Talk:Manifold/Archive 6

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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)

Hmmm, we begin a thread on accessibility, and end on concerns about whether this article accounts for manifolds which are not even locally Frechet? That's absurd. Anyway, wikipedia IS an encyclopedia. You can take your ideas of generalized manifolds to Springer or to a blog, and eventually, if it is accepted it will end up here. Wikipedia is probably not the best place for original perspectives in mathematics.MotherFunctor (talk) 21:34, 10 October 2008 (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)

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)

Manifold and Topological manifold

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

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?

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)

I would like to answer the question at the top of this section. I apologize if someone fully addressed this issue, but I couldn't tell from gleaning the various answers. The word "topological manifold" is used to describe topological spaces which are locally HOMEOMORPHIC to Euclidean space. The word "manifold" by itself has several possible interpretations, but I usually read it to mean "smooth manifold." The difference is this: on a smooth manifold, the transition functions are required to be smooth (infinitely differentiable, or at least C^k), whereas on a topological manifold, we only require that the transition functions be continuous. All smooth manifolds are of course topological manifolds, but not all topological manifolds are smooth.

Of course, "manifold" is a pretty open-ended term, and depending on who you are, it can mean even more things. For example, some people take "manifold" to mean any topological space which looks locally like a possibly infinite dimensional vectorspace. I think it's a mistake to pursue that kind of generality here too vigorously--it's a great way to make the article useless to everyone. Lay people looking up the word "manifold" undoubtedly came across it in the finite-dimensional smooth context. Experts or graduate students who are looking up things like classifying spaces or Banach spaces are not going to be searching in the Wikipedia manifold article.Miaka314 08:21, 2 May 2007 (UTC)

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)

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)

Orientability (proposed revision)

Möbius strip

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.

The Klein bottle immersed in 3D space.

The fundamental polygon of the projective plane. The arrows indicate how to glue the border to itself.

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. --C S (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. --C S (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. --C S (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)

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)

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)

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)

Some equations are not displayed correctly (LaTeX source code is shown instead). Marozols (talk) 18:21, 21 November 2008 (UTC)

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)

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? —Preceding unsigned comment 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)

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.

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)

On the "Other curves" section

I don't see the value of this section; the example of a nonmanifold is somewhat helpful, but I don't see any point in giving a bunch of example curves. It would be better to just point out that any curve which does not intersect itself is a manifold (and any curve which does intersect itself is not). Ben Standeven 05:56, 9 January 2007 (UTC)

Please find this explained in the archives of this talk page. I really don't feel like repeating the same discussion. Briefly, each example says in plain English why it is important. Perhaps you have already acquired an understanding of manifolds which makes these considerations (connected, closed, finite, or not) obvious or trivial; but I can assure you that most of our readers who are just learning what a manifold may be, and struggling to build the right mental model, benefit greatly from seeing these examples.

Let me illustrate by pointing out problems with your claim that "any curve which does not intersect itself is a manifold". What does the word "curve" mean? How many readers will think that a pair of disjoint circles is a curve? (Few!) Perhaps they will think of a hyperbola as a single curve; perhaps not. Will they understand that although an isolated point is — or may be part of — an "algebraic curve", such a curve cannot be a 1-manifold? (No!) Sometimes "curve" means a continuous mapping from the unit interval, [0,1], to the plane; but to get a manifold without boundary we cannot include the endpoints, and even then algebraic geometry shows there is no such mapping for the example cubic curve, which is nonetheless a 1-manifold. Your proposal may seem natural given your understanding, but it is mathematically deficient and a pedagogical disaster. --KSmrq^T 10:00, 9 January 2007 (UTC)

The paragraph needs to be vastly expanded, in that case; for starters it needs to explain what a "line segment without its endpoints" means (our target audience will not be familiar with this concept); and it should discuss the concept of boundaries right up front, instead of dodging the issue. And the paragraph completely ignores the most important features of a manifold: that it does not have to be embedded in Euclidean space, and can be of any dimension. I do agree we need to point out that manifolds are not required to be connected, now that you mention it (even though as a complex analyist, I do require manifolds to be connected.) I'll take the liberty of fixing one of these problems right now, though; now we have a section on a line segment with its endpoints.

Now that I think about it, we should also mention a square as an example manifold, showing how the corner resembles a Euclidean space. Ben Standeven 05:02, 11 January 2007 (UTC)

It's done. I put it after the "enriched circle" section, so I could mention that the square is a differentiable manifold (rather the point of including it). But I bet the section isn't very clear, and would benefit from a diagram. Ben Standeven 05:21, 11 January 2007 (UTC)

I know you mean well, but this section is titled "Motivational examples". We editors have previously considered and deliberately decided to exclude introduction of the boundary concept this early. This is not the section in which we rigorously define manifolds of all dimensions, nor even 1-manifolds. We are trying to build a little intuition without mentioning everything up front. There is a darn good reason why we don't state the classification theorem for 1-manifolds here! Please try to appreciate the big picture. We have worked very hard to produce a gradation of topics and rigor, and you are introducing far too much explicit detail far too early. At this point a general reader should not have to learn that a square is a differentiable manifold! On this particular detail I am quite clear, because we had discussions about the subtleties of this (now in the talk archives), and many trained editors were surprised at some of the facts. Basically, while I think you now understand some of the point of the "Other curves" section, I don't think you yet appreciate the larger structure in which it resides. Sorry; I'm sure it's frustrating. --KSmrq^T 07:34, 11 January 2007 (UTC)

Comments from a differential geometer

General remarks

This is a very nice article. I have some comments which I hope will be helpful. I'm also willing to do some editing, if you agree with my comments. Geometry guy 19:26, 8 February 2007 (UTC)

First, I completely agree with all those who think that this article should be a general article, encompassing all possible ideas of what the term manifold means, in as non-technical way as possible.

