Talk:Homotopy groups of spheres

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	Homotopy groups of spheres has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do. If it no longer meets these criteria, you can delist it, or ask for a reassessment.
Article milestones Date Process Result May 25, 2006 Good article nominee Listed October 6, 2007 Good article reassessment Kept Current Status: Good article

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	GA Class	Mid Priority	Field: Topology
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. The recent improvement drive has introduced quite a lot of new content. In light of this fact, the article would probably benefit from a layman peer review and copyedit. There also seem to be a number of as yet unresolved issues on the talk page, yet most of these seem to be nearing resolution. Once these have been worked out to everyone's satisfaction, a trip to FAC seems to be the next logical step. Great work everyone. —Cronholm144 23:06, 23 October 2007 (UTC)

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/Archive 1

[edit] Consistent notation

We should be consistent about the notation for the groups in the table. For example, we have 24+3 and 8+2²+3². I suggest 24+3 and 24+6+2 respectively. Put the biggest possible number first, and recurse. Adam1729 21:47, 25 August 2007 (UTC).

See Finitely_generated_abelian_group for the two standard notations. I think my suggestion is usually the shorter of the two. Adam1729 08:23, 27 August 2007 (UTC)

[edit] GA on hold

This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed. There are not enough references to cover all the keypoints. I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. Regards, OhanaUnited^{Talk page} 21:45, 8 September 2007 (UTC)

While I typically do not involve myself in the GA process, I would like to point out that the number of references varies based on the amount of information in each reference. One of the references listed is a textbook on algebraic topology. I would imagine that this, along with others, provide adequate reliable sourcing for the content, and the reader could easily verify the content in such a text. If we need an in-line citation for every definition, statement, or formula (maybe we do, I don't know), then be sure to point that out - it's not the number of references that needs to be increased, it's the number of in-text citations. However, I can imagine that so many in-text citations would be frivolous, distracting, or otherwise poorly styled. Could you clarify - do you believe that this article is not reliably sourced? Do you believe one cannot verify its claims? Or do you just want more citations? --Cheeser1 04:17, 9 September 2007 (UTC)

The article is fine. I've added it to the list of checked articles and retained the GA. Geometry guy 15:54, 15 September 2007 (UTC)

Having said this, I reread the article, and realised that while the verifiability was fine, the tone was not very encyclopedic, and the lead was not an adequate summary of the article. I have attempted to fix these issues. Geometry guy 16:50, 6 October 2007 (UTC)

[edit] No pattern?

Under the "General theory" section, can phrasing like

is often complicated and the results are surprisingly complex, with no pattern discerned to date.

be modified to be more accurate? Surely there are patterns- the next paragraph gives two different patterns. I think what is meant is something like "there is no known algorithm for computing π_n(S^m) for all m and n", though that's perhaps a bit confusing to a layperson (though I'm sure they wouldn't have gotten this far), and has entirely a different feel to it. Staecker 16:12, 8 October 2007 (UTC)

Good point. I suggest you be bold! Geometry guy 21:46, 8 October 2007 (UTC)

I couldn't really think of a rewrite that I like... and I still can't... Staecker 22:27, 8 October 2007 (UTC)

[edit] Table

I'm not completely convinced by some of R.e.b.'s recent changes to the table. In particular, I think listing the homotopy groups of S⁰ is not such a good idea, as the zero sphere is not connected, so we would have to start discussing base-point issues. Also, I find the stabilization along the diagonal less striking now that all the entries are filled in. Views? Geometry guy 15:48, 10 October 2007 (UTC)

Also, why the change of the trivial group notation to 1 instead of 0 which is inconsistent with the rest of the article?--agr 17:02, 10 October 2007 (UTC)

We should be consistent across the article. As a related issue, we should decide whether we want to use multiplicative or additive notation for abelian groups. Essentially R.e.b. has changed the tables to multiplicative notation, although Z is still used for the infinite cyclic group, which is really an additive notation. Geometry guy 17:49, 10 October 2007 (UTC)

If you are using m as an abbreviation for the cyclic group of order m, then the trivial group is 1, not 0, as it has 1 element. Moreover one should then use a.b rather than a+b for the sum of the two groups, as this gives the correct order. I agree there is a good case for changing Z to ∞ but I couldnt be bothered to do this, since Z seemed harmless. R.e.b. 22:11, 10 October 2007 (UTC)

About S⁰: the article defines homotopy groups incorrectly. The way to fix this is not to eliminate mention of the perfectly respectable sphere S⁰, but to fix the definition. R.e.b. 22:37, 10 October 2007 (UTC)

The homotopy classes do not form a group under composition, unlike for For S^n>0. Tompw (talk) (review) 11:56, 13 October 2007 (UTC)

I've been thinking about the table, and I made a version that I like on a user subpage. It has the following features:

• Standard notation (no need for an explanation at the beginning)
• Simple layout, with one column for each π_n and one row for each sphere
• Colors to show the stable homotopy groups, with a dark line showing the stable/unstable barrier
• Serre's infinite groups are highlighted in red.

The disadvantage is that it contains less information than the present table, although I'm not sure that such an extensive list is necessary for a Wikipedia article. Possibly it could be extended to include more data. Jim 07:01, 11 October 2007 (UTC)

I like the layout of Jim's table much better than what's presently in the article. And I also don't think that it's necessary to reproduce here everything that is known about the homotopy groups of spheres. Arcfrk 07:37, 11 October 2007 (UTC)

The notation for abelian groups in the new table is much better, and putting spheres in rows is also an improvement; these were problems I inherited from the original version. The main problem is that the new table discards most of the information, which is a bad idea as it no longer goes far enough to show many of the more interesting phenomena. There are also good reasons why almost every published table arranges by π_n+k rather then by π_n: this makes several subtle pattern much easier to see. R.e.b. 15:20, 11 October 2007 (UTC)

The two tables have complementary advantages, and the article could use both: Jim's table is easier for beginners and could go in the section on low dimensional homotopy groups, while the large compressed table could stay at the end for more serious readers. R.e.b. 16:16, 11 October 2007 (UTC)

I've added it to the beginning of the "general theory" section, with some explanation of the coloring immediately below. Jim 23:24, 11 October 2007 (UTC)

The table is nice, however the colors are really hard to distinguish. Perhaps something less bland? KSmrq has at his user page a couple of colors which are very well contrasting. Jakob.scholbach 21:00, 12 October 2007 (UTC)

