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work on intro according to peer review

Dig up some old intros:

• [1]

In mathematics, a manifold is a space which looks in a close-up view like a specific simple space. For instance, the earth looks flat when you are standing on its surface. So, a sphere looks locally like a plane, which makes it a manifold. However, the global structure is quite different: if you walk over the surface of a sphere in a fixed direction, you eventually return to your starting point.

We navigate on the earth using flat maps, collected in an atlas. Similarly, we can describe a manifold using mathematical maps, called coordinate charts, collected in a mathematical atlas. Unfortunately, it is generally not possible to describe the manifold with just one chart. When using multiple charts which together cover the manifold, we need to pay attention to the regions where they overlap. A point in such a region will be represented by more than one chart, and this is where the name manifold comes from.

There are many different kinds of manifolds. The simplest are topological manifolds, which look locally like some Euclidean space. If the work manifold is used without qualification, then most likely a differentiable manifold is meant. Other types include algebraic varieties and schemes.

• [2]

In mathematics, a manifold is a space that looks locally like a specific space. For example a topological manifold looks locally like Euclidean space. The simplest example of a manifold is the space it locally looks like itself. Thus the simplest example of a topological manifold is Euclidean space itself. The surface of a sphere such as the Earth provides a more complicated example of a topological manifold. A general manifold can be obtained by bending and gluing together "the simplest example".

There are many different kinds of manifolds.

Differentiable manifolds are used in mathematics to describe geometrical objects and they provide the natural arena to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity. They also occur as configuration spaces. For example phase space is the configuration space of a particle and the torus is the configuration space of the double pendulum.

current old intro:

In mathematics, a manifold generalizes the idea of a surface. Technically, it can be constructed using multiple overlapping pieces to form a whole and is, in this sense, like a patchwork. On a small scale manifolds are always simple; on a large scale, they have rich flexibility.

Much of the terminology is inspired by cartography, which uses flat drawings to depict features on the Earth, as in navigational charts and city maps. Thus, for example, we speak of an atlas of local charts.

In the remainder of this article we will give precise definitions and explore a few of the many and diverse examples of manifolds.

new

In mathematics, a manifold is a space which is constructed, like a patchwork, by gluing and bending together copies of spaces. For example gluing the ends of a line to another line constructs a circle. In principle any spaces may be chosen as patches, but even using relatively simple spaces, quite complicated manifolds can be built. By choosing different spaces as base material, different kinds of manifold can be constructed. Much of the terminology connected with manifolds is inspired by cartography.

In physics, differentiable manifolds serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.

The new intro is false in several respects -- it is not true that any spaces may be chosen as patches. A line has no ends. The patches of a manifold must overlap. I'm going to take a shot at it. Rick Norwood 12:19, 24 October 2005 (UTC)

Of course any spaces may be chosen. From topological to Euclidean to Banach spaces to rings to whatever to half spaces, which immediately handles the manifold with boundary. A line does have ends, or whatever name is appropriate; one on one side and the other on the other side and you can glue them together. The patches of a manifold must not overlap. They are glued together. Patches which already overlap are not patches, but one patch.--MarSch 13:22, 24 October 2005 (UTC)

I'm not sure where you are coming from, here, but you are wrong on several points, according to, for example, Topology by Munkres and Algebraic Topology -- An Introduction by Massey. As a simple example, non-Hausdorff spaces may not be used in forming a manifold, nor can spaces of different dimensions.

A line does not have ends. A line segment has ends, a line is infinite and has no ends. Also, the sides of a line are different from the ends of a line.

Finally, the patches must overlap. Again, see any textbook in Topology. The atlas given in the article, where the north polar patch and the south polar patch do not overlap except at the equator, contradicts, for example, the definition in Spanier, Algebraic Topology. Rick Norwood 14:05, 24 October 2005 (UTC)

Since manifolds with boundary are mentioned in the body of the article, I've added a brief paragraph about them to the introduction. I notice that there are a number of terms that need to be added to the topology glossary. I'll work on that. Rick Norwood 12:59, 24 October 2005 (UTC)

this is a technicality which is already handled as explained above. Your rigorous definition in the lead duplicates the introduction which has that purpose. The lead should be as jargonfree as possible.--MarSch 13:22, 24 October 2005 (UTC)

The level of jargon in Wikipedia mathematical articles differs widely from article to article. For example, the article on Topological Spaces begins with jargon. I agree that jargon should be kept to a minimum in the introduction, but not at the expense of making false statements. Rick Norwood 14:05, 24 October 2005 (UTC)

Rick, I assume you have read all the old talk pages. There are many structures called manifold, including topological manifolds, differential manifolds (locally homeomorphic to R^n), manifolds with boundary, complex manifolds, Banach manifolds (infinite dimensional). We agreed that this article will serve as an introduction to these kinds of manifolds and will be kept as simple as possible. Technicalities are to go in separate articles like topological manifold, manifold with boundary, etc. The definition of manifolds with boundary can safely be kept out of the lead section without making false statements.

It is clear to me that "ends of a line" refers to half-lines or rays, but this formulation is rather confusing. However, after changing the line to an open interval, it does not match anymore with the example in the article.

Patches do not have to overlap in the sense that not every patch needs to overlap with every other patch. The sphere is covered by six patches, which each cover half of the sphere, and not only the two patches shown in the pictures. This is a standard example, which is I think treated in Hirsch. -- Jitse Niesen (talk) 16:09, 24 October 2005 (UTC)

I now see that "topological glossary" is not the place to look for definitions of topolotical terms -- they are in the article "topological space", but not in the article "topology" which refers the reader to "topolotical glossary". It seems that some organizational work is needed, and it may be that "topological glossery" should be deleted, and the brief material there moved to one of the other topology articles. Rick Norwood 13:12, 24 October 2005 (UTC)

I put "manifold with boundary" in the intro because if you click on the link you gave for "manifold with boundary" you will find it is a circular link that redirects you right back to manifold. I am well aware that many different kinds of spaces can form manifolds, but not every kind of space can form a manifold. Two patches must either be disjoint or overlap. In particular, the intersection of two patches must be an open set. In the six disk covering of a sphere, that condition is satisfied. But a simpler covering has only two disks that overlap in a neighborhood of the equator. I have never heard "ends of a line" refer to rays, and in any case rays are not open and therefore cannot be used as coverings of manifolds, only for manifolds with boundary. But a ray is homeomorphic to a half open interval, so what is the point of bringing them up here. Rick Norwood 16:20, 24 October 2005 (UTC)

reverted

I've reverted to my original lead and changed space to mathematical space, which is surprisingly still pristine.

about overlapping

A manifold is contructed from copies of a patch. The images of those patches will overlap in the manifold, but the copies are separate. Thus the present text is correct.

about ends of a line

I quote Rick Norwood: "Also, the sides of a line are different from the ends of a line." Apparently you do agree that a line has ends. For me if the line is ]-∞, ∞[, then any set ]-∞, a[ with a in ]-∞, ∞[ is one end of the line. The other end is any non-overlapping set from the other infinity. The line has these things, but perhaps "end" is not the best name for it. What better name do you suggest?--MarSch 16:06, 28 October 2005 (UTC)

about Hausdorffness

this is a technical requirement which is _often_ assumed, but by no means necessary. Prove to me that the gluing is not possible for non-Hausdorff spaces? Ever heard of schemes? --MarSch 16:09, 28 October 2005 (UTC)

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in ).

I've cut and pasted the definition of manifold from mathworld.wolfram.com. Note the "locally Euclidean" requirement. This is standard. Encyclopedia articles should contain standard definitions and terminology. Rick Norwood 22:24, 28 October 2005 (UTC)

Rick, an explicit decision was reached long ago to make this page an introduction and branching off point for all flavors of manifold, not just topological ones. Perhaps that should be stated clearly at the top of the page. It is certainly reflected in the content.

Certainly, there are many kinds of manifolds, of which the topological and the differentiable are the most important, but both begin with Euclidean space and add structure. Rick Norwood 23:33, 29 October 2005 (UTC)

Throughout the process of refinement, the opening has been a constant target of revision. That shows no sign of changing. Why? I have no good explanation. There has never been consensus on the wording, but restricting to the MathWorld definition is not consistent with the agreed aims of the article. --KSmrq^T 00:23, 29 October 2005 (UTC)

Please note, as above, that I am not trying to restrict the definition of manifold, but only keep it consistant with the way the word is used by mathematicians. Can you cite one book on the subject that does not being with compact open subsets of Euclidean space and add structure from that beginning? Rick Norwood 23:33, 29 October 2005 (UTC)

I thought you (=Rick) agreed above that different spaces can be used to construct a manifold, not just R^n? You are confusing me. But I agree, it is probably not true that all spaces can be used to construct a manifold, and we should not suggest it.

The lead of the article (the part above the first section heading) is the most difficult part to write. A lot of work and discussion went already into it, in an effort to make it short and intelligible to as many people as possible. To be honest, I don't see that the latest version achieves this (for instance, I have real difficulties understanding the example with balloons). More generally, I do not think we should put a definition in the lead.

It is hard to me imagine a lead that does not include a definition. I have no objection if you want to remove the balloon example. I may be overly fond of science fiction. A simple statement that manifolds may have more than three dimensions will suffice. Rick Norwood 23:33, 29 October 2005 (UTC)

I think that the lead as listed under ===new=== above is not that bad, except for "any spaces may be chosen as patches". It may well be better to replace the "ends of a line" by "ends of a line segment", that would also overcome the problem that some readers may have of how to make a circle, which has finite length, from a line, which has infinite lengths. However, we then lose the connection with the example further down.

I am not sure exactly which part of that introduction Rick's complaint that the patches must overlap, refers to. It seems that MarSch thinks of patches as subsets of the "simple" space (usually R^n) and Rick thinks of them as subsets of the manifold. The books that I consulted, didn't actually use the word patch, they only talked about overlapping charts.

Chart is the correct word in the case of a differentiable manifold, open ball in the case of a topological manifold. I stuck with "patch" because I wanted to change as little as possible and still bring the article in line with current mathematical usage. Rick Norwood 23:33, 29 October 2005 (UTC)

Another possible starting point is the lead MarSch quotes at Wikipedia:Peer review/Manifold/archive1:

"Imagine you have a few sheets of paper and some glue. The paper is of a special high-quality kind that can be strechted and molded into whatever shape you want and it never tears. You could cover the Earth with just two such sheets. One strechted over the North pole all the way down to Antarctica and another stretched over the South pole all the way up to Greenland, with a bit of glue at the tropics where they overlap. You have just proved that the surface of the Earth is a paper manifold!"