In particular, concerning the question raised several times above: is every manifold a topological manifold? My answer would be no. The definition of a topological manifold is precise, and therefore technical and limiting. A topological manifold is a second countable Hausdorff topological space which is locally euclidean, i.e., every point has a neighbourhood which is homeomorphic to an open subset of euclidean space.

Thus topological manifolds do not include infinite dimensional manifolds, or non-Hausdorff manifolds, both of which arise naturally in mathematics. Examples such as the very long line do not arise so naturally, but they are still manifolds in some sense.

For this reason, I propose that the definition of a topological manifold should be moved from this article to the topological manifold article (which, as many have noted, needs much more work than this one).

Instead, this article should define manifolds in terms of charts, atlases and transition functions. Although these are not my favourite definitions, they have the advantage of being flexible and accessible. Underlying this flexibility is the categorical approach that has been mentioned already on this talk page. Different classes of manifolds can be defined by restrictions on the model category: for instance, continuous functions between euclidean, Hilbert, Banach or Frechet spaces, or differentiable (C¹, C^k or smooth) functions between euclidean spaces. Restricting the transition functions to belong to these categories define different classes of manifolds. The accessibility of this approach has not been emphasised, however: the reason it is accessible is because one does not need to know what a topological space is; instead the charts in the atlas can be used (implicitly) to transport the topology of the model space (such as euclidean space) to the manifold. Geometry guy 19:49, 8 February 2007 (UTC)

Some other, more specific suggestions.

1. The notion of a differentiable manifold which is not smooth is a minority interest these days, and the term smooth manifold is much more common. I think the structure of the articles on wikipedia should reflect this and the differentiable manifold article should only give definitions (both using atlases and sheaves of functions) of different classes of manifolds: the rest should be moved to a Smooth manifold article. Such an emphasis affects the formulation of this article in one or two places.

2. The notion of a manifold with boundary, as opposed to a closed or open manifold, should be introduced earlier, not in the constructions section.

3. The "Construction"(s) of manifolds section needs some work. There are three basic constructions: submanifolds (e.g. of euclidean space), parameterizations (and patching), and gluing (quotients). All of these can be illustrated with the example of the circle, which is a submanifold of the plane, a union of two coordinate patches, and a quotient of the real line. Of course, cartesian products provide a further construction, which should be mentioned (and is).

4. The "Classes of manifolds" section should distinguish classes obtained by restricting the transition functions from those classes requiring extra structure. Differentiable, smooth, real analytic, complex, and symplectic manifolds are important examples of the first kind, while riemannian and Finsler manifolds are of the second kind.

According to wikipedia boldness, I should just try some of this, but this is a highly evolved article, with many links, and I am quite busy, so it seems more sensible to express my views to see if they strike a chord. Geometry guy 20:08, 8 February 2007 (UTC)

Discussion

This subject has been discussed before, at length. Those of us who were introduced to manifolds via point set topology (as in Munkres) have a gut feeling that this is what manifolds are, and that the differential structure is an overlay. Those of us who were introduced to manifolds via the differential structure (as in Spivak) have a gut feeling that that is what manifolds are. The discussion is as unlikely to produce results as the discussion of whether rings must have a multiplicative identity.

Your objection to, for example, the requirement of hausdorff, obviously applies to differentiable manifolds as well as to topological manifolds. Every differentiable manifold is a topological manifold. For example (the first book that comes to hand) Wolfe, Spaces of Constant Curvature, "A differentiable manifold of dimension n is a separable hausdorff space M together with ..."

The consensus of the previous discussion was to begin by discussing the various forms of manifolds, but to require all manifolds to be separable and hausdorff. Rick Norwood 13:50, 9 February 2007 (UTC)

Hmmm... I did read the previous discussion before posting, and I did not find consensus, but rather debate and varied points of view. Your first paragraph reflects this. The term manifold means different things to different people. In algebraic topology and homotopy theory, manifolds and topological manifolds are synonymous, whereas in my field (differential geometry), manifolds are always smooth (and Hausdorff, as you rightly point out): any lack of smoothness is nearly always emphasised by talking e.g., about topological manifolds, C^k manifolds, Lipschitz manifolds etc. In geometric analysis, manifolds can be infinite dimensional (hence not separable).

I don't understand why this is a problem, however. An article on rings may say that they are (usually) assumed to have a multiplicative identity, but it might mention that in operator theory, this is often not the case.

I realise I've come in on an old and possibly emotive thread, and perhaps I did not express my view very well. It was not my point to raise objections to Hausdorff-ness, nor to insist that manifolds need not be topological manifolds. I simply think that as this issue is a technical one, and as there is not universal agreement, it doesn't need to be mentioned in the (wide-ranging, non-technical) manifolds article. Alternatively, one could say that manifolds are nearly always assumed to be Hausdorff and second countable, referring to the topological manifolds article for details. Geometry guy 15:49, 9 February 2007 (UTC)

Actually, you express yourself very well, and I'm sure you can make valuable contributions to this article. I'm glad you are aware that there is some history here.