Strong colours can't be used as background colours for accessibility reasons. Already some of these colours may be too dark for the visually impaired. (Continued below.) Geometry guy 22:25, 12 October 2007 (UTC)

OK. Well, I'm not visually impaired, but nonetheless (with a laptop screen) the colors are distinguishable only in a small angular range. Perhaps one could try hatching the cells in a different way. Or somehow color the border of every cell (this could be done with a strong color). Jakob.scholbach 07:49, 13 October 2007 (UTC)

On my cheap (and common) flat screen, the paler blue is very hard to tell form the blue. I've also increased the cellpadding because IE wasn't taking accoutn of the subscripts in the left-hand column. Tompw (talk) (review) 11:56, 13 October 2007 (UTC)

Could you find some other way to fix the problem? The cell padding makes the table too wide to fit on standard browser screens.R.e.b. 13:14, 13 October 2007 (UTC)

In the table I made, I fixed the subscript problem by setting the height of each row manually (using the first cell in each row). I also found in necessary to use valign="top" for cells containing superscripts for them to display correctly in IE. Jim 17:36, 13 October 2007 (UTC)

I have tried modifying the colour scheme to make the contrast (red/blue) between the unstable and stable groups more striking. The contrast between even and odd rows is now very little, but I think that is less important. How does it look on a cheap flat screen? Geometry guy 15:10, 14 October 2007 (UTC)

(Continued from above.) I don't like the new tables. I think there should be only one table, and I think it should combine the merits of the two tables that there are now. In particular, the period three colouration has no meaning: periods 2,4 and 8, are relevant, but not period 3. Geometry guy 22:25, 12 October 2007 (UTC)

I'm worried about the period 3 myself. I used it because it is the smallest period that highlights the diagonals effectively. Possibly it would be better if each of the diagonals were a different color?

In any case, I'm not too attached to any particular coloring scheme. But I really think that the basic table of homotopy groups in this article should be arranged with π_k(Sⁿ) in position k,n, and should use standard notation for groups in the table cells. Jim 17:36, 13 October 2007 (UTC)

Period 4 would work for me (since it brings out the 3 mod 4 issue). I think we only need one of these two tables, plus a separate, much simpler, table of the stable groups. I have mixed feelings about which approach I prefer. I like your table, but I see the benefits of the n+k approach, and we really should follow the literature. If you have a source for the (k,n) version, that might swing the balance. Geometry guy 18:06, 13 October 2007 (UTC)

The (k, n) version is used in Hatcher's Algebraic Topology book (Chapter 4, pg. 339). Jim 19:10, 14 October 2007 (UTC)

I've modified the first table with more satuated colours, using Web-safe color. Hopefully sufficient contrast for disabled and between colours. I had to fiddle with my monitor to actually see the previous colours. The period is still three, which helps keep same level of satuation. --Salix alba (talk) 15:44, 14 October 2007 (UTC)

@Geometryguy: on my cheap screen the table looks royal :-). Much better than before. Jakob.scholbach 17:44, 14 October 2007 (UTC)

Where to jump in? I'm curious to see if I can come up with a single complete table; I'll experiment privately.

• Meanwhile, using the wikEd tool and a little hand cleanup I've been able to convert the original HTML table markup to wiki pipe markup. Like Jim's table, it's much easier to deal with!
• Using 1 instead of 0 is inconsistent with the article itself, and with most literature. Better still, leave the trivial group cells blank; we have no missing data so the interpretation would be unambiguous.
• One way to address Jim's preference for "standard notation", meaning Z×Z₂³ instead of ∞.2³ or Z+2³, might be to make a conceptual shift. Can the table list group orders with p-primary factorization, rather than the groups themselves?
• With so many patterns connecting table entries, it's tough to imagine any one view that can visually highlight them all. For example,
  1. Using Jim's π_i(Sⁿ) table, with row n and column i, the equality of the S² and S³ groups jumps out.
  2. Using row n and column k, group n+k, the stable groups are more salient.
  3. Even and odd patterns are important; they take different forms in the two layouts.
  4. Other patterns, such as π_i(S²ⁿ) = π_i−1(S²ⁿ⁻¹)×π_i(S⁴ⁿ⁻¹) modulo 2-torsion, are probably beyond visual cues.
• The maximum table width should not require horizontal scrolling on an 800×600 screen. (See W3C statistics.)
• Ooh, cotton candy colors! The new background colors of the big table might be too dark for good readability contrast. Web-safe colors were never all that safe, especially not for those using 15-bit color depth; and they were chosen with CRTs in mind, whereas LCDs have become increasingly common. This tool is one way to get a feel for how color combinations look, including simulation of various forms of color blindness. A good rule of thumb is to use color as a secondary cue; for example, traffic light standards try to accomodate the 5% of men who have difficulty distinguishing red from green. (Vischeck is a fun way to experiment.)

It might be helpful to compose a wish-list of facts we'd like to indicate in the table, so we can choose the ones that we feel are most important and most practical to show. --KSmrq^T 06:36, 15 October 2007 (UTC)

PS: Two brief remarks about table notation:

• Although many people working on homotopy groups would be comfortable with Z₂, the trend seems to be towards Z/2Z or Z/(2) or even Z/2. For example, Toda (in an older publication) uses Z₂ while Ravenel (much more recently) uses Z/(2). The reason, of course, is to avoid confusion with p-adic numbers.
• The fundamental theorem of finite abelian groups tells us that the group for n = 6, k = 8 that Toda describes as Z₂₄+Z₂, having order 48, can be expressed as the direct sum of Z₃ and groups of the form Z_2^m. Since the same applies to Z₂₄ itself, it is not necessarily a contradiction for another author to list this group as Z₆+Z₈. (See Consistent notation above.) But we do care. For example, Ravenel (after Toda) lists (n = 13,k = 19) as 264.2³, and lists (n = 14,k = 19) as 264.4.2. I therefore assume these are different groups of the same order. Please note that the Jie Wu table is not a direct copy of Toda in notation; he lists (6,8) as 8+2+3, (13,19) as 8+2³+3+11, and (14,19) as 8+4+2+3+11.

Our choice of notation can affect visibility of patterns, as well as accuracy. We currently have a footnote that says "There are a few differences between Ravenal's tables and Toda's tables"; is this really true, regarding substance? (And note that "Ravenal" is a misspelling.) --KSmrq^T 11:19, 15 October 2007 (UTC)

I think I can live with two tables now, as long as the first one is as short as possible. I've adjusted the colours (inspired by KSmrq's palette). I'm still unhappy about the period 3, but I hope that otherwise, this works for everyone. Geometry guy 23:04, 21 October 2007 (UTC)

[edit] Use?