I like that. Rick Norwood 23:33, 29 October 2005 (UTC)

Me too. Perhaps this is clear enough to replace the example of the line-ends-circle? Note that the sheets of paper overlap _after_ you glue. The copies are separate. --MarSch 13:56, 31 October 2005 (UTC)

Or we can start from the first paragraph of the section titled "Introduction", as Dethomas suggests. Whatever we take, I think we should try to make the lead explain two things: roughly, what is a manifold, and why is it important. Perhaps we should start with forming a consensus about what we want to treat in the lead section. -- Jitse Niesen (talk) 00:41, 29 October 2005 (UTC) (via copy conflict)

It seems to me that the lead should do two things. It should inform the layperson what a manifold is, in such a way that the statements are not technical, but neither are they incorrect. Then it should include the mathematical definition, and a link to the various technical articles. I'm not sure what the objection to the current introduction is, except for the balloon example, which is easily removed. Rick Norwood 23:33, 29 October 2005 (UTC)

My main problems with the lead are that it is too long and too technical. It is uncommon to include a rigorous definition in the lead; see for instance Wikipedia:Manual of Style (mathematics).

A book that "does not begin with compact open subsets of Euclidean space and add structure from that beginning", but starts the discussion on manifold with infinite dimensional manifolds, is "Manifolds, tensor analysis, and applications" by Abraham, Marsden and Ratiu. But I agree that that is an exception. -- Jitse Niesen (talk) 23:40, 30 October 2005 (UTC)

I like Jitse's edits to my lead, removing the "every space".--MarSch 13:57, 31 October 2005 (UTC)

And my reaction.

One more thing. Rick, you should be less bold. Both here and at mathematics, there was an insane amount of discussion of many people for months and if not longer about how to best write things. As such, please be more conservative with big rewrites. Thanks. Oleg Alexandrov (talk) 02:19, 29 October 2005 (UTC)

Yes, I would not like to see things change so hastily without a thorough discussion beforehand. To future editors: please make sure you read the old talk archives before making serious changes to this page. There have been literally months of careful debate over various parts of this article, and we can all be more productive if we discuss major changes here on the talk page before they are made. - Gauge 07:21, 29 October 2005 (UTC)

I agree. Rick Norwood 23:34, 29 October 2005 (UTC)

This part is unclear and missleading

"One model of the universe visualizes it as a 3-manifold. This is not easy to picture -- you can't build a universe in a classroom -- but imagine the universe as the interior of two balloons, and then imagine one of the balloons turned inside out and the skin put inside the other balloon. One strange result is that you can start at the center of one balloon, travel through the overlap into the other balloon, cross the other balloon, reenter the overlap, and return where you started. In other words, if this model of the universe is correct it would be theoretically possible to circumnavigate the universe. Many science fiction stories have been based on this idea."

This looks like a broken attempt to describe a klein bottle, but the wording doesn't really make sense.

At this prominent position of the intro and together with the sphere and the circle as examples we get the impression that all manifolds would be bounded, which is not true as every euclidean space is a manifold, too. 84.160.236.106 17:50, 30 October 2005 (UTC)

Both good points. The balloon example was mine -- an attempt to describe the four dimensional sphere which I now see as unsuccessful. I'll remove it, if someone else has not already done so. I'll also add something so that the introduction does not give the impression that every manifold is compact. Rick Norwood 22:10, 30 October 2005 (UTC)

opening, observations

Endless edits have not made the opening better, and appear to have made it worse. Meanwhile, the rest of the article continues to get no serious attention. I'm beginning to notice a pattern about openings both here and in other articles, a hidden tension between two competing goals: (1) A tradition of dictionaries, encyclopedias, and mathematical papers is to begin with a succinct definition, then explore the consequences. (2) A tradition of tutorials and mathematical teaching is to begin with a provocative "hook" and motivational background, then formalize. We are getting the worst of both worlds. Editors would do well to decide the intent of the opening in light of the aims of this article, here on the talk page, before rewriting it yet again. (Frankly, the goals of this article, including the opening, were already discussed and agreed in the archives, now ignored.) Surely mathematicians, above all others, can respect this kind of discipline. Otherwise, the destructive thrashing is likely continue indefinitely. --KSmrq^T 23:30, 30 October 2005 (UTC)

It seems clear to me that the introduction should do the following:

1) Give a non-mathematician some idea of what a manifold is, which branches of mathematics it comes from (geometry and analysis), and where it leads (differentiable manifolds, physics).

2) Give a definition that is correct but uses as little technical language as possible.

I think that the tension you remark on, KSmrq, is between these two goals. I see no reason why we cannot satisfy both. Just what is your objection to the current introduction? Rick Norwood 00:13, 31 October 2005 (UTC)

Giving a precise definition using very little or no technical language would be challenging. In my opinion it would be best if the pedagogy came first and the formal definition appeared later in the article. This appears to be the standard for many math articles on WP. The formal definition should be clearly identified whereever it appears. - Gauge 04:45, 7 November 2005 (UTC)

Here's a reminder of some versions:

• The version stabilized at the end of the rewrite collaboration:

---

In mathematics, a manifold generalizes the idea of a surface. Technically, it can be constructed using multiple overlapping pieces to form a whole and is, in this sense, like a patchwork. On a small scale manifolds are always simple; on a large scale, they have rich flexibility.

Much of the terminology is inspired by cartography, which uses flat drawings to depict features on the Earth, as in navigational charts and city maps. Thus, for example, we speak of an atlas of local charts.

In the remainder of this article we will give precise definitions and explore a few of the many and diverse examples of manifolds.

---

• A (too) long version some time before that:

---

In mathematics, a manifold generalizes the idea of a surface. Technically, it is like a patchwork quilt, with multiple overlapping pieces joined into a seamless whole. Forcing the pieces to overlap, and not merely abut, is essential to the construction. Benefits include local convenience and global consistency, balancing flexibility with familiarity.

Much of the terminology is inspired by cartography, which uses flat drawings to depict features on the Earth, as in navigational charts and city maps. Thus, for example, we speak of an atlas of local charts.

Flat drawings are two-dimensional, as is appropriate for the surface of a sphere, a 2-manifold; but we can also have a 1-manifold like a closed loop of string, a 2-manifold which is not a sphere (such as a Klein bottle), a 3-manifold such as the set of rotations in three-dimensional space, and so on. In fact, most of the manifolds that arise in practice are abstract spaces like rotation space.

Manifolds often come with additional structure needed in their use. For example, a differentiable manifold supports not just topology, but differential and integral calculus. The idea of a Riemannian manifold led to the mathematics of general relativity, describing a space-time continuum with curvature.

The consistency of manifolds is a strong demand. For example, we cannot dangle a string (a 1-manifold) from a sphere (a 2-manifold) and call the whole a manifold. In a sense, every tiny piece must look identical, not string-like here and sphere-like there, nor a little of both (as at the attachment point). Topology has other tools, such as the CW-complex, to deal with such objects. But the demand also prevents milder violations like a cylindrical strip, because the edge of the strip looks different. For these cases it is convenient to introduce a manifold with boundary.

In the remainder of this article we will give precise definitions and explore a few of the many and varied examples of manifolds.

---

• A version at the beginning of the rewrite:

---

In mathematics, a manifold is a space that looks locally like the Euclidean space Rⁿ, and the Euclidean space indeed provides the simplest example of a manifold. The surface of a sphere such as the Earth provides a more complicated example. A general manifold can be obtained by bending and gluing together flat regions.

Manifolds are used in mathematics to describe geometrical objects and they provide the natural arena to study differentiability. In physics, manifolds serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds are used to model the spacetime in general relativity. They also occur as configuration spaces. The torus is the configuration space of the double pendulum.

---

• The current version, violating many thoughtful agreements:

---

In mathematics, a manifold is a topological space which is locally exactly like a piece of one of the familiar Euclidean spaces but which may be quite different globally. Each manifold has a dimension and a precise definition is that in a manifold every point in the space has an open neighborhood homeomorphic to (think: very much like) an n-dimensional Euclidian open ball. In many cases, manifolds are constructed like a patchwork, by bending and gluing pieces of Euclidean space. An example of a one-dimensional manifold, or 1-manifold, is a circle, formed by taking two open intervals, bending them into arcs, and allowing their ends to overlap. An example of a 2-manifold is a sphere, which can be formed from two patches -- think of one patch covering the northern hemisphere and the other patch covering the southern hemisphere -- which overlap in a neighborhood of the equator. There are higher dimensional analogs of these constructions, and there are infinite examples as well as finite. In fact, Euclidean space itself is a manifold.

The concept of a manifold is often extended to the idea of a manifold with boundary. In a manifold with boundary, a second type of "patch" is allowed, a piece of Euclidean space together with its boundary, for example a closed interval or a disk together with the circle that bounds it. The dimension of these patches with boundaries must be the same as the dimension of the other patches.

There are many specialized kinds of manifolds, which have additional structure, one of the most important of which is the differentiable manifold. Much of the terminology connected with differentiable manifolds is inspired by cartography.

In physics, differentiable manifolds serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.

---

Some of the agreements violated?

1. Keep it short.
2. Avoid use of mathematical technical term "space".
3. A manifold does not always have a fixed dimension.
4. Neighborhoods can be spaces other than Rⁿ.
5. Defer mention of boundary to intro.

(This and more can be found in the history of the article and its talk archives.) The current version is worse than any of the previous short versions. --KSmrq^T 01:56, 31 October 2005 (UTC)

1) Keep it short. I agree.

2) Avoid the word "space". Why on earth? "Space" is one of the few mathematical words most people have at least heard.

3) I have never seen any reference that calls something a manifold when it doesn't have a fixed dimension. A space pasted together of parts of different dimensions is called a CW-complex, not a manifold. The only thing I can think of that might fit your description is that in a manifold with boundary, the dimension of the boundary is one less than the dimension of the manifold. I need to see a reference on this one.

4) Neighborhoods can be spaces other than Rⁿ. Give me a reference. Of course, there are manifolds that have more structure than Euclidean space, but less? Show me.

5) I'm not sure what you mean here, but I assume you mean defer mention of boundary until after intro. There is no problem with this. The problem was that in the previous version there was no definition of manifolds with boundary, and the link to "Manifolds with boundary" was a circular link that took you right back to the "Manifolds" page, which talked about manifolds with boundary without actually defining them.