I have just two suggestions. First, following Wiki conventions, that we take this discussion to a new section at the bottom of the page. And second that you make a few edits and see how they are received rather than attempting a complete rewrite of the entire article. Rick Norwood 16:27, 9 February 2007 (UTC)

Okay, discussion moved (hopefully without damaging anything else!). I certainly have no intention to rewrite the article! It is already a pretty polished piece of work. Geometry guy 16:47, 9 February 2007 (UTC)

I do agree with you that the "Mathematical definition" section of this article (which presently gives a definition of a topological manifold) is out of place. This article is, as you say, meant to be a overview of the manifold concept. Specific definitions belong in the more specialized articles.

Some more history for you: I did a complete rewrite of the topological manifold article last November, and it is in much better shape than it was prior to that. It is far from complete however. In particular it needs mentioning of when such manifolds admit smoothings, triangulations, etc. Also, I left the definition of a topological manifold rather general, only requiring that they be Hausdorff and locally Euclidean. The reasons for addition constraints such as paracompactness and second countability are discussed in the article.

Lastly, if you are looking for something to do, our differentiable manifold article needs a whole lot of work. It is mostly a cut and paste job right now. -- Fropuff 17:19, 9 February 2007 (UTC)

I'm glad you agree: I hope I have managed to pick up the feel of this article from the talk page. Thanks also for the further information: I certainly didn't mean to be critical of the work done so far on topological manifolds. As for your last point, in fact, my main mission is to help to get a differentiable (or rather smooth) manifolds article into reasonable shape, but this requires some coordination (and a feeling for how various articles fit together). I have left some comments on that page too, and also on the differential geometry and topology page. Geometry guy 19:01, 9 February 2007 (UTC)

Great! I'm glad someone is willing to tackle the smooth manifolds material. It is in dire need of attention. -- Fropuff 19:15, 9 February 2007 (UTC)

This article evolved over a long period, and many of the seminal discussions are archived. With regard to smooth manifolds, there was mention of a theorem by Whitney (as I recall) that says a C¹ manifold can always be made into a smooth manifold. (Not so for C⁰ manifolds.) This would seem to partly account for the emphasis on smooth manifolds.

As for editing suggestions, history suggests that people edit what they see first, rather than what needs the most work. For example, the introduction of this article was revised sometimes several times a day, even while the rest of the article languished half written. Today, this article, meant to be a gateway, still draws excess editing attention while the detailed technical articles cry for a champion. Therefore, I make the obvious suggestion: Suppress the urge to edit this one, and come back to it after lavishing some tender loving care on smooth/differentiable manifolds. --KSmrq^T 20:51, 9 February 2007 (UTC)

I've seen the Whitney result somewhere on wikipedia, but I don't remember where just now. However, this does not really account for the emphasis on smooth manifolds. The subject has moved on from foundational questions, to more open-ended ones. The emphasis on smooth manifolds is simply a matter of pragmatism: most differentiable manifolds we are interested in are (naturally, not just equivalently) smooth manifolds, and smooth manifolds are much easier to work with.

As regards editing, I agree completely. In fact, I am taking a bottom-up approach, and editing some nuts-and-bolts articles right now, not even the smooth/differentiable manifolds articles. I just wanted to touch base here, and also with the topological manifolds article, which is a vital underpinning for the smooth stuff. Geometry guy 22:41, 9 February 2007 (UTC)

I've re-written the definition Manifold#Mathematical_definition to include both the stress on topological and differentiable manifolds (and note on different structures), and a word on generalizations (together with some technical notes on what exactly locally homeomorphic means in this context). I've moved the discussion of non-Hausdorff manifolds to its own page, and am writing a page on Categories of manifolds to discuss in detail various notions of manifold. Nbarth 16:03, 11 November 2007 (UTC)

History

After I have added some material to the History section, it appears that this article has more comprehensive history than the supposedly 'main article' History of manifolds and varieties. I am not sure what to do it about it, but it appears that the other article has fairly spotty coverage (and almost none as far as varieties are concerned). Also, they are no longer in sync, as had been the case before. But what worries me most is that the topology of manifolds subsection may come a little too early in the course of the article. On the other hand, I wasn't able to find any other place on Wikipedia with comparable material, even for linking purposes. Arcfrk 06:25, 17 April 2007 (UTC)

Generalization

In engineering design, non-manifold models are common. For example, the union of a plane and a line segment, or two intersecting planes. Neither of these are manifolds, but locally they look like manifolds. You can define functions over them and you can deform them. Is there a word for such a thing? Non-manifold? —Ben FrantzDale 02:02, 5 May 2007 (UTC)

It looks like what I'm talking about is homeomorphic to a simplicial complex. —Ben FrantzDale 16:30, 6 May 2007 (UTC)

CW complexes and immersed submanifolds (see also immersion (mathematics) which is just a stub right now) are two other notions which cover some of these generalizations. Geometry guy 21:49, 6 May 2007 (UTC)

Cool. Thanks. —Ben FrantzDale 22:57, 6 May 2007 (UTC)