Is it possible to tell why people are desperately trying to compute these groups? (I mean, beyond the "spheres are some of the most simple objects, we should know their homotopy groups"). Are there concrete applications of the knowledge of π₁₇₀(S¹⁴⁰)? Jakob.scholbach 21:07, 12 October 2007 (UTC)

The real reason, in my understanding, for trying to compute them is that they are difficult to compute, and so the endeavour inspires the development of new techniques in algebraic topology. As for concrete applications, I don't know any, although π₃^S is loosely related to the appearance of the number 24 in string theory, according to Baez. Geometry guy 13:28, 13 October 2007 (UTC)

Also, studying homotopy groups of spheres is the same as studying critical points of map R^m → Rⁿ (and more generally maps between m-manifolds and n-manifolds). For example, the map f(z) = zⁿ on the complex plane sends the unit circle to itself with degree n, and a typical critical point for a map between surfaces locally looks like zⁿ for some value of n. In general, the geometry near a critical point of a map R^m → Rⁿ can be described by an element of π_m−1(Sⁿ⁻¹), namely the way that the m−1 sphere around the critical point maps to the n−1 sphere around the critical value. Jim 19:25, 14 October 2007 (UTC)

OK. Interesting. Once, I saw a proof of the fundamental theorem of algebra using pi_1(S^1)=Z. These things should be mentionned in the article. Jakob.scholbach 20:26, 14 October 2007 (UTC)

Fantastic! These will be great additions to the article. Geometry guy 22:03, 14 October 2007 (UTC)

A little concerned that there might be some circular argument here, i.e. can pi_1(S^1)=Z be proved without using the fundamental theorem of algebra? A citation for this part would be most excellent. --Salix alba (talk) 19:39, 16 October 2007 (UTC)

?? You can calculate pi_1(S^1) with Hurwitz' theorem, for example. I don't know the proof of Hurwitz by heart, but as far as I know, it does not use something like the fundamental theorem of algebra. Another way is the universal cover of S^1 = R/Z, namely R. The deck transformation group is Z. (You can come up with complex numbers of absolute value 1 and use exp (2pi i z) etc., but nobody forces you to do so). Jakob.scholbach 20:19, 16 October 2007 (UTC)

Maybe I'm overlooking some well-known thing, but I've never seen a calculation of the fundamental group of the circle that even uses the fundamental theorem of algebra. I'm hard-pressed to understand how this could even be possible. --Horoball 20:41, 16 October 2007 (UTC)

the fundamental theorem of algebra can perhaps be removed from the applications section. it's cute and one can certainly mention it in the fundamental theorem of algebra article. but it's not really relevant here, and, unlike other items in that section, it certainly does nothing to convince the reader that π_k(Sⁿ), and the machinery used to compute them, is useful. it looks more like a trivial footnote. Mct mht 10:30, 3 November 2007 (UTC)

I think it migth be worth expanding on degree and π_nSⁿ)=Z. There a sentence in the π₂S²)=Z section:

These two results generalize: for all n > 0, πn(Sn) = Z (see below).

which has been bugging me for some time as it not clear where the (see below) should point to. Expanding on this could neatly tie up fundamental theorem, degree and related results. --Salix alba (talk) 13:53, 3 November 2007 (UTC)

[edit] More mathematics?

In order to eventually attain FA standard, the article should perhaps contain a complete computation of one homotopy group, thus conveying some of the effects etc, which one encounters when dealing with the problem. Currently the article names a lot of tools, but hardly tells how they work. Jakob.scholbach 21:18, 12 October 2007 (UTC)

For example, the subsection "Finiteness and torsion" are a mere repetition of what is shown in the tables (plus the information that it was Serre who showed this in his paper). Jakob.scholbach 21:28, 12 October 2007 (UTC)

The article has been much expanded recently and contains a certain amount of repetition. This needs to be fixed and I intend to do that. In terms of computing an example, π₁(S¹) might be the most obvious case. What more should be said? Geometry guy 22:17, 12 October 2007 (UTC)

For π₁(S¹) we have θ mapping to mθ, for m∈Z. We can do the same thing explicitly for π₂(S²) with φ as latitude and θ as longitude, and then explain that π_n(Sⁿ) = Z in the same way for all n. (Aside: I'm happy to use "=", but perhaps we should somewhere say that this is a convention for what might more properly be called an isomorphism.)

We can exploit an opportunity with the Hopf fibration, p:S³→S². First, we should explain why the long exact sequence of the fibration breaks up into short ones. Then we should explain how a short exact sequence forces a monomorphism and an epimorphism, and how shortening further will combine those into an isomorphism. This begins to give some feel for calculations with sequences, and helps explain the telling comment (under "Computational methods") that "the spectral sequences below have fewer non-zero differentials and so give more information". The longer the sequence, the longer the chain of reasoning, and the more we must work to understand the maps in the sequence. This is, after all, a huge part of the game. --KSmrq^T 00:26, 13 October 2007 (UTC)

Yes. Explaining why certain sequences split is certainly the right thing to do. If it is possible without too much trouble, I would like to find an explanation of pi_5(S^3), say, i.e. something which gives non-expected results and which is specifically using the sphere situation. pi_n(S^n) is just the Hurwitz theorem and less specific to the sphere case. But a decent explanation of Hurwitz has its merits, obviously.

Also, is there a link between K-theory and these groups? The existence of these Hopf fibrations seems to point in this direction. Jakob.scholbach 07:45, 13 October 2007 (UTC)

I have to admit, when I posted at the mathematics round table I had in mind that a version of the article close to its state at the time could be tried out at FAC. Since then the article has doubled in size! You guys are way more ambitious than me! This is no bad thing of course, and I am deeply impressed by efforts to make this article so comprehensive.