I gather you disagree with my requirement #2 -- that the intro should have a correct definition of manifold -- and think that the intro should just give a layman's idea of what a manifold is. Rick Norwood 13:23, 31 October 2005 (UTC)

Some comments on the earlier introductions.

Intro 1 -- the use of "surface" gives the false impression that all manifolds are two-dimensional. The phrase "rich flexibility" is vague and misleading. The use of charts is only in differentiable manifolds (and those with more structure than differentiability).

I agree with your observations. I don't understand why you say that top. manif. have no charts. From one of your earlier comments, perhaps you mean that every point should have a neighbourhood homeomorphic to a Euclidean open ball. Since the ball is homeomorphic with the whole space, this means exactly that you have a chart.--MarSch 13:51, 31 October 2005 (UTC)

Intro 2 -- too long, as you note. It gets right the requirement that patches must overlap, and that the dimension must be fixed. Does not limit charts to differentiable manifolds. And the sentence "Flat drawings are two-dimensional, as is appropriate for the surface of a sphere, a 2-manifold..." is confusing.

Intro 3 -- Not bad -- and rather close to the current version, which essentially just adds a few examples. Rick Norwood 13:31, 31 October 2005 (UTC)

Ad 3) If you had read the old discussion pages, specifically, Talk:Manifold/Archive2#Time to merge the stuff?, you would have seen that I there give the reference Abraham, Marsden and Ratiu, Manifold, Tensor Analysis, and Applications, Definition 3.1.7.

Ad 4) The same book starts with defining manifolds modeled on Banach spaces with possibly infinite dimensions (Definition 3.1.3).

-- Jitse Niesen (talk) 14:06, 31 October 2005 (UTC)

Please. So long as some editors push towards a formal definition and others push towards informal intuition neither will be satisfied. Every time an informal phrase is used, the one camp objects that it's misleading or wrong. Every time a technical term or definition is used, the other camp objects that it defeats the aim of speaking to a general audience. For example, it is ironic that Rick says "why on earth" avoid the word space. What we decided was that the general public is familiar with the word "space", only they understand it in phrases like "the space program", not in its technical mathematical sense. The word "surface" avoids this problem; but even when combined with "generalizes" it offends a formalist. This is madness. Detailed comments on specific versions or specific violations waste everyone's time; the tension must be addressed. --KSmrq^T 19:55, 31 October 2005 (UTC)

We've been here before.

"A (topological) n-manifold is a separable metric space each of whose points has an open neighborhood homeomorphic to either Rⁿ or to Rⁿ₊..." 3-Manifolds, John Hempel.

Clearly the requirement that the neighborhoods be open means that in a connected space, such as the circle written as the union of two open neighborhoods, those neighborhoods must have open, non-empty intersection.

I could cite half a dozen other books that say the same thing. Rick Norwood 15:17, 31 October 2005 (UTC)

I'm guessing here that this comment is directed at me. I completely agree. The problem I'm having is with your blindly putting "overlapping" in, without regard for context. When gluing a manifold together the pieces you start with are completely separate. Only after you've glued them together do they overlap anywhere. This also implies that you CANNOT glue them together where they overlap and accomplish anything. This would be a non-action. When you glue you specify where you glue and this can also be done by MAKING the separate copies overlap and then glue them together where they overlap. It is a clumsy way of saying it, but at least it is correct and uses the word "overlapping" which should make you happy. --MarSch 12:01, 1 November 2005 (UTC)

Abraham, Marsden, and Ratiu

"Ad 3) If you had read the old discussion pages, specifically, Talk:Manifold/Archive2#Time to merge the stuff?, you would have seen that I there give the reference Abraham, Marsden and Ratiu, Manifold, Tensor Analysis, and Applications, Definition 3.1.7. Ad 4) The same book starts with defining manifolds modeled on Banach spaces with possibly infinite dimensions (Definition 3.1.3). -- Jitse Niesen (talk) 14:06, 31 October 2005 (UTC)"

As I'm sure you know, the book you cite is notorious for the number of errors it contains. However, both of the definitions you cite are correct, and both agree with my points. Definition 3.1.7 says that a manifold is made up of open subsets of a Banach space. A Banach space is a normed linear space and a linear space is Rⁿ with additional structure added, so every manifold is made up of pieces homeomorphic to Rⁿ. Definition 3.1.3 mentions that the dimension may be infinite. That is not a problem, but you cannot have some of the pieces of finite dimension and some of infinite dimension, nor can you have some pieces of countably infinite dimension and some of uncountably infinite dimension. (I do not think we need to go into the difference between the topological dimension and the banach space dimension.) The important thing is that, according to your own reference, every piece has the same dimension and if pieces intersect, they intersect in open sets. Rick Norwood 18:43, 31 October 2005 (UTC)

Somebody had better say 'connected' very quickly. Charles Matthews 18:54, 31 October 2005 (UTC)

I didn't know about the bad reputation (I only took out the book from the library when writing this article); thanks for the warning. I don't think the definitions agree with your points: Defn 3.1.3 says that a neighbourhood can be a space other than R^n, namely an infinite-dimensional Banach space, and Defn 3.1.7 says that the dimension is not fixed. However, and this is indeed the important thing, I do agree with "every piece has the same dimension and if pieces intersect, they intersect in open sets", in the interpretation: the dimension is the same at every point in a connected component, and the intersection of two charts is open. -- Jitse Niesen (talk) 19:05, 31 October 2005 (UTC) (via edit conflict, but I did mention "connected")

Good. We agree. (It was to take care of the possibility of a manifold that was not connected that I included the qualifier "if pieces intersect".) The introduction should certainly include the possibility that the "n" in Rⁿ could be ω. The introduction should mention the infinite dimensional case. With all this in mind, I am going to attempt another rewrite. I'll try to change as little as possible and keep it short. Another thing that needs fixing -- the word "chart" is only used in the differentiable case, in the topological case, use "open n-ball". But I'll leave that for later. One thing at a time. Rick Norwood 22:24, 31 October 2005 (UTC)

(Had an edit conflict. Sorry about the ordering of comments.) The correct title uses the plural: "Manifolds, Tensor Analysis, and Applications", ISBN 0387967907. I can't speak to the errors from direct experience, not having a copy at hand; however, the Amazon reviews complain of typos. One mathematics professor describes it to his students as "A very well respected, application oriented text that immediately works in infinite dimensions." There is a SIAM professional review, but I can't read my downloaded copy for some reason. Marsden has a web page saying a third edition is in preparation; perhaps it will have fewer typos. My personal experience of Ralph Abraham is positive, and Marsden is exceptional. It is hardly likely they would make major mistakes on definitions at the core of the book, especially aspects that are essential to the applications they discuss. Our prior discussion proposed that we distinguish n-manifolds, which are mainstream, while acknowledging these more general manifolds with mixed and infinite dimension, which are outliers but valid and notable. --KSmrq^T 22:53, 31 October 2005 (UTC)

Latest rewrite

I've done a careful rewrite, based on the agreement above that every manifold has a fixed, but perhaps infinite, dimension.

I'll pause here, before tackling the fourth paragraph of the introduction, which needs only a brief mention that "charts" are only used in the case of differentiable manifolds. Rick Norwood 22:51, 31 October 2005 (UTC)

Why will you not listen? Your rewrite will not stand. No rewrite will until the underlying tension is addressed. Beyond that, you are mistaken about the fixed dimension; each connected component has fixed dimension, but not the complete manifold (in the Abraham et al. book). --KSmrq^T 22:57, 31 October 2005 (UTC)

I have the book right here in front of me. Definition 3.1.7 "A differentiable manifold M is an n-manifold when every chart has values in an n-dimensional vector space." You are simply wrong on this; I could quote a dozen different books that say the same thing. Dimension is an essential characteristic of a manifold. A space with parts of differing dimensions is not a manifold but rather is a CW complex. I won't listen to you because you not only disagree with all of the standard books on the subject, you even disagree with the non-standard and error ridden book you yourself cite as a reference. In short, you don't get to tell me No. You don't have the authority. Rick Norwood 01:09, 1 November 2005 (UTC)

Please understand me. I applaud your enthusiasm for improving the article. Honestly and sincerely, I do. My remarks are not meant to belittle you, nor to thwart your efforts. Quite the contrary, I am trying to help. I am not your adversary here; I am your fellow traveler. I do not regard you as stupid or ignorant. (Moreover, I abhor argument from authority, rather than from evidence, and do my best to stifle the temptation.) I do think your behavior indicates lack of Wikipedia experience.

You continue to ignore a fundamental disagreement about criteria the opening should satisfy. (And by the way, I understand your opinion about n-manifolds; read the archives and you'll find me making similar arguments, though I have since been persuaded otherwise.) It is a courtesy that I discuss this with you, because I do have the authority — as does anyone — to revert or rewrite anything you write. For example, you use the word "space" despite being told of a thoughtful agreement to avoid that, and why. In doing so, you seem to hold yourself above mathematicians who have worked hard on the article so far, and beg to be reverted. Perhaps that was inadvertent; I hope so. We're not all fools here; it won't hurt to listen more and write less. All of us want a great article. I look forward to working with you to achieve that. --KSmrq^T 02:50, 1 November 2005 (UTC)

Rick, most people disagree that disconnected parts of a manifold should have the same dimension, although Jitse just contradicted himself so severely that I'm not sure where he stands anymore. Furthermore since this is supposed to be a non-technical article such technical details should be left out. The fact that some people define a topological manifold as having one dimension and others allow more dimensions should be mentioned, but for this the terminology "manifold of dimension n" was invented. There is no reason to make such a restiction in this article. --MarSch 12:12, 1 November 2005 (UTC)

Topological manifolds do have charts. And transition maps. Otherwise the def of diff. manif. you quote does not work. Open ball neighbourhoods ARE exactly charts.--MarSch 12:18, 1 November 2005 (UTC)

To be clear about where I stand: different connected components may have a different dimension (at least according to some authors), so for instance

$\{(x,y,z)\in \mathbf {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\}\cup \{(x,y,z)\in \mathbf {R} ^{3}\,|\,x=0{\mbox{ and }}z=2\}$

(the disjoint union of a sphere and a line) is a manifold. This is inconvenient in many cases, so sometimes it is stipulated that manifolds are connected, or that the dimension of every connected component is the same.