There is another object called a stratified space, which might be a closer fit for the kind of object you're interested in. CW-complexes and simplicial complexes are more oriented towards pure-mathematics. Stratified spaces tend to occur much more often in physical sciences. Rybu (talk) 10:24, 16 December 2007 (UTC)

Hamilton-Perelman Solution of the Poincaré Conjecture

I moved the Hamilton-Perelman link out of the See also section. Still, editors here may want to have a look at it, since it needs a bit of work to bring it into shape. Silly rabbit 00:16, 22 June 2007 (UTC)

I've restored the link, and am at a loss to understand your objection to it. It is certainly important work in manifold theory. Rick Norwood 15:05, 22 June 2007 (UTC)

My objection is that it opens the floodgates for adding everything conceivably related to manifolds under See also, especially since the result pertains only to 3-manifolds. That's what categories and lists are for. But I am willing to be overruled on the matter, if there is sufficient consensus. Silly rabbit 15:09, 22 June 2007 (UTC)

I agree with Rabbit, this link is more appropriate at Poincaré Conjecture. Arcfrk 15:35, 22 June 2007 (UTC)

I agree with Rabbit as well. I've added a link at 3-manifold (there is already one at Poincaré conjecture, of course), but I don't think it is encyclopedic to link it from the see also section here. Geometry guy 16:44, 22 June 2007 (UTC)

I can only say that for my entire career as a researcher in manifold theory, the Poincare conjecture has been the holy grail of manifold theory. I still recall the excitement when the four dimenional case was finally solved. To object to a link here seems strange to me. Rick Norwood 19:04, 22 June 2007 (UTC)

I have linked it from the section discussing low-dimensional manifolds. I hope all parties are satisfied. Silly rabbit 19:27, 22 June 2007 (UTC)

I had in the back of my mind exactly this kind of solution when I wrote "from the see also section here". Perfect! Geometry guy 19:59, 22 June 2007 (UTC)

Leading Illustration

Can someone explain why my edit changing the illustration of spherical trigonometry was reverted? As mentioned in November, Image:Triangle on globe.jpg is not a real (spherical) triangle. As no change had been made, I thought it would be appropriate to replace it with the correct Image:Triangles (spherical geometry).jpg. This paragraph currently misleads the reader twice, by assuming the image illustrates both a sphere and a triangle. I hope we can do something about this. –Sarregouset (Talk) 18:35, 16 September 2007 (UTC)

Has the talk about an atlas become meaningless?

I quote the current bit about atlases.

The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of charts. The atlas containing all possible charts consistent with a given atlas is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is a very abstract object and not used directly (e.g. in calculations).

Does that convey any meaningful information to you? It doesn't say anything to me. Why is there any discussion about uniqueness of an atlas before compatibility and various structures is discussed? Basically, you're talking about "consistency" before you've defined it. You need to discuss structures before talking about compatibility/consistency. Rybu (talk) 10:33, 16 December 2007 (UTC)

The very first line

A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space...

Has me a bit confused. Should this be every point is a neighborhood, exists in a neighborhood, or has a neighborhood operating on it or what...? I find 'has' a bit poorly defined... --Ceriel Nosforit (talk) 00:22, 19 March 2008 (UTC)

See Neighbourhood (mathematics). silly rabbit (talk) 02:21, 19 March 2008 (UTC)

This is a good example of a problem discussed in the section "manifolds for laymen" below. It is not at all poorly defined to anyone understanding the topological concept of a neighbourhood; as for the problems experienced by anyone without that understanding, see my detailed comment below. Nevertheless, I have tried to come up with a form of words which might be a little more helpful to a layman. You may like to let me know whether you think I have succeeded. JamesBWatson (talk) 12:19, 20 March 2009 (UTC)

Manifolds for laymen

Is there a way that some one could describe the word Manifold without words like topology, homeomorphic, Euclidean space and Topological space, when I click on these links I am confused with even more words I don't understand. Could someone kindly put a simple definition that all idiots including myself can understand, its annoying when words that are used every where and yet they are not understood. I looked Manifold up in a dictionary and it only gave me the Wikipedian definition in the mathematical sense. Thanks I think such a definition would greatly benefit this article. --Zaharous (talk) 03:50, 19 January 2009 (UTC)

Not much can be said if we aren't allowed to use terms like "Euclidean space", or "real coordinate space". A surface (that is, a 2-manifold) can probably be described. Any sufficiently small neighborhood of a point in a surface looks like a rubber sheet which can be stretched onto a part of the plane. The same holds in higher dimensions, but it does require some sophistication to grasp. siℓℓy rabbit (talk) 03:59, 19 January 2009 (UTC)

Thanks anyways Silly Rabbit --Zaharous (talk) 04:03, 19 January 2009 (UTC)

Note that some courses in university on differential topology don't assume much prior topological knowledge. I.e, they only assume that you know what "Hausdorff", "second countable", "compact" and "connected" are. I don't really agree with this method of teaching because it is good to have an important grasp of the various important topological properties of a manifold. However, if you want you can always search up some familiar universities with "manifold", and you should get something useful. --PST 08:35, 19 January 2009 (UTC)

The introduction to this article is considerably better I think than most maths article on wikipedia. It gives a reasonable definition without being too mathematical, plus it gives a nice example. I can't say the same of some of the articles it links to though. Perhaps it could be polished up a bit more but I'd much prefer to see the effort spent where it would do more good. Dmcq (talk) 14:57, 19 January 2009 (UTC)