I just have one word of caution though: we are writing an encyclopedia article, not a textbook, and we should either avoid unnecessary detail, or farm out such detail to subarticles (such as J-homomorphism). I'm not saying any of the above suggestions are a bad idea; I'm just reminding us of our goal. I'm willing to do the trimming and copyediting to achieve encyclopedic style, but am worried that the article could grow indefinitely, because there are so many interesting things one can say about these groups. Geometry guy 18:23, 13 October 2007 (UTC)

Yes. Perhaps, as a guiding motto: somehow, the emphasis should be both on symmetry (of the single spheres) and on the sequence of the spheres S^n, n \in \N (for example, expressing pi^n(S^m) in terms of a simpler (probably lower-dimensional) pi_n'(S^m')). Right now, the loop space does not even occur in the article, a gadget which is probably of high importance (computationwise and conceptwise) together with smash product, such as S^n \wedge S^1 = S^{n+1}. Jakob.scholbach 14:09, 14 October 2007 (UTC)

[edit] Homotopy theory of schemes

In algebraic geometry, there is also a notion of homotopy groups. There, there are two types of spheres, though. The first is a simplicial sphere T=Δ¹ / boundary, the other one is the multiplicative group G_m. This can be found in a paper of Morel and Voevodsky [1]. Should something in this direction be included here? (As far as I know, it is really hard to compute anything, though). Jakob.scholbach 12:30, 15 October 2007 (UTC)

Even though I'm a fan of algebraic geometry, I think this is probably a bridge too far. Geometry guy 18:58, 16 October 2007 (UTC)

This sort of material is the kind that would generate its own article, which would probably then be full of red links. Now, that is not 'forbidden'. But in a sense the approach of creating some preliminary articles first is normal. This is work from about 1999. It will be a great day when we can say "any twentieth-century mathematics fits into Wikipedia". Maybe three more years' work? After all this article is really about 1950s/1960s things. Charles Matthews 21:48, 16 October 2007 (UTC)

[edit] Definitions section

In my opinion, the "Definitions" section is not very well-written. I would strive for a more concise and (at a time) brief section.

1. It defines a lot of terms which are not really necessary to define here. Somebody who wants to know about the homotopy groups of spheres will know what a mapping of topological spaces (esp. a one-point mapping) is. Likewise, the notation S^n does not need to be explained in this length. A link to hypersphere together with a short definition should be enough.
2. Saying "The i-th homotopy group of a topological space X is essentially the set of homotopy classes of mappings from the i-sphere Si to X." is imprecise. I believe a solid definition not using any "essential"-type words is needed. (This would be a repetition from fundamental group, too, but here some redundancy is perhaps OK).
3. Linked to the item above: Currently, the intention seems to be to talk about the fundamental groupoid, too. This is, however not the purpose of the article (and is also not covered in the text). Discussing the relation between the groupoid and the group should take place at fundamental groupoid.

Jakob.scholbach 12:43, 20 October 2007 (UTC)

I partly agree, but not entirely. (1) Do you really think so? Why should a reader interested in maps between spheres even know what a topological space is? If you read through the talk page (in particular, some of the GA reviews) you will notice that some of this explanatory material came from a desire to make at least some of the article accessible to a curious general reader. There is no point in trying for FA unless we do that. Note that this section occurs before the "low-dimensional examples" section, and so it should be as non-technical and self-explanatory as possible in my opinion. (2) The problem is that the precise definition is a bit complicated and needs motivation. I agree that the current formulation needs improvement. (3) If there is an intention to discuss the fundamental groupoid, it is news to me. I agree, such discussion is not relevant here. Geometry guy 13:49, 20 October 2007 (UTC)

(edit conflict)

We began an exercise to see if we could polish the article to achieve Featured Article status, and to that end I believe we should make the definitions meaningful for as many readers as possible. The material R.e.b. has been adding is hopelessly technical for most readers, including mathematicians not working in this speciality. If we skew the whole article too far in that direction, we will fail not only FA, but also most readers. Nobody said this would be easy; in fact, the point is to tackle something hard. If one extra sentence explains Sⁿ to almost anyone, given that the whole article concerns spheres that seems a good use of a sentence. It is not uncommon for a mathematics undergraduate to have a course in general topology, but much less likely that homotopy groups will be familiar.

Let's consider what we need to define.

1. Topological space. The general definition is in terms of open sets, but all we need are spheres and intervals so we could cheat.
2. Continuous map. The inverse image of an open neighborhood is open — does that stimulate a mental image?
   • The category Top. Yeah, we wish.
     • We can say something like: points which are close remain so when applying the map.Jakob.scholbach 15:10, 20 October 2007 (UTC)
3. Homotopy. Given two continuous maps ƒ, g:Y→X (where for us Y = Sⁱ and X = Sⁿ), formally we need a continuous map h:Y×I→X such that h(y,0) = ƒ(y) and h(y,1) = g(y). Oh well, we have a picture.
4. Homotopy equivalence. Two maps are equivalent if they are homotopic, so we get equivalence classes.
5. Homotopy group. Do we know what a group is? (Do nothing, do in succession, undo.) Anyway, we need to define "addition", and then either just assert we have a group or try to make that plausible. Life would be simpler if we could quickly specialize to abelian groups.
   • The category Grp (or Ab), and the functor from Top to Grp (or Ab). Just kidding.

A few other items would also be nice.

1. Suspension. Freudenthal is pretty important, and suspension is easier for spheres than in general.
2. Exact sequences (for anything technical). Composable maps with image equal to kernel; but motivation and intuition would be nice, too. In homology we have chains (boundary of boundary is zero), as with calculus.
3. Spectral sequences. All the bits R.e.b. wants to throw in depend on this, and I'm at a loss how we're going to explain.
   • I think this article should just be an invitation à la danse avec une suite spectrale, so to say, not the dance itself. I.e. if we are able to convince a (math) reader that s.s. are the tool to calculate these groups he/she will get a thirsty feeling to jump right here. Jakob.scholbach 15:15, 20 October 2007 (UTC)

I'd really prefer to stay away from pointed spaces, especially since they buy us nothing of interest for our big topic. We note the important role of fibrations, especially Hopf's S³→S²; how much background do we give?