In my reading, the definition "A differentiable manifold M is an n-manifold when every chart has values in an n-dimensional vector space" means that there is a class of differentiable manifolds called "n-manifold" in which the dimension is fixed (namely n), but it does not say that all differentiable manifolds have a fixed dimension. In fact, a few lines further down, in the same defintion, the authors say "n can vary with the component" (I also have the book in front of me, though a different copy).

About topological manifolds having charts, I am going to cite another book, which you hopefully like better: "Differential Topology" by Morris W. Hirsch (Springer, 1976). At the very start of Section 1.1, he says:

"A topological space M is called an n-dimensional manifold if it is locally homeomorphic to R^n. That is, there is an open cover {U_i}_{i∈I} of M such that for each i∈I there is a map \phi_i : U_i \to \R^n which maps U_i homeomorphically onto an open subset of R^n. We call (\phi_i,U_i) a chart (or coordinate system) with domain U_i."

(Obviously, Hirsch uses proper TeX for the formulas.) -- Jitse Niesen (talk) 13:02, 1 November 2005 (UTC)

Ksmrq -- I got a little hot under the collar when I read your "Will you not listen?" I apologize.

Jitse -- I now see that comment in Abraham. I can only say that this is not standard, but since it is used by some mathematicians, then the article should mention both points of view. (I notice a similar problem in the Wikipedia article on Rings, where it requires rings to have a multiplicative identity, and then mentions an alternative definition which agrees with what I have always understood, that rings may lack a multiplicative identity.) As for calling the n-balls in a topological manifold "charts", the book you quote is a book on differentiable manifolds. Books on topological manifolds that I have read never call the n-balls charts. But, once again, we can present both points of view.

I think the reason for the strong disagreements here is that some of us work in topological manifolds and some in differentiable manifolds and these areas have grown apart over the last thirty years or so, and now use the same terms with different meanings. I once heard a series of lectures by Chern on differentiable manifolds, and the definitions I know mostly came from those lectures, but that was a long, long time ago, in a galaxy far, far away. Rick Norwood 15:29, 1 November 2005 (UTC)

Apology accepted; not a problem.

Slightly off-topic: Wikipedia presents a number of challenges, some social (the joys of vandalism and cranks and consensus), some technical. The ring identity question is easily handled in a paper or a book, because the author can assert whichever convention best serves their purpose; not so here. The diversity of authors and articles makes wiki-wide consistency difficult, if not impossible. (We do try for self-consistency within an article.) The identity convention has been discussed, and I believe I've seen it asserted as a standard for Wikipedia (though I don't find it in our piece of the Manual of Style.)

I can't imagine an n-ball being called a chart; the chart is a bijection, the ball is an open set. Or am I missing something? --KSmrq^T 19:23, 1 November 2005 (UTC)

Agree with KSmrq. I have never seen charts confused with open balls before; they aren't even the same type of object (one is a mapping, the other an open set). I'm not even sure what an "open ball" means in a topological space without a metric. - Gauge 04:45, 7 November 2005 (UTC)

Moving right along

I have noticed that one problem in Wikipedia, not just here but in many other articles as well, is that it is easy to spend countless hours on the opening paragraph and give not nearly enough attention to the rest of the article. I am going to move on, and fix the last two paragraphs of the introduction. Rick Norwood 01:12, 1 November 2005 (UTC)

I've worked on section three. It was a good section already, but occasionally too technical and occasionally it repeated itself. Also, since it sometimes refers to "charts" as "maps" I think we should avoid the use of "map" meaning function, and so I've changed "transition maps" to "transition functions".

The next task, it seems to me, is doing something about the extensive repetition. Rick Norwood 01:40, 1 November 2005 (UTC)

I prefer the term "map" for a function. "Transition function" sounds awkward, which means that I probably haven't heard people use it often/ever. If there are places where "map" should be replaced with "chart", please do so, keeping in mind that charts are maps. - Gauge 04:45, 7 November 2005 (UTC)

Once again, we have a difference in usage between those who work in topological manifolds and those who work in differentiable manifolds. In topological manifolds, you rarely hear either "map" or "chart". In differentiable manifolds, you hear those two words all the time. But we are writing an article that is, at least in part, for the non-mathematician, who, hearing "map", thinks of a picture of nations and cities, which is more appropriate to the idea of a chart than to the idea of a diffeomorphism. Rick Norwood 13:14, 7 November 2005 (UTC)

In most of my experiences with topology, functions between topological spaces (differentiable or not) have been called maps. Whether a "map" conjures up geographical notions is in the eye of the beholder, I suppose, but personally it never occurred to me to make this association. We can link to the word map easily enough. If the term "function" really is preferred to "map" in topological manifolds, then we can make a special note about that case, but in general usage I almost always have seen "map" used instead. By the way, could you give some references to this usage of "function" in topological manifold theory? - Gauge 04:12, 9 November 2005 (UTC)

Actually, the word used in topological manifolds is homeomorphism. The article is a kind of sometimes awkward compromise between using words the way mathematicians use them and using words the way the layman uses them. The word "map" is a perfectly good mathematical word, and I use it all the time when writing papers. In this article, which is much more apt to be read by a layperson than by a mathematican, who already knows everything in the article, I think "map" gives the wrong impressing. But, mathematically, there is nothing on earth wrong with it. Rick Norwood 23:03, 9 November 2005 (UTC)

Recent rewrites

MarSch's favorite "Antarctica gluing" thing which was rejected a while ago found its way back in the article. This article clearly suffers from editorial creep, and I suspect the current article version is not the best one.

MarSch, you better keep yourself busy finishing topological manifold and differential manifold which for now are just random cut&paste edits by you. Oleg Alexandrov (talk) 20:33, 17 November 2005 (UTC)

One thing at a time Oleg. And when will you learn not to tell people what to work on? It got itself edited into oblivion, but I don't remember much discussion about that. Besides it would make for a good picture for the intro.--MarSch 23:57, 17 November 2005 (UTC)

Vb's edit is an improvement. MarSch's edit is, at best, a mixed bag. For example, differentiable manifolds are NOT simpler than topological manifolds. Differentiable manifolds ARE topological manifolds, with additional structure added. And the sentence "People once believed the Earth was flat," has nothing to do with mathematics or the article or anything else, and is debatable -- certainly intelligent people knew the Earth is round from the earliest times, by observing the shadow of the Earth on the moon during an eclipse. Anyway, it has no place in the article. Rick Norwood 23:09, 17 November 2005 (UTC)

Topological manifolds and diff. manifolds are of about the same complexity, but diff. manifolds are closer to intuition, which is the kind of simplicity which is more important than lack of structure. As you can see from the history this was not my first reaction, but the fact that topologically there is no difference between a cone and a plane convinced me that they are less intuitive. The flat earth piece is not new, but something I salvaged from the introduction. I didn't introduce it and I don't care if it goes. Actually I would support that.--MarSch 23:57, 17 November 2005 (UTC)

At first, I was going to try to work with MarSch's edit instead of reverting it, but the first three or four changes I checked were totally bad, dogmatically POV. MarSch, if you read this, please discuss what you are trying to accomplish here.

From your above comment it is clear you didn't really check what I did. --MarSch 23:57, 17 November 2005 (UTC)

I was tempted to revert all the Antarctica stuff, but one thing at a time. Rick Norwood 23:29, 17 November 2005 (UTC)

Ok, MarSch, I won't revert this time. Instead I'll encourage someone else to revert. If nobody does, we can take it from there. Rick Norwood 00:57, 18 November 2005 (UTC)

MarSch, you attempted to rewrite manifold once in June, and second time around September. Both times your edits were deeemed inferior and rejected by the community. A more sensible person would have learned the lesson. Oleg Alexandrov (talk) 02:26, 18 November 2005 (UTC)

If you had any real arguments you would have used them.--MarSch 11:44, 18 November 2005 (UTC)

I agree all changes made by MarSch were not optimum but some ideas were good. Such as in the lead

Viewing a complicated space as glued together from simpler spaces and thus as a manifold is a powerful way of gaining a better understanding of that complicated space. It makes it possible to transport mathematical structures defined on the simpler space to the more complicated space. By choosing different simpler spaces, different kinds of manifold can be constructed, such as topological manifolds, differentiable manifolds and analytic manifolds.

I therfore think as Oleg, MarSch was a bit too bold. But Oleg also, his revert was a bit bold too. Moreover it is clear this article is crying for a figure in the lead. Vb

I might as well waste some more time and try and explain why I rewrote much of the beginning of the article. FOllowing Vb's peer review comments I found that the section about charts atlases and transition maps was incorrect and arbitrary. Furthermore the introduction was almost completely a copy of the lead. So I merged lead and intro as also someone suggested in peer review and rewrote the other section. The example in the lead didn't survive, because people didn't agree with the material which should be used. Not paper, not rubber, not cloth. Well who cares what exactly is used. The example itself is graphic, simple and intuitive and allows for a nice picture in the lead. What more can you want? --MarSch 11:44, 18 November 2005 (UTC)

There is no 16:22 12 Nov so I'm not sure exactly how much has been reverted. I'm going to try to put back in some of the good stuff, from Vb and, yes, from MarSch. But, MarSch, your rewrite was totally unilateral, massive, and undiscussed. You must have known it would get reverted. For one thing, differential manifolds seem natural to you, but 95% of the human race has no idea what "differentiable" means, while everyone can understand gluing two pieces of paper together. Rick Norwood 13:51, 18 November 2005 (UTC)

PS: The revert is to 23:22 12 November. Rick Norwood 13:57, 18 November 2005 (UTC)

Working on the introduction, I took out the "people thought the earth was flat" bit again, which nobody seems to like, and eliminated some of the duplication with the lead, which MarSch rightly complains about. I will pause now for comments before doing anything else. Rick Norwood 14:11, 18 November 2005 (UTC)

Every rewrite is of necessity unilateral and massive. I was addressing some duplication and mistakes, so I don't feel it needs to be discussed that they be fixed. Just because an edit is big and you don't want to spend time checking changes is not a valid reason for reverting. Do you think topological is more clear to most people than differentiable? If not then what do you mean? smoothly deformed is something which people will understand so at least now the evil math word gets explained and is seen to correspond with intuition. This is what I tried to do first to topological, but I found it does not correspond to intuition, since it allows continuous deformation, so you wouldn't be able to see any difference between a smooth piece of paper and a piece with a crease, which is not reality.