I believe that the lede can be considerably better. At the moment it is intended only for laymen (it is not even up to par in that respect as this thread shows) but what about the more advanced readers who are familiar with point-set topology (and perhaps algebraic topology). Where do homotopy groups of spheres come into it? Or even homology theory. I guess that over time this article will expand. But there has to be some agreement with regards to the triviality of the article. We shouldn't make it trivial and neither should we make it overly complex. But as an encyclopedia, we write every mathematics article from scratch. I think it is safe enough to assume that the reader knows point-set topology for otherwise the article is meaningless to him/her. --PST 17:28, 19 January 2009 (UTC)

Please read WP:LEAD. What you are outlining would I believe directly contradict wikipedia style guidelines. More advanced bits can fit lower down in the article but the first priority to cater for people who are new to them in the first instance. We should not assume they already know most of the answer. How about looking at topology for instance instead, it can't even give a decent description of the subject in the leader and would turn even someone who knows what it is about away? Dmcq (talk) 17:44, 19 January 2009 (UTC)

The article could use some expansion. But as I have said previously, the expansion should come from the bottom up rather than the top down. It is inappropriate to mention, for instance, homotopy groups of spheres in the lead if they aren't even mentioned in the body of the article. As far as whether a reader should be expected to know some topology, I think that is a reasonable assumption. However, the lead is supposed to give context for the rest of the subject of the article in as non-technical a manner as possible (see MOS:LEAD), and to summarize the contents of the article. Currently, an additional paragraph needs to be added to the lead, probably summarizing certain aspects of the topology of manifolds (as discussed in the body of the article). But beyond that, there isn't much else to do as far as the lead goes. siℓℓy rabbit (talk) 17:52, 19 January 2009 (UTC)

I fully sympathise with Zaharous: you turn to an article about manifold to find out what one is, only to find it is defined in terms of other concepts which mean nothing to you, which are themselves defined in terms of other concepts you don't know ... However, this is, unfortunately, what mathematics is like. You have to have some knowledge of numbers before you can understand multiplication; you have to know what "right angle" means before you can understand Pythagoras's theorem; you have to have some knowledge of topology before you can understand topological concepts, and manifold is a topological concept. To me the idea of defining "Manifold" without words like topology seems a little like defining "dairy farming" without words like cow and milk. Of course we should make the ideas as accessible to lay people as possible, but anyone who has no idea what topological means is not going to understand what manifold means, no matter how it is phrased. The present from of the article does its best by saying things such as "every point has a neighborhood which "resembles" ... Euclidean space", as well as giving the technical word "homeomorphic" to specify what "resembles" means. I am not sure how much this helps a layman: is "resembles" too vague to be helpful? If so, then how would one do better? Is there a way of describing more clearly what is meant without using topological terms? If there is then it would be good to do so, but it is not clear to me that there is. I will give the matter some thought, but I fear there is a danger that any attempt to do so may merely add more unhelpful words, making the issue more confused. In topology, perhaps more than in other branches of mathematics, there is a long history of avoiding technical jargon with the laudable intention of making information accessible to lay people, but with the result of producing confusion and misinformation. For example when I was a teenager I read more than one book which used the expression "topologically equivalent to" in places apparently meaning "homeomorphic to", and in other places clearly meaning "continuously and invertibly deformable into", in such a way that the (lay) audience for which the books were intended would be likely to finish up quite misled: I know I was. Something not unlike this occurs, I think, in the discussion above, where C S says that a knot "is of course topologically a circle", and Elroch says it is "misleading to say a knot is a circle. What makes a knot a knot is the way it is embedded in 3-dimensional space...". Probably C S meant that a knot is homeomorphic to a circle, but the difference between a knot and a circle is a topological one, so that to say that the one is "topologically" identical to the other is subject to being interpreted in other ways than "homeomorphic to". Now try going one step further than C S, by banishing not only "homeomorphic", but also "topologically", "toplological", etc, and see how easily you can paraphrase C S's sentence without losing the meaning completely. Certainly I am prepared to accept that in the interest of accessibility to the layman it may be acceptable to give an account which is not 100% rigorous. However, I am simply trying to express the fear that in attempting to do so there is a danger of throwing the baby out with the bath water. To reiterate my essential point: manifold is a topological concept, and is not likely to be very comprehensible to anyone who has no idea what topolgy is. Better to define it in a way which conveys something to someone with a little understanding of topology than, by trying to make it accessible to someone without any, finishing with a definition which does not mean much to anyone unless they already have enough knowledge that they don't actually need our definition. JamesBWatson (talk) 15:40, 17 March 2009 (UTC)

My comment above does not preclude the possibility of minor improvements to make the information more accessible. I always found 'every point has a neighborhood which "resembles" Euclidean space' unsatisfactory: what does "resembles" mean? To say that it is defined by the gloss "is homeomorphic to" is no answer, as anyone understanding "homeomorphic" does not need "resembles". Having given some thought to this question I decided "is locally indistinguishable from" was better. JamesBWatson (talk contribs) 12:12, 20 March 2009

In my opinion, "resembles" is infinitely better than "locally indistinguishable from". We do not need more jargon, least of all, in the first line. Arcfrk (talk) 17:54, 20 March 2009 (UTC)