Anyway, we can see that a definitions section that does no more than get us to homotopy group, specialized for spheres, requires either a lot of explanation or a lot of handwaving or a lot of inspiration. I think we can do a decent job, and my standards are pretty high. Caveat: I'm afraid R.e.b. has shown almost no consideration for readers who are not experts; we must do better than that in the definitions and other early sections. --KSmrq^T 14:28, 20 October 2007 (UTC)

Is there really a need to disparage a contributor (R.e.b.) for perceived lack of "consideration"? Sometimes it can be hard to understand how others see what you write. Perhaps R.e.b. puts care into being clear in his way and doesn't realize that to some it may appear to be inaccessible. Everyone has different strengths and contributes in a unique way. Perhaps he is willing to leave general exposition to those he feels is more capable at that. Let's not get into this kind of personal remark, which does not in anyway advance the discussion in a positive manner. --Horoball 10:27, 23 October 2007 (UTC)

I don't believe KSmrq meant these remarks in a disparaging way. I certainly value, and I think he does too, editors adding technical content without consideration for making it accessible to the general reader. Indeed, most math articles are built that way initially. As you say, this is a collective endeavour, so we can all bring complementary skills to improve the article. R.e.b. has done a fantastic job in adding deep content. Now it is up to others to make that content as accessible as possible. Geometry guy 21:37, 23 October 2007 (UTC)

(edit conflict. I just reply to Geometryguy): I see your point. Perhaps I exaggerated a bit. Some ideas/remarks:

1. Perhaps we can have a separate section called "Introduction" or something, which is accessible to a general reader and a "Definition" section which would give the information for a math reader. Currently, the style merges high-level maths ("The number n is the intrinsic dimension of the sphere as an independent topological space") with "low"-level style of explanation ("Two such mappings are considered to be equivalent if one can be continuously deformed into the other, a process called homotopy.").
2. An Intro-section could come up with some explicit picture of S^1 and S^2, and could graphically show a non-trivial element of pi_1(S^1). This might be a golden moment for an animated gif. I find the static image of 2 \in pi_1(S^1) a bit difficult to understand at first sight. (A layman will not give us the chance of taking a pencil and follow the black line along the screen).
3. Phrases like "As such, it is a 2-dimensional manifold" are probably repelling a layman, therefore should be deferred to a more formal (sub)section.
4. At the moment, the lead of the article has quite an overlap in terms of content and style of exposition (i.e. accessibility).
5. In the more formal "Definitions" section, which concepts should be (re)explained here and which ones should just be linked? We do have separate articles for all concepts "defined" in the section, such as continuous function (topology), circle, sphere, and hyperspheres, homotopy groups.
6. The reader interested in a rigorous definition currently needs to refer to fundamental group and homotopy class to get the full formal definition. (Too far, IMO, given that the article talks about nothing else than this)Jakob.scholbach 14:39, 20 October 2007 (UTC)

* Just a brief remark: ironically, two of the points you mention ("The number n is the intrinsic dimension of the sphere as an independent topological space" and "As such, it is a 2-dimensional manifold") were originally introduced to deal with the common misapprehension that the 2-sphere is a 3-dimensional object! Geometry guy 16:24, 20 October 2007 (UTC)

I know from experience talking about "intrinsic dimension" and "independent..." is just jargon to most people. If we want to be accessible, what is wrong with something such as "the number n is the number of degrees of freedom". I've had more success with that. Or "number of parameters needed to give coordinates". Something like that. --Horoball 10:33, 23 October 2007 (UTC)

I agree. We need to find a way to express the fact that the 2-sphere is 2-dimensional, not 3-dimensional, without resorting to jargon. Geometry guy 21:32, 23 October 2007 (UTC)

* Another remark: the overlap between lead and article was partly because I was incorporating lead material into the article, and had not yet trimmed the lead. I've trimmed it now. There is a tension in the lead between conflicting forces: it should be a concise summary of the article, but the lead is also supposed to be the introduction, and it needs to be accessible. I've been striving towards a compromise between these forces. Geometry guy 16:56, 20 October 2007 (UTC)

* General comment: I agree with KSmrq that we should make the definitions meaningful for as many readers as possible. I am more concerned about failing readers than failing FA. I made the FA suggestion not just to test out ability to produce technical featured content, but to see whether Wikipedia as a whole can handle such content. As I already mentioned, I did not have in mind the recent expansions, but the R.e.b. came along, and we suddenly have a much richer article, albeit at the expense of introducing a lot more technical content. I agree with KSmrq also that the key is to maintain a balance between exposition/accessibility and comprehensiveness/technical content. I think we are doing not so badly thus far. We are not going to achieve KSmrq's dream that a curious general reader will come away with an idea of what a spectral sequence is, but neither are we going to have an article that is entirely satisfactory in its conciseness and precision for the mathematical reader. Some of the ideas above are worth trying. Let's keep editing away: I think we are getting closer to the FAC stage now. Geometry guy 15:24, 21 October 2007 (UTC)

[edit] Impossible dream

Finding myself inspired to tilt at windmills, I have tried to write a definitions section that honestly covers the basics for a general reader. The results are provocative; see Talk:Homotopy groups of spheres/Definitions.

Obviously it would need to be filled out with interwiki links and a reference or two, but that's premature. I'm curious to hear impressions. I don't want to bias comments, so I'll just say that aspects of interest include its length, its completeness, and how well it might actually serve someone who knows nothing about homotopy groups (nor topology).

The implicit question is, are we willing to do what it takes to address a general reader? Or do we say, in effect, that if you don't already know something about topology and homotopy groups then this article is not for you? I can live with either choice. I think it's important that we make an explicit decision and write accordingly. --KSmrq^T 09:09, 23 October 2007 (UTC)

I reckon you're close to routing those windmills ;)

Seriously, I don't think this is an either/or: I would like to see an article out of which a wide variety of readers can get something. Obviously, the more mathematically trained the reader, the more they will get. We ought to convey an idea of what the subject is about to a general reader, but I don't think we should hold their hand. The interested general reader will have to accept that a lot of background is needed to really understand the subject. KSmrq has written a nice textbook introduction, which captures the key points rather well, but (and he will not be surprised to hear me say this!) I don't find it very encyclopedic. Here I refer both to the style (e.g., "our 'glueing the ends' construction of a circle is instructive") and the content (e.g., it is primarily the job of homotopy to motivate and explain the term: this article just needs to give the idea and provide a wikilink). One approach for communicating technical content encyclopedically is to present the background informally, using wikilinks to provide the precision. (Piping is one way to do this: e.g., in a medical article, write "cell growth" instead of "anabolism".) That is the sort of thing I would favour for the early sections of this article. Geometry guy 18:47, 23 October 2007 (UTC)

Regardless of what we decide to do for the article, I'm tempted to save the subpage to show those annoying editors who whine about us insensitive, unlettered mathematics geeks not writing articles they can understand. Say: "Here's the mathematics background necessary to understand just the title of this article, never mind the content." And then we mumble inaudibly: "Now scurry on back to your Pokémon." :-D

