1. The precise definition of orientability is given in a fully detailed, separate entry.
2. The previous material on the Mobius strip, Klein bottle, and the projective line duplicates material found in the parent entries.
3. There seems to be a consensus that we should take an expository tone in the present entry, and to relegate more technical details to other entries.
4. I replaced the technical definition and constructions with informal exposition.

To User:Rick_Norwood as a point regarding wikipedia etiquette: we will all do better to revise rather than revert. If my edits have errors of grammar and usage as you indicate, you should copyedit appropriately. Removal of a substantial contribution should be peformed only if it is deemed to be incompatible with broad concensus.

I realize that my own edits removed a fair bit of material. However, as far as I can tell, the constructions and definitions that I removed are found elsewhere. I indicated as much in my introductory paragraph and provided the appropriate links. If you feel that some of this material should be preserved, by all means restore and integrate it into the entry. It would be helpful, however, if you were to address the issue of duplication. Rmilson

Good points. But sometimes revising is too much pain and making huge changes at once is not a good idea. People spent a huge amount of time on this article, that's why big things better be discussed here first. Oleg Alexandrov (talk) 17:50, 3 May 2006 (UTC)

Very well. My proposed changes are found in the subsection below. I have integrated some previous exposition, added some expository material, but excised the more technical material. For your own part, you may want to address the duplication issues raised by me above. Rmilson 18:05, 3 May 2006 (UTC)

[edit] Orientability (proposed revision)

In dimensions two and higher, a simple but important invariant criterion is orientability. A non-orientable n-manifold has the curious property that an n-dimensional body can undergo a continuous motion so as to becomes its own mirror image. If this can't happen, the manifold is called orientable.

To be more precise, overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their 'handedness'; these are the orientable manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable. Thus, for the case of a 2-dimensional surface embedded in 3-dimensional space, orientability can be understood by saying that an orientable surface has two sides while an non-orientable surfacethe surface has just one! For the case of 3-dimensional manifold, orientability means that there is a well defined chirality, a sense of ordering that distinguishes between the right and the left-handed. See the 'Alice universe' entry for more on the physical implications of non-orientability.

We illustrate this counter-intuitive phenomenon with some informally presented examples. Follow the links for more information and for the formal constructions of the objects under discussion. Arguably, the most famous non-orientable surface is the Möbius strip , a band formed by twisting and gluing the ends of a long rectangle (see illustration) . Another famous example is the Klein bottle , a non-oreintable surface without borders that looks like a deformed inner tube (see illustration). A Klein bottle is formed by gluing together the ends of a long hose in such a way that a clockwise motion around one end corresponds to a counter-clockwise motion around the other end. In order to accomplish such a feat within the confines of 3-dimensional space, we must allow for the Klein bottle to have self-intersections; the hose has to pierce its own surface so that the moving end joins the stationary end from the inside of the hose.

Another good example of a closed, non-orientable surface is the real projective plane. This configuration is obtained by gluing shut a hemisphere along the equator, but doing so in a way that attaches each equatorial position to its antipode (see diagram). Again, in 3-dimensional space self-intersections will be required to achieve this construction: for more information see Boy's surface and Roman surface. The n-dimensional generalization of this construction leads to real projective space, a noteworthy example — if n is even — of a higher-dimensional, non-orientable manifold.

In this revision it says

A non-orientable n-manifold has the curious property that an n-dimensional body can undergo a continuous motion so as to becomes its own mirror image. If this can't happen, the manifold is called orientable.

I think this is a missleading definition and possibly wrong. For example the sphere eversion can make the inside of a sphere the outside and a sphere IS is own mirror image.