I certainly agree that we don't need more jargon, but which bit of this is jargon? "Indistinguishable from" is certainly not jargon: the jargon expression this tries to mirror is "homeomorphic to". I suppose that you could just about consider "locally" to be jargon. However, even if it is jargon, is that fact enough to outweigh its advantage? "Resembles" just means "is somewhat like", and does not specify in what way it is like; the point about homeomorphism is that it means more than just "resembles": it means that the topology is identical. What I was trying to do was to convey this concept of being identical while avoiding the jargon. I am by no means 100% happy with the attempt, not least because the meaning of "locally" is not made clear, but I still feel there is an advantage in at least trying to convey the idea of being essentially identical, rather than vaguely suggesting the neighbourhood is somehow like Euclidean space, but giving no indication of how or to what extent. If anyone can come up with a better way of attempting to do this I shall be delighted, but I am not convinced that using as vague an expression as "resembles" actually conveys any more to the layman than just saying "is homeomorphic to": neither really tells him what the point is. It is possible that simply leaving out the word "locally" would be an improvement, if that would make the sentence easier for a layman to follow. I would be interested in other people's opinions. JamesBWatson (talk) 19:55, 20 March 2009 (UTC)

I don't see any advantage to having the word "locally" in the sentence. For someone who knows what the word means, it is redundant since we are already talking about a neighbourhood. For a lay person, it conveys nothings, and may be a potential source of confusion. The term is not usually used in this way outside of mathematics. I have no strong opinions on the "resembles" versus "indistinguishable" question. siℓℓy rabbit (talk) 20:11, 20 March 2009 (UTC)

Indeed, "locally" is currently redundant. What about:

"In mathematics, more specifically topology, a manifold is a space that on a small enough scale resembles usual space (i.e. is locally homeomorphic to Euclidean space)."

The parenthetical could also simply explain "usual space" as "(i.e. Euclidean space)" without explaining "on a small enough scale" to avoid "locally homeomorphic". Also, there might be a better word than "usual". Thoughts? RobHar (talk) 22:47, 20 March 2009 (UTC)

I'll cast my vote tentatively in favor of your suggestion. It looks better than what we have here now. siℓℓy rabbit (talk) 04:46, 21 March 2009 (UTC)

Good: definitely an improvement. I think that "on a small enough scale" is clearer to a layman than either "locally" or "neighborhood": it does the job I was clumsily trying to do with "locally" much more successfully. In the parenthetical gloss the single word "locally" is more succinct than a whole phrase including "neighbourhood", but I have no strong preference. I am still not happy with "resembles", though, and would prefer some form of words that makes it clear that the resemblance is 100%. Maybe something like "... a space that on a small enough scale is exactly like ...", if there is a consensus against "... a space that on a small enough scale is indistinguishable from ...", which is still my preference, but I am not going to insist on it. Also "usual space" is not totally satisfactory, but at the moment I don't have a better layman-friendly term to use. My main objection to "usual" is its general vagueness, but also it possibly might be read as implying "3-dimensional". I hope we can find a way that makes this clearer, but if we can't I would marginally prefer "ordinary space" to "usual space": it just seems to me like a more natural expression, but that is a very small detail. JamesBWatson (talk) 17:08, 21 March 2009 (UTC)

Once again, my chief objection is against the superfluous and potentially confusing word "locally", which you have yet to present a convincing argument in favor of. Absent such an argument, I suggest that the word should be removed, as I have already done once. siℓℓy rabbit (talk) 23:59, 21 March 2009 (UTC)

New lead

Rather than recycling the well worn arguments of mathematical precision vs general accessibility, I went ahead and edited the lead a way that, I hope, balances both of them. Arcfrk (talk) 01:53, 22 March 2009 (UTC)

I like the new lead. Good work! --PST 02:13, 22 March 2009 (UTC)

Thank you, Arcfrk. This is definitely an improvement. I think even the use of the word "resembles", to which I had objected, is less objectionable in the context of the sentence you have written. You have kept "Euclidean space", which was one of the expressions which Zaharous objected to as being opaque to the layman, but, as I wrote above at (perhaps excessive) length, there really seems to be a limit to how far you can explain mathematical concepts without mathematical terms, and maybe it is best to leave this one in. JamesBWatson (talk) 18:20, 22 March 2009 (UTC)

Surface of a sphere

There seems to be a slow running argument about 'surface of a sphere', and 2 dimensional sphere. I'm not sure this article should use the phrase at all. Surface=boundary as far as I'm concerned when working in an area like this and the boundary of the sphere is null. However there is the very important fact that outside of topology people are quite happy to talk about for instance 'the volume of a sphere'. My two cents is that the article should just grasp the nettle and say that a sphere is the surface of a ball in this area of mathematics at the top of the article and not refer to the surface of a sphere. Dmcq (talk) 17:31, 6 November 2009 (UTC)

In my opinion, we are wasting our times arguing about trivial sentences. The main goal now is to improve the article as a whole, rather than correcting a few four word phrases around the place. The IP's edits are good and he is possibly the first person in the recent past to take initiative in improving the article. This should not be discouraged by other editors. That said, I completely agree with all your points - it is just, in my opinion, that WP is becoming less productive by the day (reverts and deletion of content should be last priority (unless of course, in the case of purely incorrect content or vandalizm); rather, people should work to add content). --PST 02:06, 7 November 2009 (UTC)

gluing manifolds with boundary

I've added a "dubious" tag to the section on gluing a manifold along the boundary. For smooth manifolds, it is not true that it is enough to specify a smooth bijection between the boundaries of the pieces in order to define a smooth manifold structure on the whole. A simple example would be the gluing of two intervals: [0,1] and [1,2] along their common boundary, 1.