On a more philosophical note, I do think it's good to give people the confidence that they can understand if they try, but maybe it's not so good to create the illusion that there are simple explanations for everything. I believe I could eventually explain a cohomology spectral sequence to a lay reader, but I wonder how many would have the patience to work through it. Isn't perseverance a big part of what has been called "mathematical maturity"? --KSmrq^T 12:41, 24 October 2007 (UTC)

 :-) Yes, why not save the subpage, but, as you probably know, subpages of mainspace are disabled in the mediawiki software for Wikipedia, so you have actually created two new articles with slashes in the title (this intro, and the experimental tables). I think it would be better if they were subpages of this talk page. As for the philosophy, I'm with you 100%, but there is a limit to what we can achieve here, and mathematical maturity is a rare commodity! Geometry guy 17:08, 24 October 2007 (UTC)

Thanks for doing that: could you {{db-auth}} the redirects too? They've already been picked up by bots as new articles, and I've fixed the link above. Geometry guy 23:13, 25 October 2007 (UTC)

I did not want to comment until others had their say. Here's my view of my "Definitions" and what's in the article. The current text of Definitions in the article is an impossible mix of levels. As well, it tries to combine definitions — which should be precise — with tutorial — which should be informal — and ends up doing neither well. My essay is a sprint, with the finish line being a precise topological definition of n-sphere and a precise definition of homotopy group. We have articles on both of these, and so could argue that we need not — and should not — repeat those definitions here. In this view, we don't need a Definitions section! I will not go that far, but neither would I give a tutorial on topological spaces and continuous maps.

The point of my Definitions subpage was not that we should put all that into the article (we absolutely should not!), but to spread out the infrastructure to better decide what we must have and what we can do without.

• My single paragraph on n-spheres, which also introduces "suspension", is at just the right level of detail (IMHO), and should replace what's in the article on that.

• Defining a homotopy group takes most of the space in my essay, and should be brutally trimmed. The steps I included — each with a paragraph! — are: (1) continuous map and constant map: (2) homotopy and null homotopic; (3) equivalence classes under homotopy; (4) addition of maps and homotopy classes; (5) group structure for fundamental group; and (6) generalization to higher homotopy groups with role of base point. I have nice SVG pictures to go with these (forced to be PNG by MediaWiki's choice of the most awful renderer available). I do not define exact sequence, and I still think we ought to help the mathematicians who do not know this central tool.

If we are serious about speaking to a broad audience, we must do a better job of defining/introducing the homotopy groups. We cannot leave it to other articles. And I want editors besides me to grapple with the tough questions.

• I want Geometry guy to better demonstrate how to tutor without sounding tutorial, and to decide how much tutorial he really wants. I want Jakob.scholbach to stop writing a Wikibook and to start thinking in terms of one or two brief paragraphs to say everything.
• I am especially impatient with the contradiction between "be inclusive" (accessible for the lay reader) and "be encyclopedic" (formal and concise for the expert). Maybe it's only a tension, and a happy compromise is possible. Maybe I'm frustrated because I don't see that I'm getting much help making it happen.
• Yes, I am perfectly capable of doing a good job on my own, but where's the Wikifun in that? Besides, Wikiprocess works better if everyone "owns" the result, and the only way to appreciate the writing choices is to be forced to confront them.
• Use the text mostly to speak to the mathematicians, and use the pictures to speak to the broader audience. Focus on two essential details of the homotopy group: (1) pointed homotopy with null homotopy, and (2) addition via equator pinching. I've already inserted the necessary homotopy-of-pointed-circle-maps picture; I'll shortly provide a pinch picture to go with it. And if no one else has the courage to tackle exact sequences, maybe I'll see what I can do.

By the way, those garish polygonal pixelated spheres must be replaced. They are an amateurish eyesore, and I wince every time I see them. While we're at it, we should picture a S²→S¹ map, and a S³→S² Hopf fiber bundle (showing both S³ and S²).

Last word: I'd like to have fun with this. If we can actually pull off getting this through FA, I'll be thrilled. If we can't do that but can bridge the gap between freshman calculus and the stuff R.e.b. has been injecting, I'll be awed. And failing that, if we can convince a handful of readers not only that this stuff is interesting, but also that they might be able to understand it if they work at it, I'll be more than satisfied. --KSmrq^T 12:46, 29 October 2007 (UTC)

Well said - especially the last word! There is indeed a tension between being accessible and encyclopedic, and the happy compromise is hard to achieve. For me, "encyclopedic" does not so much mean "formal and concise for the expert" as "authoritative", and "accessible" does not so much mean "inclusive" as "explanatory". I think it is possible to be both authoritative and explanatory, and I'm looking forward to taking up your challenge. But not tonight: I intended to make a start, but was distracted into despair by the lamentable state of our articles on the sphere and the n-spheres. Geometry guy 22:45, 29 October 2007 (UTC)

replying to KSmrq -- I don't want anybody to want anybody to do or not to do something in a collaborative work like this. Everybody contributes the way he thinks best. This is the strength of WP, not editors authoritatively claiming to follow the one and only correct path. Instead of not writing a wikibook and starting to think in certain terms, I will rather stop contributing to this article. Good luck for those who keep working. Jakob.scholbach 19:27, 2 November 2007 (UTC)

Thanks for the good wishes and for your contributions. I've made a small step to move the current section in the direction of being more explanatory, but hopefully without losing authoritative tone. The two versions on the subpage are very helpful, and I think they could also be used to improve some later parts of this article (e.g. the suspension idea) and the n-sphere article. Geometry guy 21:17, 2 November 2007 (UTC)

(edit conflict)

If you do not wish to collaborate, I'll be disappointed, but must respect your choice.

Collaboration requires that we talk about our ideas and goals. If you read what Geometry guy and I have both said, we are hoping to find an appropriate balance between two extremes. A long tutorial would certainly help more readers to understand the meaning of the title, and perhaps more of the article (but not much more). An alternative is to let the articles on spheres and homotopy groups carry most of the burden. As Geometry guy and I agree, the length and style of the "Definitions" subpage I created is not appropriate for the article. I believe you were misled by seeing my text without seeing my opinion of its implications, and have written in a style even more tutorial than my example.

The point of my remarks about what I would like to see is to exert pressure towards the middle. If you think an extended tutorial is appropriate, you are certainly welcome to make an argument. However, that does go against common practice in Wikipedia. The usual recommendation is that such material belongs in Wikibooks, not the encyclopedia.