If you want to lead by example, then you shouldn't start by unilaterally reverting. I've continued working on my version. I intend to rewrite all of the article into something coherent, easy to understand and unriddled by arbitrary facts or details or technicalities. --MarSch 14:55, 18 November 2005 (UTC)

The biggest problem here is that you say things that just are not true. For example, "every rewrite is of necessity unilateral and massive". NO NO NO NO NO! A good rewrite attempts to do a little at a time, preferably one section at a time, and to seek a consensus. Your attempts to remake the whole article all at once are just going to get reverted.

I am not the only person reverting your rewrite. Nobody else has come forward and said that they find your rewrite an improvement. If you really feal a rewrite is necessary, I suggest that you first say, here, why you feel the current version needs improvement, and then change one section and wait for comment. Otherwise we are just going to take turns reverting what you have written. Rick Norwood 15:09, 18 November 2005 (UTC)

On rewriting things

MarSch and Rick: you both reverted the article twice today. One more move and you will be in problem with the three revert rule.

Neither of you was much present around when Jitse Niesen, KSmrq, Gauge and Marcus Schmaus spent a huge amount of time writing and discussing this article. MarSch, doing such huge unilateral rewrites is wrong. You have been here long enough to learn that.

Please let us leave this article this way for a while, and see what other people have to say. Thank you. Oleg Alexandrov (talk) 16:24, 18 November 2005 (UTC)

Note that all of my reverts have been reverts of MarSch's reverts, that is, returns to the status quo. Any future reverts of MarSch that are necessary I'll leave to you. Rick Norwood]] 19:16, 18 November 2005 (UTC)

An alternative to a revert war is for MarSch to pursue a one-man variation of Jitse's approach. Start a complete rewrite on a subpage in user space, and polish it. When satisfied, propose it on WikiProject Mathematics as an alternative to the current page. This allows consensus without collaboration, strange as that may sound.

Off-topic: This conflict may reflect genetic defects in Wikipedia or its culture. Articles do not consistently progress from weaker to stronger; and as a great deal of energy is wasted, so is the knowledge, talent, and enthusiasm of desirable contributors. A peer-reviewed journal article can never get better, neither can it get worse; but it can get supplemented by a later article, perhaps from a different source. The fluidity of Wikipedia is a structural problem for its quality, avoided in open-source software by various mechanisms lacking here. It's not just misguided edits; look at the constant vigilance required to deal with blatant vandalism. --KSmrq^T 16:54, 20 November 2005 (UTC)

Yes you are right, I stated something similar at talk:Mathematics not so long ago. Charles had an interesting reply which you should read. Oleg Alexandrov (talk) 16:57, 20 November 2005 (UTC)

Mathematics is an article I instinctively avoided touching. I'm either too wise or too cowardly to jump into that snake pit. Anyway, I think Charles' enthusiasm for quick response is, contrary to his assumption, not incompatible with stability. The typical software release approach says: here's a stable version, and here's a "bleeding edge" version. Choose according to your interests and needs. If I'm a teacher in elementary school, I'll point my students at a stable version known to be reliable, NPOV, and free of vulgar vandalism. If I'm a researcher, I'll look at both versions, and also the talk page. A deeper question concerns "gatekeepers". Open-source software allows anyone to edit their private copy, but those changes must be vetted before they become part of the public copy; peer-reviewed journals are similar. This seems very much at odds with the approach of Wikipedia and blogs, though Wikipedia does have a model of quality control in the Featured Article process. But safety and security are always in tension with speed and convenience. Wouldn't it be nice to never have to worry about locking keys in a car? Wouldn't it be lovely if airport baggage screening could disappear? And wouldn't it be miraculous if police and politicians could always be trusted to act honestly and in the common good? I don't think it is possible to stifle the vandalism, wars, and ill-conceived edits that currently afflict Wikipedia without also inflicting a cost in timeliness and convenience and other less tangible qualities.

The core question is, who do you trust to do what? This is not just vandals on one side and admins on the other. I trust myself to write good articles on some topics, but I still want others looking over my shoulder because I don't trust myself to be perfectly clear and perfectly correct even in my areas of expertise. In Tolkien's Lord of the Rings, the wizard Gandalf is terrified when Frodo offers him the One Ring, knowing that even he—good as he is—cannot be trusted that far.

"You are wise and powerful. Will you not take the Ring?"

"No!" cried Gandalf, springing to his feet. "With that power I should have power too great and terrible. And over me the Ring would gain a power still greater and more deadly."

Ah, well; I fear we have strayed much too far off-topic when I begin quoting Tolkien in a discussion of manifolds! --KSmrq^T 22:09, 20 November 2005 (UTC)

Actually I think of stability here in terms of 'ballast': fundamental reference material, the structure of redirects and hyperlinks that are not in dispute, and so on. Some things swirl around in verbal clouds, but one can see an advance in sheer 'documentation' of areas of knowledge going on as well. Charles Matthews 23:08, 20 November 2005 (UTC)

To reply also a bit to KSmrq above. I also avoided editing mathematics for the same reason as you. I belive people who are really itching to get to that one first thing in the morning are either fools or very smart, and very likely the former. Oleg Alexandrov (talk) 23:25, 20 November 2005 (UTC)

gluing along boundaries

This is called a quotient space, and it sometimes, but not always, results in a manifold. The potential for misleading the reader is so great that I don't think this paragraph sould be in an introductory article. If we want it in the article, then we need to present necessary and sufficient conditions for when a quotient space is a manifold, and some examples, such as the torus formed as a quotient space of a square. Rick Norwood 19:36, 18 November 2005 (UTC)

I changed the paragraph to stress this point. This topic is so large that it really needs its own article. I suppose quotient space would be the appropriate starting point. - Gauge 02:08, 17 December 2005 (UTC)

Rick: I think it may be misleading to call a manifold resulting from a quotient construction a "quotient space". Quotient spaces in general need not be manifolds, and I think we should keep this distinction clear. Secondly, the point I was trying to make in the remark about "pathological spaces" is that quotients are not even remotely nice in general; here I think of the bad foliations arising in noncommutative geometry, for example. There are also examples of nonmeasurable sets arising naturally as quotient spaces. - Gauge 02:37, 23 December 2005 (UTC)

In my own area, knot theory, "quotient space" is the standard expresssion used when a quotient map results in a topological manifold. For example, the standard view of the torus is as a quotient space of a square -- similarly with the projective plane, the Klein bottle, Moebius strip, etc. Three manifolds are formed by gluing torii into knot compliments. I realize that other specialties may use different terminology. One thing I've discovered writing for Wikipedia is that mathematical terminology is not nearly as standardized as I once thought it was. Have you see their definition of a ring! Rick Norwood 16:38, 23 December 2005 (UTC)

A ring is an additive category with one object, of course ;-) Yes, I agree that the above things you mention are quotient spaces. My gripe is that quotient spaces need not be manifolds, and we should make this clear. This is the sentence that I think may be misleading: "There is, however, no reason to expect the result of this operation to be a manifold. If it is, it is called a quotient space." I would rather say that you get a quotient space regardless, but it may not always be a manifold. - Gauge 00:12, 25 December 2005 (UTC)

Good point! Do you want to do the rewrite, or shall I? Rick Norwood 01:49, 25 December 2005 (UTC)

Okay, I took a stab at it. Let me know what you think. - Gauge 01:46, 26 December 2005 (UTC)

Looks good. I changed one word. Rick Norwood 17:16, 26 December 2005 (UTC)

a Euclidean space

I am not a native speaker of English but wouldn't it be rather an Euclidean space? 84.160.245.39 20:13, 29 December 2005 (UTC)

Yes, it should be "an". Rick Norwood 22:54, 29 December 2005 (UTC)

Like 'an Yuletide greeting from an user'? No, it is pronounced 'yewclidean', so it is a Euclidean space. Charles Matthews 22:57, 29 December 2005 (UTC)

And an 'appy 'olliday to you. Rick Norwood 23:11, 29 December 2005 (UTC)

I think this was also discussed in the archives. My ear agrees with Charles Matthews; the correct article is "a", not "an". Also, omitting the article would be undesirable; it would shift the meaning. --KSmrq^T 05:14, 30 December 2005 (UTC)

New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as manifolds, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

Advanced math? Where?

Well, I miss parts of the old article that actually talked about "advanced math". The current article seems to have chopped out ... well, anything "advanced", or at least anything that was interesting (to me). Where's the description of the Kahler manifold, the symplectic manifold, the Calabi-Yau manifold, etc? I understand that most lay readers won't miss that content, but its a bit of a shame to see it gone. There seems little here of interest to anyone who actually took math classes in college. :( linas 07:11, 28 January 2006 (UTC)

I would welcome short descriptions of those under the "Other types and generalizations of manifolds" section, and of course each of these has its own article as well. The articles topological manifold and differentiable manifold have been designated as the places for more technical details about "manifolds" in common parlance. - Gauge 08:51, 28 January 2006 (UTC)

For what its worth I've added the three mentioned to List of manifolds. --Salix alba (talk) 10:42, 28 January 2006 (UTC)

I've removed them, because those are structures, not particular manifolds. Charles Matthews 12:24, 28 January 2006 (UTC)

OK, so those are structures. I don't know if/where we might have articles listing all of the "manifolds endowed with additional structure.".