There are many ways to give a smooth structure to the glued interval [0,2] which will be compatible with the smooth structure on the two pieces. One is the obvious structure on [0,2]. But note that [1,2] is diffeomorphic to [1,3], so another structure can be obtained if we identify [0,2] with [0,3] using the diffeomorphism of [1,2]. This is not the same structure!

Usually the way to resolve the problem is to choose a "collar" neighbourhood of both pieces.

This is a very crude explanation. If someone can write a decent explanation in the article itself that would be awesome. It is worth mentioning that the resultant smooth structures are all diffeomorphic to one another - but not the same! (as if this wasn't confusing enough already).

Amitushtush (talk) 12:51, 1 February 2010 (UTC)

I agree. The article does not explicitly state that defining the bijection between the two boundaries is enough to define the new manifold, but that is the way it is likely to be read. A rewrite is highly desirable. JamesBWatson (talk) 15:49, 1 February 2010 (UTC)

Thus a function T from (0, 1) to itself can be constructed

What does "Thus a function T from (0, 1) to itself can be constructed" mean? Wisapi (talk) 13:06, 9 July 2010 (UTC)

$T:(0,1)\to (0,1)$. PST 10:35, 10 July 2010 (UTC)

Why we always call manifold "structure"?

What is a "structure" in this case? e.g. "symplectic structure", "complex structure", etc. Is those "structure" differential structure? --虞海 (Yú Hǎi) (talk) 09:57, 24 July 2010 (UTC)

Yes, for a smooth manifold. For a topological manifold, you need to consider the pseudogroup of homeomorphisms. Arcfrk (talk) 00:23, 25 July 2010 (UTC)

misleading use of "Euclidean"

Before I start criticizing, let me say that I think this article does a superduperfantastic job of explaining a highly technical subject in an accessible way. I love how it introduces motivating examples before giving the general definition. What I want to criticize is the misleading use of the word "Euclidean." The lead says, "In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold." To me this is a definition of a Riemannian manifold, and, as written, excludes pseudo-Riemannian manifolds. I suppose it comes down to what you mean by "resembles." To someone who *already* knows what a manifold is, it's clear that a manifold doesn't come equipped with a metric, so "resembles" is very loose, and does not rule out spaces with a non-Euclidean signature. But to someone who doesn't already know what a manifold is, this very strongly implies something that isn't true: that manifolds always come equipped with a Riemannian metric (which we normally think of as part and parcel of a Euclidean space). I'll take a shot at rectifying this.--75.83.69.196 (talk) 03:46, 4 October 2010 (UTC)

Untitled

lay people? some of with degrees in math who did masters in other areas are still as befuddled. the details here (trees ) are great but the forest is not covered well! — Preceding unsigned comment added by 130.76.64.118 (talk) 21:51, 27 October 2011 (UTC)

I moved this comment from the very top of the page to the bottom. I wonder if it was meant as a reply to the above discussion at Talk:Manifold#Accessibility concerns. By the way, does anyone know how to set up automatic archiving of this talk page (there are various bots that can do it)? We've still got old discussion from 2006 hanging around. Jowa fan (talk) 04:53, 28 October 2011 (UTC)

smooth and riemannian contexts

Both in the smooth context and in the Riemannian context, the cumbersome definition with atlases is unnecessary. The basic object can be defined as a subset of Euclidean space. I started to implement the necessary correction in the lede. This should be done in the individual sections as well. Tkuvho (talk) 10:18, 8 February 2012 (UTC)

An editor apparently feels that the graph definition is misleading, but did not explain why. Tkuvho (talk) 17:17, 29 May 2012 (UTC)

Hi Tkuvho! the 2-sphere is one of the first examples of a manifold. Please explain how you would like to describe the 2-sphere as the graph of a function. — Preceding unsigned comment added by Lost-n-translation (talk • contribs) 12:56, 30 May 2012 (UTC)

I suggest you stop making edits to the article until you understand this point. The claim is not that the sphere is the graph of a function. The claim is that the sphere is locally the graph of a function. Namely, you can choose (say, orthonormal) coordinates near every point so that a neighborhood on the sphere is the graph of a suitable function. Tkuvho (talk) 14:07, 30 May 2012 (UTC)

I was actually first trying to see if you understood your own `point'. (There are different levels of misunderstanding.) After more that 20 years of teaching this concept, I immediately see how novices might be misled "by a subset of Euclidean space which is locally the graph of a smooth (perhaps vector-valued) function". The novice would not realize immediately that the choice of coordinates would depend on the point. At the time that most students encounter `manifold' they are most comfortable thinking about one coordinate system at a time. In your beginning paragraph, you do not give an explanation of `local' even by example. The first paragraph of a Wikipedia page should be directed at the most general audience with deference given to the reader who has no prior experience with the subject of the article. One should not presume that the word `local' should automatically trigger the response: `Oh yeah, that means I should choose a local coordinate system as well as a labeling of some of the variables as range variables and the rest as domain variables, and I guess I oughta make sure that the range variables are transverse to the submanifold or something like that." You have hidden the essence of the concept of a manifold in the word `local'.