I would like you to continue to contribute to this article. Here's what worries me. I do not begrudge the time I spent writing all those Definition paragraphs, even though I expect to discard 90% of it. As an experienced writer, I have come to terms with the need to "murder my darlings" (though I can still feel a pang of regret when I do so). I see you polishing material that I believe is not likely to be used. I'm not sure you realize most of your work may be discarded along with mine. It may hurt to hear that now; how much more so later? --KSmrq^T 21:33, 2 November 2007 (UTC)

[edit] Singularity theory

I'm trying to get my head around the link to singularity theory, and wondering how much information you can really get from HGoS's. Looking at say maps from R¹ to R¹, normal forms of singularities will be xⁿ, n>2. However the homotopy groups here π₀(S⁰) does not contain much information of use, indeed it does not seem to distinguish between singular x^3 and non singular cases x.

I still think the whole applications section deserves an unreferenced tag, specific citation for all of these would help those wanting to read more on these connections. --Salix alba (talk) 11:56, 22 October 2007 (UTC)

You need to complexify to get the full information out of the homotopy groups. Then the map sending z to zⁿ has winding number n in π₁(S¹). I've asked Jim if he can find a suitable ref. I guess Arnold must have discussed this somewhere, but I haven't been able to track it down. Geometry guy 18:02, 23 October 2007 (UTC)

I asked a friend of mine who works in singularity theroy about this and this was his reply

My quick and not very well considered opinion: Not sure! I think the point is to say something about homotopy groups using singularities, not the other way round. And I don't think I've never been to a talk where someone did either task. Besides, why would a sphere in R^m map into one in R^n? I'll have to think about this!

From a quick google search I didn't come up with much linking the two, some very old work and some very new, this might be more general homotopy than specifically spheres. I think one of the core problems is that equivilence relation used in ST is diffeomorphism which is more stringent than homotopy. Yes complexification could solve the R¹ to R¹ problems, things get more interesting with R² to R¹ when the umbilics start to appear: x^2 y+y^2.

I see Jim has replied

The idea is most commonly applied to π_n(Sⁿ), e.g. in the definition of the index of an isolated zero of a vector field, or in the definition of the ramification index for a map between Riemann surfaces.

which seems a more modest claim, which maight be more appropriate to include here. --Salix alba (talk) 12:52, 24 October 2007 (UTC)

Yeah, I suggest you modify it along those lines: those facts can be sourced to numerous textbooks. Geometry guy 17:02, 24 October 2007 (UTC)

[edit] Balance

I think the article is unbalanced insofar as it contains plenty of tables whose content I consider less encyclopaedic. Seriously, isn't the value of pi_30(S_20) something we could refer the reader to a (printed or online) reference or to an appropriate subarticle? Unless we do a more thorough discussion of the values, I feel it is not that much sense in merely listing the values. For example, I like the table of stable h.g.- everybody will notice the kind of periodicity in the complexity of the groups. But the subsection fails a little bit to explain these. I don't know if it is somehow (perhaps vaguely) possible to give indications on this. Is it? If so, it has to be there.

Another thing: a while ago I added a sentence relating the existence of Hopf fibrations to the theory of division algebras over R. KSmrq deleted this calling it a "cleanup". I don't know/understand why he did so. I still think this kind of remark is actually more worthy than listing values whose principal effect is to convey to the reader a feeling of helplessness.

Another way to tailor down the amount of table values: the three tables contain a considerable overlap. I know there are arguments for either presentation, but altogether the article contains more letters than spirit, so to say. Jakob.scholbach 12:59, 26 October 2007 (UTC)

As the diff shows, my edit was a cleanup, of a number of scattered items. I decided to delete the sentence in question

• "The occurrence of these fibrations and the fact that there are no similar higher-dimensional analoga is explained by the theory of division algebras over the real numbers, thus exhibiting a link between topology and abstract algebra and number theory."

because it struck me as an unhelpful digression. (It drew my attention because of "analoga", presumably a typo.) In fact, "is explained" overstates the case, since division algebras are just one possible approach.

The content of the article will be what we make it. At the moment, "we" is mostly R.e.b. (talk · contribs) adding substance, with a little polishing by Geometry guy (talk · contribs) and me. I did write a first draft of a more substantial Definitions section, but when I asked for comments only Geometry guy responded. I expect to rework the Definitions, and perhaps flesh out the discussion of lower examples a little. The major idea I would like to see amplified is how we use sequences, exact or spectral, to compute groups. Although I did put a little in concerning the fibration sequences (as we discussed not long ago), we could really use a section with a more thorough discussion of the process. Also, if you want to work on polishing and structure, I believe R.e.b. is working mostly from Hatcher's online books, so you could read up and add a little more kindness for our readers. --KSmrq^T 19:23, 26 October 2007 (UTC)

I believe the tables are encyclopedic, especially as Wikipedia's goal is to contain the "sum of all human knowledge" (subject, of course, to WP:N and WP:NOT). So the real question is, should they be in this article, or are they unnecessary detail, which could be filled in by an article on Tables of homotopy groups of spheres? My feeling is that they should stay, especially as they are much more appealing that the technically advanced paragraphs. I agree though, that they should be accompanied by more explanation, and I think R.e.b. has been adding some.

As for the division algebras sentence, I would have just re-added it, maybe rephrased. If KSmrq really thinks it is a big deal (which I doubt), he would have removed it again with a more specific edit summary... (In general, reverting a revert is not impolite, IMO, but after the second revert it is time to talk!) Geometry guy 20:01, 26 October 2007 (UTC)

OK, I tried to not only criticise, but also give some (more) constructive input. The results of this nocturnal effort are here (2nd section). I thoroughly read your, KSmrq, proposal, which I like insofar as it is accessible, but IMO brings too much notions which are not necessary to spell out that explicitly (homeomorphic, manifold, equivalence classes, associativity, suspension). I tried to give an introduction as layman-friendly as I could come up with. Compared to KSmrq's proposal, more emphasis is put on the general mathematical concept of category (this is an everyday word, fortunately :-)) and the impact of a functor (without telling that we have a functor). I (would) have some more images illustrating the story. Most of this is pretty intuitive, so I think images can convey almost all of the content. In addition to such an "Introduction" section, I would write a more formal and also much shorter "definitions" section.