The current "advanced" section lists four types: is this the complete list? I can certainly imagine taking any fairly-well understood topology, and using it as a base space which supports charts, defining glueing, etc. Is there a category that does this? i.e. a "chart" category? (Brain fart: at least part of what chaos/fractal people try to do is to understand the map from the Cantor set into the dynamical system they have at hand, although the language of charts/glueing is rarely used). linas 16:15, 28 January 2006 (UTC)

Manifold with boundary

Ithink we need to have a brief section on what a manifold with boundary is. --Salix alba (talk) 10:30, 28 January 2006 (UTC)

Comments

I saw this article on WP:FAC. It doesn't look like it's going to pass, but I have two comments so it can be improved:

1. The first sentence, while technically accurate, does little to capture the spirit of what a manifold is for people without a strong mathematical background. People don't think of manifolds as glued together collections of charts, or as topological spaces with a complete atlas of charts, they think of manifolds as spaces locally homeomorphic to n-dimensional Euclidean space. So why not, something like "A manifold is a mathematical space that in a small neighborhood of any point on the manifold looks like n-dimensional Euclidean space, while the global structure looks quite different. An example is the torus..." That was pretty cumbersome, but you get the idea. Why mention charts? They seem more fundamental to the definition than to the idea.
2. Why not discuss the classification of 2-manifolds in terms of Euler characteristic (or genus for orientable manifolds)? I realize that this is a page about all kinds of manifolds, but it is a very easily described, famous result that non-mathematicians might find it enlightening to hear about. –Joke 15:03, 31 January 2006 (UTC)

Yes sounds good to me. How about a manifold is a mathematical space which looks locally like a n-dimensional Euclidean space. Probably worth avoiding neigbourhood in first line. --Salix alba (talk) 15:17, 31 January 2006 (UTC)

I agree. I now see that this has been discussed above in opening, observations. The old version [now at Talk:Manifold/old version] also seems to have a less cumbersome introduction. I think making charts the first thing discussed about a manifold is like making the Peano axioms the fundamental thing discussed on the natural numbers page. "The natural numbers are a collection of objects beginning with "0" which have the property that if a is a natural number, then so is a+1..." It's missing the forest for the trees. Or the atlas for the charts. Or... –Joke 15:39, 31 January 2006 (UTC)

Thanks for stopping by to play. An uncountable number of people have spent an uncountable number of months considering an uncountable number of variations of the first line and the rest of the intro. Well, that's a slight exaggeration; but it has been beaten to death, as a review of the talk page and its archives will show. The best way to improve the article at this point is to concentrate attention on the substance, the organization and content of the body (which has been beaten on, but far less). --KSmrq^T 15:37, 31 January 2006 (UTC)

Perhaps, but I don't think it's any good as it is. The responses on the WP:FAC page seem to indicate that it is not accessible enough, and frankly, I agree. Such a fundamental topic in mathematics deserves a description that anyone can understand relatively easily. –Joke 15:42, 31 January 2006 (UTC)

The article as it stands is a compromise between people who came to the idea of a manifold via point set topology and people who came to the idea of a manifold via calculus on manifolds. The opening paragraph was a compromise that I don't think either side was entirely satisfied with, but at least it was something both sides could accept, after a long and heated debate. Maybe it is time to revisit the issue, especially in the light of the negative reaction from non-mathematician reviewers (as contrasted with the view of people with a little math (even if they are in 10th grade) which is that this article is about as readable as an article on manifolds gets). Spivak, anyone? Rick Norwood 20:26, 31 January 2006 (UTC)

Consider

I don't claim to be an expert on grammer, but there seem to be a lot of Consider .... statements about, is this good style? --Salix alba (talk) 16:22, 31 January 2006 (UTC)

It is bad style. That's chatty mathematicsese. –Joke 02:22, 2 February 2006 (UTC)

New intro

I would like to suggest that we work on a new intro here, and then take it to the article, rather than doing our rewrite in the article. Here is what I would like to see. Comments?

In mathematics, a manifold is a geometric object which is simple locally but which may be complicated when viewed as a whole, and which may have additional structure. A sphere -- an idealized version of the surface of the Earth -- is a manifold. Locally, the Earth seems to be flat, but viewed as a whole, it is round. And the surface of the Earth can be given additional structure: lines of latitude and longitude, for example.

Every manifold has a dimension. One-dimensional manifolds are called 1-manifolds. A straight line is a 1-manifold. So is a circle. Two-dimensional manifolds are called 2-manifolds. A sphere is a 2-manifold. So is the surface of a donut, a torus. A solid ball is a 3-manifold. So is the universe we live in (except for black holes). Einstein suggested that space-time may be a four-dimensional manifold.

Examples of manifolds with additional structure include differentiable manifolds, symplectic manifolds which serve as the phase space in classical mechanics, and four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.

A technical mathematical definition of a manifold is given below.

Rick Norwood 22:05, 31 January 2006 (UTC)

Mani-fold, mani-spindle, mani-mutilate

• JA: Perhaps it would be better to write a couple of entirely independent and separate articles. A couple of title suggestions may be found in the head. Jon Awbrey 22:26, 31 January 2006 (UTC)
  • Apologies for tangentiality, but if I ever write scifi I'm so including the word "manibend" in the technobabble. --Kizor 23:16, 31 January 2006 (UTC)

Comments II

I've been thinking about this page over the past couple of days. Here are some ideas that I think are good:

• History should be at the top (i.e. the first section or the second section, after overview). This is an encyclopedia article, not a math textbook, and historical context is important.
• I can see that there has been tension between the topological manifold people and the differentiable manifold people. Both definitions need to be mentioned, and should probably be stated just as they would be in a math textbook, and repeated in more informal language. Also, it would be nice to give an example of a topological manifold that does not admit any differential structure and maybe mention that in dimensions four and up topological manifolds can have inequivalent differential structures (I don't have the paper for this handy...).
• Why is there a whole section about orientability? I agree it is important, but it is surely doesn't deserve as much space as the history section, differentiable manifolds and topological manifolds combined. It is, in my opinion, no more important than mentioning, say, the Hausdorff property, compactness, separability, etc...
• The section on Charts, Atlases and Transition Maps could probably go with differentiable manifold.
• Perhaps the litany of examples can go near the end? And also, can we include a mention of the classification of 2-manifolds there?
• The sentences "... or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional "string" to three dimensional "ball" makes an object called a CW complex, not a manifold." in the overview are unfortunate, because they inundate a reader with technical detail before it is necessary. Why not move most of this information to the end, with "other types and generalizations of manifolds"?

Anyways, I'm going to poke away at doing some editing. Hopefully I won't raise too many hackles. –Joke 23:00, 1 February 2006 (UTC)

• • Orientability - I think these examples are important. They are the first time we see non-trival manifolds. Maybe we could rename the section to Examples and include a few more. Some 3-manifolds for shape of the universe could be nice. Some odd topoligical one perhaps? Orientability could lead into a discussion of topological properties of manifolds, homotopy groups etc.
  • Charts, etc. are common to all manifolds, not just the differential ones. It might be nice to illustrate the difference between topoligical and differental manifolds. Came apond a nice examples illustrating the difference: the filled upper-half hemisphere, this is a topological manifold with boundary, but not a differential manifold with boundary.
  • Math-text book/encyclopedia dare I say it but I'd ver on the side of the text book, that is a clear introductory text on the topic. --Salix alba (talk) 23:21, 1 February 2006 (UTC)

I just tried to reorganize and rewrite much of it to incorporate some of my comments above. It was rough going, so I don't know what anyone will think of this. Here are some comments

1. Maybe we should have a section called "properties of manifolds" where we briefly describe compactness, orientability, etc...
2. I think the examples section needs to be reorganized and rewritten for clarity. The more I think about it, the more it seems pointless to try and describe the difference between a topological manifold and a differentiable manifold to the layman, other than to say that you can do calculus on a differentiable manifold (or, at least it has a tangent bundle). This is for the simple reason that there is no real difference in dimensions lower than four (at least according to my recollections... I don't have any books at all handy).

As for the comment above, re charts, most books define topological manifolds without charts: all you need to know is that a homeomorphism exists, you don't have to specify it when you specify the manifold. Re textbook/encyclopedia, well, this is an encyclopedia. That's not to say there oughtn't be pedagogical examples, merely that it is preferable to try maintain an encyclopedic style. –Joke 02:21, 2 February 2006 (UTC)

Let me respond to as many of these points as I can:

1. History: While historical context is important, it is not important enough to delay defining and describing what these things are. People will only want to know the history of these things once they understand a little about why they are important and interesting. Why would someone want to know the history of something unimportant or uninteresting to them?
2. Topological manifolds and differentiable manifolds: We decided long ago that these concepts really need their own articles. The consensus was that this article should be the least technical of the bunch, implying that most technical definitions and details should be placed in those articles rather than this one. As for homeomorphic topological manifolds with differing differential structure, I think that is way too advanced for the level of reader we are pitching this article to. Such things belong in differentiable manifold.
3. Orientability: This topic has the advantage over Hausdorff assumption, separability, etc. that it is intuitively much easier to understand and benefits newcomers almost immediately, without having to invoke lots of technical details.
4. Charts, atlases, and transition maps are part of the definition of any manifold, not just differentiable ones, so I don't understand why you would want to place this in the differentiable section. Okay, that's a bit of a lie: it turns out that two topological manifolds are homeomorphic as topological spaces iff they are homeomorphic as manifolds. Still, I think this concept is important enough to the subject to belong in this article.
5. The "litany of examples" serve to make the article more understandable. Moving them to the end would only make matters worse for the lay reader.
6. Regarding the writing about dimensions and relation to CW complexes, I agree that this is a bit heavy and probably we would do much better with just a picture here showing how we don't allow strings attached to balls to be called "manifolds".
7. "The more I think about it, the more it seems pointless to try and describe the difference between a topological manifold and a differentiable manifold to the layman..." — I agree with this.

-Gauge 03:10, 2 February 2006 (UTC)

While I appreciate your efforts, Joke, these changes are coming much too fast and with way too little discussion. Considering that we literally spent months trying to get the article into its current state, doing this rewrite all at once is asking to get reverted. Instead, please be less bold and we will discuss your proposals one at a time. - Gauge 02:34, 2 February 2006 (UTC)

Well, I'm never sure how bold to be. I thought a little would be required if this were, in fact, to make it to featured article status. In any case, those that are interested can see the diff here. I understand that a lot of horse-trading went into the page as it stands now, but it can clearly use improvement. Reverting is principally a tool to fight vandalism. If you didn't like the edit at all, you ought to improve it, or at least say why. –Joke 03:08, 2 February 2006 (UTC)

Please see above for some specific criticisms of your suggestions. When I have some more time I'll take a look at the diff to see what may be salvageable. - Gauge 03:21, 2 February 2006 (UTC)

...Salvageable!? So here is my reasoning.

1. For the history, if the article is not meant to be technical, the history is far and away the most important thing. The development of the manifold in the 19th and 20th centuries, along with the development of other abstact notions of spaces, was one of the most important things that happened.
2. I agree that the topological and diffentiable manifold stuff may be too technical to put in this article, but as it is it is not very good. It doesn't do well as a vague definition because it says things like "topological space" and "homeomorphic", and it doesn't do well as a formal definition, because, well, it's not. So I thought the formal definitions might as well be done properly.
3. It is true that orientability is a much more appropriate topic for this article than separation and countability axioms. However, I maintain that other properties of manifolds, such as compactness or (algebraic) topology are equally importanty.
4. See my comment below.
5. Upon reflection, I agree, unless it is possible to persuade the lay reader to move to the end. I just find the haphazard way in which technical speak and examples are mixed up in this article. It's too confusing for the lay reader, and useless for a technical reader.