Other than than I've long felt orientability has been given too large a section in the article. I'd much rather a short description of orientability and have a seperate examples section where we should a wide range of different manifolds to really give the user a feel for the topic. We are very poor on examples at the moment only circle, sphere, real projective plane, and torus. How about double torus to illustrate genus better (I can provide a picture of this), it might be posible to try and illustrate some three manifold, I've seen some lovely animations of hyperbolic three manifolds, a few lie groups like the rotation group, would widen the understanding of the reader so they do not leave thinking manifold are just embedded surfaces. --Salix alba (talk) 19:47, 3 May 2006 (UTC)

Yeah, it's very unclear what is meant here. For a knot (which is of course topologically a circle and thus orientable), there are ones which are not isotopic to their "mirror image" and ones that are. Also, there are right handed and left handed Mobius strips. What's going on with the separate article on orientability anyway? Maybe just make that the main article on this stuff? As for examples of 3-manifolds, particularly illustrable ones, I would suggest starting with the 3-torus first, as it's pretty easy for people to pick up (thinking of it as a cube with sides identified, in analogy with the 2-torus being a square with side identifications). It's also easy to modify these side gluings to get more examples with interesting phenomena, e.g. a 2-sided Klein bottle in a non-orientable 3-manifold. --Chan-Ho (Talk) 12:55, 6 May 2006 (UTC)

I think it's a little misleading to say a knot is a circle. What makes a knot a knot is the way it is embedded in 3-dimensional space or the 3-sphere, and it is more useful to identify a knot with the three manifold with boundary that is the complement of a small open neighbourhood of the knot. An interesting question (to which I'm not familiar with the answer) is "what property of this 3-manifold (if any) is equivalent to whether a knot is isotopic to its mirror image?" Elroch 14:49, 6 May 2006 (UTC)

Sure :-) I guess I said it badly, but I think you understand the point that knots can be oriented. Anyway, I don't know if it is really more useful to think of a knot as a knot complement. Sometimes it is, but other times, it isn't. Some invariants only make sense when thinking of a knot as an object in the 3-sphere; as far as I know, trying to figure out crossing number, or say bridge number, from a description of a knot complement (without using the knot) is a seriously deep question. In general, trying to relate diagrammatic properties of knots to properties of its complement is a big endeavour with limited progress.

In answer to your question, there's a simple property of the knot complement equivalent to a knot being achiral, which is that the complement should admit an orientation-reversing homeomorphism. I guess that answers your question, as stated. However, that's not a satisfactory answer, as the question then becomes, "How do you tell if a knot complement admits an orientation reversing homeomorphism?" Hyperbolic knot complements admit a canonical decomposition into polyhedra and symmetries of the complement are just the combinatorial symmetries of this decomposition. This is easily computed by programs like SnapPea. So SnapPea can tell you if a hyperbolic knot is chiral or not. For the other kinds of knots, i.e. satellite and torus, something might be known, but I don't know it.

Additionally, I wanted to mention that in terms of practical utility, examining the knot complement can be pretty computationally expensive. Generally, 3-manifolds are inputted into computers as a bunch of tetrahedra glued together, and a knot complement might have to be divided up into very many tetrahedra. So sometimes for computing certain kinds of things, it's faster to compute from the diagram directly, without translating to the knot complement. --Chan-Ho (Talk) 18:46, 6 May 2006 (UTC)

Yes, it was my turn to be rash in suggesting identifying a knot with its complement. I would guess two knot complements are homeomorphic iff the knots are isotopic, but I'm not personally aware that that is known, and my knowledge of this stuff is way out of date. Anyhow, nice answer :-) Elroch 23:15, 6 May 2006 (UTC)

Actually it's a theorem of Gordon and Luecke that two knots are isotopic precisely when their knot complements have an orientation-preserving homeomorphism; if the complements are just homeomorphic (by something orientation reversing), then you get an extension of the homeomorphism to the 3-sphere (not obvious! and also part of Gordon-Luecke's theorem) sending one knot to the other but they won't be isotopic unless the knot is achiral. I was just pointing out that an interesting aspect of knot theory is that one can approach it from the view of 3-manifolds but other approaches apply and the interaction of these different perspectives is one of the interesting things about it. --Chan-Ho (Talk) 05:04, 7 May 2006 (UTC)

Yes, of course orientation is a property of embeddings that can't be addressed by homeomorphism. Nice to know that the correct version of that theorem has been proved since I studied this stuff officially! Elroch 11:18, 7 May 2006 (UTC)