Next, although it is true that every smooth manifold can be embedded in some Euclidean space Whitney), it is misleading to define all (smooth) manifolds as a subset of some Euclidean space. Indeed, many if not most manifolds that arise in applications do not come to us as subsets of Euclidean space. For example, the phase spaces and configuration spaces often do not come to us as subsets of Euclidean spaces. The universe in which we live may not come to us as a subset of Euclidean space. It is unnatural, therefore, to limit the reader's imagination by giving them a Euclidean crutch.Lost-n-translation (talk) 11:13, 31 May 2012 (UTC)

It may be true that some "manifolds that arise in applications do not come to us as subsets of Euclidean spaces", but they certainly don't come to as as collections of charts, either. Readers should certainly be given every crutch possible when it comes to a technical field such as mathematics, and Arnold's crutch is a great one. Tkuvho (talk) 12:21, 31 May 2012 (UTC)

There is no doubt that crutches are good, but the reader should be alerted to the fact that he is being given a crutch. Normally, we see if someone can stand on their own feet before giving them a crutch. The reader shouldn't be misled into thinking that the crutch is the norm. A submanifold is a wonderful example of a manifold, perhaps the most intuitive. But pretending that every manifold arises as a (smooth) submanifold shortchanges the reader interested in both physics and mathematics...

And why do you insist on using graphs to define charts? Don't you like, for example, spherical and cylindrical coordinates? The usual purpose of a differentiable manifold is to be able to do analysis (e.g. analyze fluctuations in the temperature of the surface of the earth). Must we always use graphs to locally translate analysis on the (sub)manifold into analysis in Euclidean space?

And, yes, unless you're a topologist, they don't often come with coordinate systems (charts). The applied analyst often simply wishes to differentiate a function defined on the manifold. In order to do so, he must impose coordinates. Differentiation is a Euclidean construct and essentially, by definition, a smooth manifold is an object on which one can do calculus. But in order to actually do the calculus, you have to have coordinates.

I don't think that my version over-emphasizes the role of atlases as is done in some more abstract treatments. I think it would be natural for some to write: `An n-dimensional manifold is a second countable Hausdorff topological space such that each point has a neighborhood homeomorphic to n-dimensional Euclidean space.' They would be quite right, of course, but the novice would get nothing from it. On the other extreme, to try to eliminate local coordinates from definition is impossible. To suppress local coordinates from the discussion is misleading.

Arnol'd? Can you give me a specific citation?Lost-n-translation (talk) 13:43, 31 May 2012 (UTC)

I think Lost-n-translation's version is more readable than Tkuvho's version, but I'm not really happy with either. "Locally a graph" doesn't work, because a lemniscate is a graph but not a manifold. And the ocean is, at best a manifold with boundary, not a manifold. This subject has been discussed at length before. Everyone wants manifolds defined the way the manifolds they work with are defined. The fundamental question seems to be whether manifolds are fundamentally geometric objects (point sets) or fundamentally algebraic objects (defined by functions). The more geometric definition is both more accessable to the non-specialist and has the advantage of including all types of manifold, instead of only, say, differentiable manifolds.

How about this definition?

"An n-dimensional manifold is locally like ordinary n-dimensional Euclidean space, (and also satisfies two additional technical requirements discussed below). One-dimensional manifolds (1-manifolds) include lines and circles. Two-dimensional manifolds (2-manifolds) include the Euclidean plane and also the surface of a sphere and the surface of a torus."

The two technical requirements are Hausdorff and first countable, but I think both are too technical to include in the lead. Rick Norwood (talk) 14:31, 30 May 2012 (UTC)

Before you make editing suggestions, please try to understand the material better. The lemniscate is not a graph because if it were a graph it would be a manifold. Namely, there is no function f(x) such that the lemniscate is a graph thereof near the central intersection point. The definition I cited comes from Arnold. It is difficult to imagine that there would be an error here though of course this is possible. Tkuvho (talk) 16:49, 30 May 2012 (UTC)

Hi Rick! I wasn't thinking of the surface of the ocean as necessarily a manifold with boundary. The image of an open set in the plane under a smooth embedding into R^3 is a manifold without boundary. The closure is usually a manifold with boundary.

I only used the surface of the ocean example because it leads naturally to the use of the word `chart'. Perhaps the surface of the ocean is a bad choice. Another is the surface of the earth. Examples of manifold are ubiquitous, and so I think it's natural to include some in the introductory paragraph.

I'm assuming that every reader has used a map, and perhaps I'm assuming too much when I expect them to have thought of a map as a flat representation of a non-flat object (Columbus).

Your proposal for the introductory paragraph is short and sweet, but perhaps too brief. It sort of says `a manifold is like Euclidean space, but if you really want to know what's going on you're going to have to slog through what's written below.'

By most standards, I'm a pretty broad mathematician who has dabbled in mathematical physics. I've worked with manifolds in physics, topology, dynamical systems, differential geometry, complex analysis....Lost-n-translation (talk) 11:13, 31 May 2012 (UTC)

I respect your expertise in these fields but content disputes should not be resolved through repeated reverts (4 of them). Tkuvho (talk) 12:22, 31 May 2012 (UTC)