So, to me, it sounds like one reasonable thing to do would be to move all technical content to the differentiable manifold and topological manifold articles, and leave this page as a page principally about submanifolds of Euclidean space, with, perhaps, some discussion of other techniques for constructing differentiable manifolds. –Joke 03:36, 2 February 2006 (UTC)

So I picked out some changes from my edit which I hope (against hope) should be relatively uncontroversial. Please don't revert the whole thing en bloc. –Joke

(Gauge) Another list follows:

1. You edited the lead section again, which is generally the most controversial part of the entire article. Secondly, by making proposals on the talk page, the point is to allow interested parties to critique the content. You have given people (besides me) almost no time to react before modifying the article with your new ideas.
2. The term n-manifold is no longer defined, you got rid of the explanation regarding connectedness entirely, along with the important explanation of a string attached to a ball.
3. The "Riemannian manifolds" entry misses the most important point about them: that you can measure distance. I know it's implied, but why not just say it? The section on generalizations is not intended for rigorous definitions; it is supposed to indicate (roughly!) what these generalizations are and why they are important.
4. Mentioning that symplectic manifolds are directly linked to classical mechanics should be enough. Symplectic manifold already gives a rigorous definition; we shouldn't be repeating it here.
5. The entry on "Lie groups" was very agreeable to me in contrast to the other ones; it is short, sweet, and not too complicated.
6. If a CW complex is "not a manifold", we should at least say why they are important at all.

I'm not going to revert again, but I'm guessing someone else will. Too many changes with no consensus on the talk page helps no one. - Gauge 05:43, 2 February 2006 (UTC)

I think I responded to all of your concerns, except the CW complex one. All I know is they're used in algebraic topology – I don't have the faintest idea why. I did rephrase the sentence about physics in the intro to make it less technical. –Joke 14:22, 2 February 2006 (UTC)

I'd much prefer if we discuss first on talk page wait a day for responses and then change, rather than big changes first. --Salix alba (talk) 09:00, 2 February 2006 (UTC)

Fair enough. Feel free to comment on the above discussion. –Joke 14:09, 2 February 2006 (UTC)

"We"

Is there a style convention regarding the inclusive "we" in math articles here? Certainly many, many math articles published in journals use "we" to include the reader in the discussion, so why get rid of it? Would people be upset if every math article eliminated "we" entirely? I might be... - Gauge 02:40, 2 February 2006 (UTC)

We is used all over the sciences, not just in math. It's the dispassionate science we, meant to include everyone. It's generally thought that its use should be minimized in formal writing (such as papers), and probably in encyclopedia articles as well. –Joke 03:00, 2 February 2006 (UTC)

The use of "We" has been one of the main complaints in the featured article candidate page. If we are seriuos about getting this status then We has to go. In general I think we can be avoided without much consequence in most article, not something I'm particularly concerned about. I'm not sure I'd take mathematical journals as a guide to good style. (Also "consider" as above) --Salix alba (talk) 08:43, 2 February 2006 (UTC)

Further, one of my math profs was very hot on only using "We" to indicate opinions of the authors, i.e. POV statments. I got a lot of red ink for various "we"s used else where. It was a simple job to tranform every occurence into non "we" statements (I think there is probably an algorithm for this tansformation). After the changes it read just a well and I'd now say "We" is lazy style. --Salix alba (talk) 08:58, 2 February 2006 (UTC)

"We" is indeed used all over mathematics. My experience, contrary to Joke's, is that it is recommended in many style guides for technical writing (might just be a maths thing though). Nevertheless, it seems that most Wikipedians consider it unsuitable for encyclopaedias (I like "we", by the way, but can't really be bothered about it). -- Jitse Niesen (talk) 13:36, 2 February 2006 (UTC)

Ok, that might be the relevant maths. It certainly isn't forbidden in physics, and is used liberally in abstracts and the introduction, but its use should be minimized. –Joke 14:08, 2 February 2006 (UTC)

"We few, we happy few." Paul Halmos is generally considered one of the best writers of mathematical prose, and his book Naive Set Theory, one of the best written mathematics books. I quote from page one, "We shall have occasion to refer to this again a little later."

The people who object to the use of the editorial "we" in encyclopedia articles as way too stuffy, IMHO. Rick Norwood 19:50, 2 February 2006 (UTC)

Yes, I agree that that is a great book. I also agree that it should be OK to use we. I'm just saying that it should be minimized, and some editors of non-mathematics articles will look askance at it. –Joke 14:10, 3 February 2006 (UTC)

Small Quibble

I really don't like the opening sentence about "patchwork." A N-dimensional manifold is a second countable Hausdorf space with the property that every point has a neighborhood homeomorphic to an open ball in N-dimensional Euclidian space. The most important property that distinguishes a manifold is that it locally looks like some specific space at every point. As opposed to, for instance, an analytic variety, which is the solution space of a system of M polynomials in N variables, and may have cusps and other interesting things on it. I am not going to leap in and edit a "featured article" which has had peer review, but I feel strongly that the opening sentence should be changed to something meaning "locally like some other space" and the "patchwork" metaphor eliminated, which could mean lots of things, many of them non-manifold-like. Hermitian 03:22, 2 February 2006 (UTC)

The article is not featured. You're absolutely right. There seems to be some misconception in the article and here on talk that a topological manifold requires a notion of "charts" and an "atlas." This is false: all you need to know is that every point has a neighborhood homeomorphic to an open set in Rⁿ. –Joke 03:24, 2 February 2006 (UTC)

Yes, I read too quickly. It was a "former featured article candidate." The basic notion of manifold is that of a topological manifold, which is locally n-dimensional. We can add a differentiable structure (not necessarily uniquely up to diffeomorphism) and get a differentiable manifold. We can also add other things, like a connection, a metric, a continuous group operation, or whatever, and get various other refinements. Take a look at the PlanetMath [3] article on manifolds. It is concise, includes everything necessary, and nothing in it is wrong or misleading. We should strive for something similar, with additional English text explanation of the various concepts suitable for the non-technical reader, and the incorporation of additional topics of interest. A paragraph on "Exotic Spheres" would be nice.Hermitian 20:05, 2 February 2006 (UTC)

There is something to be said for how transition functions describe what happens at the overlaps, otherwise it is hard to say how these Euclidean spaces are fitting together. I am inclined to agree that a "patchwork" is probably more difficult to understand than a space locally homeomorphic to Rⁿ or otherwise, but I would be conservative and propose your changes here before making them in the article, and then give adequate time (1-2 days at least) for a reply from interested parties. - Gauge 03:43, 2 February 2006 (UTC)

Well, the fact that they're (the maps into Rⁿ) homeomorphisms means that the compositions $f\circ g^{-1}$ and $g\circ f^{-1}$ must also be homeomorphisms. These are the transition functions, but I don't think I've ever seen it defined in this way. The difference is that a topological manifold doesn't need to come equipped with an atlas – it is sufficient to prove homeomorphisms exist – whereas a differentiable manifold must come with an atlas. If you're temporarily satisfied with the way the article is now, then I'm happy for the time being, although, as I said above, I really think something needs to be done to even out the tone. –Joke 03:58, 2 February 2006 (UTC)

Prerequisites

Using Prerequisites is okay in a technical article, as is written in Make Technical Articles Accessible. I'm not saying don't even try to read this, simply stating the obvious fact that the ordinary lay person will not get head or tail of the even easier sections of the article without a strong familiarity with High School math. Just look at the concepts referred to in Motivational Example. Feel free to clear up the prerequisites section, but please don't remove it. Loom91 07:37, 2 February 2006 (UTC)

I don't agree. Putting a "there be dragons here" sign at the top of an article about one of the most important topics in mathematics (and mathematical physics) over the past two centuries is not acceptable. In fact, I don't think I've ever seen a technical article with a prerequisites tag, not, say Einstein field equations which needs it if anything does.–Joke

Agree with the above (unsigned) post. I've never seen a math article on wikipedia with prerequisites explicitly mentioned, and I am not convinced that we should start now, especially when much more sophisticated articles don't list them. - Gauge 18:04, 2 February 2006 (UTC)

Agree w/Joke, Gauge (if my opinion is worth anything here). There are certainly ways of making things clear and understandable without making them dumb an stupid. (which is what has happened to some of the more pop math articles). One should cater to the smart but ignorant, rather than the stupid and ignorant. linas 00:28, 3 February 2006 (UTC)

This is not a question of what is seen on other articles. Just see http://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_accessible#Articles_that_are_unavoidably_technical . How can "Not done on other articles" be a valid objection? You have got to start somewhere! Who disagrees that you don't need a comprehensive knowledge of at least precalculus to even get the Motivational Examples section? Loom91 13:44, 3 February 2006 (UTC)

I don't agree that an article about manifolds is unavoidably technical. Sure, some if it is, but I think any reader can grasp the basic idea of what a manifold is, and most readers can grasp the examples with a little thinking and high school algebra, which isn't exactly technical. –Joke 14:09, 3 February 2006 (UTC)

You will notice that I specifically stated High School ALgebra as the prerquisite. You also agree that the page requires AT LEAST high school algebra. A fifth grade child is highly unlikely to understand even the simplest possible explanation of Manifolds. There's a reason people get education.

And as you yourself say, some (in my opinion a large) part of the article HAS to be highly technical. The prerequisites I mentioned are required to understand even motivational example section. Unless the article is written in a way that DOES NOT require Precalculus to be understood, the prerequisites header is at par with Make technical articles accessible. I just don't get your problem with the prerequisite header! Please stop pointlessly reverting. Loom91 11:53, 4 February 2006 (UTC)

Summary of my reservations

In case you don't want to read through the above, this is a summary of my reservations about the article. I have to say I'm surprised they're so controversial.

• The history section is poor, and is at the end. This is an encyclopedia article about one of the most important topics in mathematics. Articles like particle physics, evolution, big bang all have prominent history sections. It is sort of expected of a major article, and this should be no different, particularly given how central the subject is to modern mathematics. I realize that nobody here may know much about the history – I certainly don't – but that's no excuse for marginalizing it.
• The article is at an inconsistent technical level. For example, it is mentioned that topological manifolds are generally required to be Hausdorff and second countable, but it is never mentioned what the compatibility condition for transition maps is.
• The idea of an atlas of charts is too prominent. In fact, it is used in the introduction when it is only relevant to a differentiable manifold. A topological manifold does not require an atlas of charts. Why not just use some kind of "locally" hand waving in the introduction?
• The organization needs to be improved. The examples are good, but it is often not clear what the motivation for the examples is, they're just thrown out there in the "construction" section. Often, the first thing you want to know about a mathematical object is not how to construct it, but what it actually is in some deep sense. Maybe it would be best to talk about submanifolds the way Riemann originally did, as surfaces in some higher dimensional Euclidean space, probably defined as zeros of functions, and then point out with examples that a lot of the associated concepts (the metric, the extrinsic geometry) are no longer part of what we think of as a manifold, but rather extra structure.