I've just been wondering what is the weakest class of manifold in which there is no isomophism between a manifold and its mirror image, wherever the distinction makes sense? I would hope it's possible to just take the definition of a topological manifold using an atlas and add the condition that the transition maps preserve orientation. The question would be whether one can define this for general continuous bijections between open subsets of Rⁿ (and if not, in which dimensions are stronger conditions required and what are the weakest conditions to add to make it definable?). Elroch 22:11, 7 May 2006 (UTC)

Oops. I eventually realised that of course no non-orientable manifold has an atlas at all if we demand that transition maps preserve orientation. I assume handedness is not an intrinsic property whatever way you look at it, but requires reference to something else (like an embedding). Elroch 23:25, 7 May 2006 (UTC)

[edit] Help at topological manifold

I have not yet reviewed all the specialized manifold articles, but topological manifold is in a shocking state of disarray. Those who are interested in improving the overall quality of Wikipedia, and who have relevant knowledge and interest, might want to divert some of the considerable energy that is going into thrashing small aspects of this article towards improving major aspects of that one. Seriously, it is an embarrassment. A little work would go a long way. --KSmrq^T 22:27, 3 May 2006 (UTC)

[edit] A question for KSmrq

On May 3, 70.249.220.45 edited your text and introduced many errors. You then added a short paragraph. I reverted the edit by 70.249.220.45, and then went back to the paragraph you worked on and tried to get it in good shape, but I made some typos. You reverted back to the error filled text of 70.249.220.45. I've reverted back to my version -- which, except for the short paragraph on Manifolds with boundary, is really your version. I've fixed the typos. If you still don't like my text for that paragraph, let me know, and I'll put your text for that paragraph back. Rick Norwood 15:02, 5 May 2006 (UTC)

Sorry, I was pressed for time when I needed to explain more at length. I'll comment on your latest effort, which is much the same.

1. Introduction of projection has previously been challenged; why do it?

   But to say that the map "is" the first coordinate is even more unclear.

2. The link to "closed manifold" is more helpful than your replacement, "compact space#Compactness of topological spaces" (which was, ironically, my first attempt).

   Either link is fine with me.

3. We do not like to use "graph (graph theory)" as the text; the parenthetical part is there for purposes of disambiguating the Wikipedia link. Instead we write [[graph (graph theory)|]], which suppresses the parenthetical component. (Note the "|".) The fact that the WikiMedia software has that built in should give a strong hint that it should be suppressed.

   The problem with "graph theory" is that this isn't "graph theory", which is the study of sets of vertices and edges, but graphs. The link is to the wrong place.

   No it isn't. Graph (graph theory) (a redirect to graph (mathematics)) is about graphs, as used in graph theory.Ben Standeven 17:57, 11 May 2006 (UTC)

4. I inserted quotation marks around "edge" in discussing boundary because I don't think most people would think of the surface of a ball as a literal edge; you removed them.

   I dislike the use of quotes around a word to indicate that it is the wrong word. If its the wrong word, we should find a better one. What's wrong with "boundary".

5. In (n−1)-manifold you replaced a correct minus sign with an incorrect hyphen.

   Not my intention. Sorry.

6. Again, "boundary (topology)" should suppress the parenthetical part.

   I agree.

7. In the "See also" remark, the period must go inside the parentheses.

   This convention dates back to the days when a period was set in movable type -- a very tiny sliver of movable type, and could be protected with a large bit of type such as a parentheses. Today, it is more reasonable to follow whatever logic dictates.

8. The new paragraph with inline mathematics had broken tags the first time; now it omits italics for variables.

   Yeah. I really need to learn how to use TEX inside Wiki. Sorry.

This is all aside from the merits of the content! So I reverted.

I would not have minded the revert if you had gone back to your version, but you went back to a worse version.

As for the content, we now mention boundaries repeatedly in the article, each time as if the first. Likewise for questions of dimension. This is not specifically your fault, but I take the opportunity to raise the concern. --KSmrq^T 20:32, 5 May 2006 (UTC)

The whole article needs to be gone over from start to finish, to address this problem. Do you want the job, or would you rather I did it? Either way is fine with me. Rick Norwood 16:24, 6 May 2006 (UTC)

As an amateur who has always wondered what a manifold is, and learnt little from Spivak's "Calculus on Manifolds", I am greatful to those who have contributed to this article.regford 22:11, 23 May 2006 (UTC)

Thanks. Nobody should attempt to read Calculus on Manifolds until they know enough about both calculus and manifolds to write the book themselves. Rick Norwood 01:01, 24 May 2006 (UTC)

[edit] On the circle motivational example

The circle example talks about coordinates, but in the figure provided there is no coordinate system. Also, there is suddently an "a" in the TeX formula that I'm not sure of where it comes from. --Abdull 18:36, 2 June 2006 (UTC)

The sentence before the TeX formula says: "Let a be any number in (0, 1)". Did you see this? - Gauge 06:47, 28 June 2006 (UTC)

Rick Norwood made a correction to this after I put up the question ( see http://en.wikipedia.org/w/index.php?title=Manifold&diff=56543355&oldid=56542975). Thanks for your help! --Abdull 19:55, 10 July 2006 (UTC)

[edit] Gluing

"A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries."

Isn't a simpler name for "a sphere with a hole in it", "a disc"? Or am I overlooking something? --80.175.250.218 13:00, 23 August 2006 (UTC)

You are correct. --KSmrq^T 13:14, 23 August 2006 (UTC)

[edit] Intersecting circles question

I read this article ro help me understand the Poincare conjecture, which is in the news lately. The article helped me greatly, but part of the article was not clear to me. Although I am not a mathematician, I am an engineer; so I probably have more knowledge of mathematics than the average reader of this article; so I think if something in the article is not clear to me then it will probably not be clear to many readers. Anyway, here are my questions:

The second paragraph of the introduction says "Examples of one-manifolds include a line, a circle, and a pair of circles." Shouldn't that say "... and a pair of NON-INTERSECTING circles" (see my next paragraph below)?

Under the section titled "Other curves", the second paragraph begins "However, we exclude examples like two touching circles that share a point to form a figure-8 ..." Would two circles intersecting at two points also be excluded as not a manifold? —The preceding unsigned comment was added by Mherndon (talk • contribs) 05:53, 2006 August 26.

You have learned well: yes, the circles must be disjoint. However, as for clarifying language, be careful what you ask for. Whenever we load the introductory material with excessive attempts at mathematical precision, we lose more readers than we illuminate. The body of the article has enough technical details to satisfy those who need it. Already the "figure-8" discussion — which is still somewhat informal — states the essential issue, the appearance of a neighborhood. --KSmrq^T 13:43, 27 August 2006 (UTC)

[edit] Impressed with this article

Since I usually nitpick on discussion pages, I just wanted to say that this article is very coherently constructed, provides good examples, and covers the topic well for a wide range of readers. Thank you to all who contributed to it. —The preceding unsigned comment was added by 24.205.231.209 (talk • contribs) 20:36, 2006 November 12.

[edit] Misleading illustration?

I don't like the illustration in the very beginning of the article. In the red "triangle", it looks like the side between β and γ is not "straight", i.e. it is not a geodesic curve. This is unfortunate. So it is no real (spherical) triangle. And the sum of the angles does not measure of the area like it does for correct triangles on a sphere.

Of course, if you use nongeodesic sides, your "triangle" can have any angular sum, independant of the curvature of the space.

It would be very nice if somebody would make another version of the image in which the triangle is geodesic. /JeppeSN 22:12, 19 November 2006 (UTC)

The side in question looks like a parallel of latitude. If it were the equator, it would be a geodesic curve, a great circle arc, like the sides which are lines of longitude. But you are correct, the figure is not a spherical triangle. Beyond that, maybe it's not the best idea to begin an article which is based on topology with a figure which is based on geometry. --KSmrq^T 00:30, 20 November 2006 (UTC)