What do you think? –Joke 14:50, 2 February 2006 (UTC)

• <pedantic>topological manifolds do require an atlas of charts exist, they just don't need to be explicitly defined</pedantic> saying local homeomorphism is equivilent to saying a chart exists for a suitable open set. Collecting all the charts together gives the atlas. Transition maps ensure things work for intersection of two of the open sets. I'm not agaist some local handwaving early on, but a bit of rigor later is good.
• History. Posibly worth mentioning ellipitical and hyperbolic spaces. (ducks)
• beware zeros of functions, things the Whitney umbrella have distinctly non-manifold topology, more a case of stratification.
• compatable condition for transition maps. Don't conditions for these just fall through as a corollary of Hausdorff assumptions? --Salix alba (talk) 16:31, 2 February 2006 (UTC)

All agreed. (For the first point, while I agree an atlas of charts exists, it needn't be presented with the manifold, unlike in the differentiable case. I have never seen a topological manifold defined using this terminology, although I'm sure somewhere someone does.) –Joke 16:47, 2 February 2006 (UTC)

While we're on the subject of reservations, let me add my own. I've always thought this article tries to be too general. The presentation makes it look like topological manifolds are a special type of manifold, and that not all manifolds need to be topological. In my mind this is fundamentally flawed; all manifolds are topological (in that they are locally Euclidean, Hausdorff topological spaces). Specializations result by adding additional structure (like an atlas of smooth or analytic functions). I've never liked having topological manifold as a separate article. -- Fropuff 17:09, 2 February 2006 (UTC)

Very good point. --Salix alba (talk) 18:05, 2 February 2006 (UTC)

Well, "all" manifolds are topological only if you allow them to be locally homeomorphic to a finite or infinite-dimensional Banach space, if you ask me. I think of this article as more about explaining the "manifold concept" in mathematics rather than the rigorous technical properties of topological manifolds. The latter article was created so that technical details that only serve to confuse the idea to newcomers can be placed there. Obviously we will have to be careful about how much detail we put on this page to reach FAC status again. - Gauge 00:04, 3 February 2006 (UTC)

Just curious, are there any good examples of ones which are not homeomorphic to a Banach space? --Salix alba (talk) 00:47, 3 February 2006 (UTC)

Well the obvious ones would be manifolds with boundary, although for pedagogical reasons it is probably better to introduce these later rather than sooner. Under the same considerations, I can agree to write as if all manifolds are locally homeomorphic to finite-dimensional Euclidean space (or half-space) so long as this article isn't inundated with technical detail that belongs at topological manifold. - Gauge 05:06, 3 February 2006 (UTC)

Well, I consider Banach manifolds to be generalizations rather than specializations of manifolds, which I assume to be finite-dimensional Euclidean spaces. Like I said I think the article tries to be too general. IMHO, of course. -- Fropuff 02:46, 3 February 2006 (UTC)

On being careful

• JA: Please discuss article-related issues at the article talk page, not my user talk page. I copy your remarks and reply here.

Please be careful in reverting Hi, you reverted my edits to the intro on manifold, which is fine. But when you do it, don't revert other edits, such as [4], which is in accordance with Wikipedia style guidelines for linking years, minor edits like [5] and [6], and this edit [7] which has nothing to do with the intro. If you disagree with these edits, you should comment on talk and explain which you are reverting and why. Don't undo other people's work simply because it is more convenient just to revert to the most obvious version. –Joke 14:06, 2 February 2006 (UTC)

• JA: I reviewed the entire recent history of the article and picked what I considered to be the optimum reversion point, just after the "we"-reconstructions and the de-"consider"-ations. Most of what you say above is just plain -- he finds himself at a loss for words -- given the pace, the ignoratio elenchi, and the low quality of the changes that you have been making, not to mention the lack of due consultation with others on what is clearly a moderately well-wrought article. I regret that others may lose time and energy interweaving with that lack of due process, but that is the train that you are conducting, not I. Jon Awbrey 14:38, 2 February 2006 (UTC)

Fine. I saw this article on WP:FAC and thought I could help out. If you are not interested in the Wikipedia process, then I am happy to go back editing other articles. –Joke 14:52, 2 February 2006 (UTC)

• JA: Fine, structure, constant, okay, I get it. But you were imposing a special not general flame of reverence, throwing out way too many babes with the bathos, and overall proceeding with undue celerity, not to mention friction. More heat than like comes of that. Jon Awbrey 16:44, 2 February 2006 (UTC)

Since going slow is no longer an option.

My suggestion that we take this slowly has not been accepted. I think the most important step, in light of the strong statement by other wikipedians that the lead is unreadable, is a new lead. I've written one.

Next, again with an eye toward an accessable learning curve, I plan to move the history section into second place, followed by a mathematical definition. Rick Norwood 20:03, 2 February 2006 (UTC)

Why does moving history there make it more accessible? I find it quite a hard section to understand, because it uses quite a lot of technical terms. -- Jitse Niesen (talk) 23:22, 2 February 2006 (UTC)

Agree with Jitse, for reasons I stated earlier. Looking at what is actually written in the "History" section makes it clear to me that any potential readers will stop reading the article at that point due to a deluge of technical details and references. Do we really expect people to know things like curvature, ambient space, surface, analytic continuation, abelian varieties, Hamiltonian and Lagrangian mechanics, differential geometry and Lie group theory? This is terrible as motivation and only serves to educate a posteriori once one has been in mathematics and has seen these things. It serves no purpose at the beginning of the article. - Gauge 00:14, 3 February 2006 (UTC)

I don't think it necessarily has to use so many technical terms, at least without a quick hand-waving explanation. The key point is that the history section should have some sense of narrative, not simply be a collection of trivia. I think characters like Felix Klein, Arthur Cayley, Hermann Grassmann, Sophus Lie, Gaspard Monge and Leonhard Euler could merit a mention. Although perhaps they didn't advance the definition of manifolds, they certainly advanced the way in which we think of problems involving manifolds. Although maybe it would be best to write a history of differential geometry rather than a history section here? I will try and work on it later in the day – no time now. –Joke 14:20, 3 February 2006 (UTC)

Solera Process at Wiki Whinery

• JA: Rick, et aliases, "We" now have a function (mathematics) article that I would not give a sixth grader to cut their dog's teeth on. Has it ever occurred to you, just maybe, to ask why people liked the way this article was written, what their mathematical and pedia-gogical reasons might have been for the design decisions that they made, that you are now trashing without a thought? Who knows, those choice may be indefensible, but you could at least lend an ear. Who knows, you might actually learn something in the process. I am sure that some of the same folks, or reasonable proxies thereof, are still around, if you give them half a chance. If this is really such a cornerstone of mathematics, then what's the rush to shift it? I'm all for pushing the envelope, do it every day, but this particular package is looking more and more like some amateur hack job every time I revisit it. Jon Awbrey 20:38, 2 February 2006 (UTC)

Why are you writing this here? Can you be clearer please, are you talking about manifold or function (mathematics)? What are "aliases"? –Joke 20:44, 2 February 2006 (UTC)

• JA: I'm alluding to a process that went on at the article on function (mathematics) that appears to have spread to manifolds. Expanding "al." as "aliaseses" was kind of a joke, Joke. I'm not normally this $\vdash$ive, but my stint as a WikiPeon has taught me a whole new meaning for the phrase "critique sans merci", and besides I'm not the one in a hurry, and I find that subtlety eats up more time than any mortal's got. Jon Awbrey 21:06, 2 February 2006 (UTC)

Jon, you do not like the way I write. You have said so on many occasions, with what you evidently consider disarming frankness. But the facts are not on your side when you accuse me of rushing and of not taking other people's opinions into account. Rick Norwood 21:44, 2 February 2006 (UTC)

• JA: I'm talking about a generic process. Cosi fan tutte. ${\mathcal {W}}\varepsilon$ all do it. Which is why ${\mathcal {W}}\varepsilon$ do math and science in groups, so's to catch each other at it. It has nothing to do with the way you write. ${\mathcal {W}}\varepsilon$ all write terrible on the first eleventy-one drafts or so. The perfectly native generic thing that everyone generally naturally does is to try and stay within their "Zone Of Comfort" (ZOC). The reason for working in groups is so's ${\mathcal {W}}\varepsilon$ can construct $\bigcup _{j=1}^{k}ZOC_{j}$ instead of $\bigcap _{j=1}^{k}ZOC_{j}$, which latter is almost guaranteed to approach $\varnothing$ as k gets large. It's one of those "initial condition highly sensitive" (ICHS) types of processes. If ${\mathcal {W}}\varepsilon$ do it right, ${\mathcal {W}}\varepsilon$ get the more perfect union, if ${\mathcal {W}}\varepsilon$ do it wrong, ${\mathcal {W}}\varepsilon$ get zilch.

Clearly, Jon, we will have to agree to disagre on the subject of which of us edits unilaterally and which of us always tries to work with others and find a consensus. Rick Norwood 23:03, 2 February 2006 (UTC)

What are your specific criticisms of Rick's work, Jon? If they are about function (mathematics), please list them over there. - Gauge 00:20, 3 February 2006 (UTC)

• JA: Rick, Gauge: I will try to put it more simply. I was not addressing Rick's "way of writing", either now or in the past -- I was not even aware of who wrote what at the time in question --Rick's deletion of some things that I thought should have been presented up front, coming on top of a series of such deletions by others, simply provided a new occasion for commenting on the general bias that I see here, where someone can disrepect other people's laborious work, just because those who went before don't happen to be paying attention right at the moment, and somewhat ironically protesting all the while that their own alterations and deletions be respected. I was not the only voice saying this, but what usually happens is that the more respectful folks just give up and go away. My reverts were respectful, they were reverts that attempted to preserve some of what is now lost. Now I will just give up and go away. Jon Awbrey 02:44, 3 February 2006 (UTC)