As for the "cleanup": I will try to find a reference backing up my statement and if I find it, restore the statement. Saying why there is a fibration for certain indices equal to 3, 7 and 15 is IMO not at all unhelpful, but there is certainly a neater way to put it than I did. (For my "analoga": we say so in German, being somewhat closer to our ancient greek roots, it seems...) Jakob.scholbach 23:22, 26 October 2007 (UTC)

A few people have asked for more images, anything in particular? --Salix alba (talk) 23:56, 26 October 2007 (UTC)

Well, if you could do that, it would be great. I'm personally not at all into animated gifs, so I would have some trouble doing this. I hope the descriptions of what I had in mind (see here) are clear enough. Thanks. Jakob.scholbach 10:55, 28 October 2007 (UTC)

I agree that it is important that we keep the tables. In a subarticle, maybe, but they are very important in homotopy theory, and represent vast amounts of human research. I would personally keep tables as large as known on the homotopy groups of spheres, Lie groups, Grassmannian manifolds, Stiefel manifolds, flag manifolds and so forth, but this may be POV. An encyclopedia is a reference work, and the homotopy groups of spheres are the periodic table of the homotopy theory world. For what it's worth, I have a PhD in topology. Adam1729 (talk) 20:48, 23 December 2007 (UTC)

I concur — in fact, I'll see you, and raise you one. The tables as they are only include part of the story — the additive structure of π_*(Sⁿ), but not the "multiplicative" structure, and other, higher-order operations. This is mentioned (too briefly, in my opinion), in the Ring structure subsection, but not fully developed in the tables, so as to make it clear(er) what it means, and how useful that is. To make them "come alive", and reveal the deeper meaning behind those numbers, one must pay attention to those operations, I think. So, instead of wringing our hands about whether to keep or not to keep those tables, POV vs NPOV (what's that got to do with this, as close to pure knowledge as one is likely to get to?), and so on, how about working on having even more spiffy tables? Cheers, — Turgidson (talk) 22:09, 23 December 2007 (UTC)

[edit] Spheres

So I don't forget, and for others to contemplate, topology can use perhaps a half-dozen different definitions of n-sphere.

1. Implicit surface, x₀²+⋯+x_n² = 1
2. Parametric surface, such as (cos φ sin θ,sin φ sin θ,cos θ)
3. Suspension of equator, Sⁿ = ΣSⁿ⁻¹
4. Interval mod boundary, Iⁿ/∂Iⁿ
5. Disk mod boundary sphere, Dⁿ/Sⁿ⁻¹
6. Spin of semi-equator, (x₀,…,x_n−2,x_n−1 cos θ,x_n−1 sin θ), x_n−1 ≥ 0

Each of these has its uses and benefits for our topic. For example, the spin version is helpful for explaining π_n(Sⁿ) = Z. --KSmrq^T 06:05, 4 November 2007 (UTC)

A handy list! I would add the 1-point compactification of Rⁿ, although this may only prove to be useful for visualizing the Hopf fibration of S³. Geometry guy 20:20, 4 November 2007 (UTC)

It occurs to me that I left out the CW-complex version (equivalent to your suggestion), which is important for the homology groups, and hence (via Hurewicz) another route to π_n(Sⁿ) = Z. --KSmrq^T 20:46, 4 November 2007 (UTC)

[edit] A-Class Review?

I'm wondering if an A-Class review would be worthwhile input at this point. We have most of the material in place now, but we are wondering whether we can improve the presentation, accessibility and balance. It is possible that other members of the Mathematics WikiProject will be able to provide some helpful input and/or stimulate some useful debate. A-Class review is not very active at the moment, but I think another post at the round table might attract some interest. Geometry guy 20:20, 4 November 2007 (UTC)

Sure, why not? One other remark/idea: currently, the article does not make clear (to me) whether the route is via the stable groups towards the non-stable ones or the other way round. The question may seem cumbersome, as in every equality a=b, a explains b and the other way round. I ask for the following reason: I don't get a feeling of the evolution of the techniques. E.g. is it true that people study stable groups because the non-stable ones are inaccessible? The greater part of the General theory section (which is very long) is devoted to stable matters. Perhaps a restructuring to something like 4. Non-stable groups (Hopf, Finiteness & torsion) , 5. Stable groups (J-homo., ring structure) is in order? We could then put the tables of values of the groups closer to the corresponding sections. For example, right now, the phenomena occurring in the stable table at the end are nicely (but pretty far) explained by the J-homomorphism section. Jakob.scholbach 21:50, 4 November 2007 (UTC)

I've done the A-Class listing now. As regards the stable vs. unstable issue, my view is the following. It is better to make the distinction between general groups and stable groups. People originally studied the groups in general, but then they realised that there are stable ones. Furthermore, understanding the stable groups is helpful prerequisite for understanding the unstable ones. Modulo this, I agree that some restructuring on the lines you suggest may be helpful. Geometry guy 19:42, 5 November 2007 (UTC)

[edit] Claim that the Hopf fibration "corresponds to complex numbers"

In the section entitled Hopf fibration, the article says that the fibration S^1 -> S^3 -> S^2 corrseponds to complex numbers and that some other fibratons of spheres correspond to the quaternions and octonions. No explanation is given of what this could even mean or why it might be true. I know there is some kernel of truth in what is being said but can't recall the connection enough to make it comprehensible. I think the statements should be clarified. Probably the best thing to do would be to flesh out the statement on the Hopf fibration page and then make the statement here a bit more vague and provide a link. Jrdodge 16:55, 7 November 2007 (UTC)

I agree with the problem, and your proposed solution. What is needed is the observation that S^2 is the complex projective line, S^4 the quaternionic projective line, and S^8 the octonionic projective line. I hope someone can fix it: I may have time at the weekend. Geometry guy 18:34, 8 November 2007 (UTC)

OK, I'll give it a shot (if I get a chance), but basically, the Hopf map S^3 -->S^2 is given by (z_1,z_2) --> (z_1:z_2), where in the source S^3 is viewed as the unit sphere in C^2, and in the target S^2 is viewed as CP^1 (as Geometry guy says), with homogeneous coordinates. Replacing C by H or O gives the other Hopf fibrations. Turgidson 22:05, 8 November 2007 (UTC)

Discussions of this family can be found in Hatcher's Algebraic Topology, p. 379, and in Steenrod's The Topology of Fibre Bundles, especially p. 109. Steenrod gives a more complete discussion of the octonion (Cayley number) case, for which the proof of being a bundle is complicated by lack of associativity.

There are many ways to describe the S³→S² fiber bundle. The one we want for these generalizations takes the entire space sphere to be pairs (z₀,z₁) with |z₀|²+|z₁|² = 1, and takes the projection to be z₀/z₁. This gives base spaces S² for z₀,z₁∈C, S⁴ for z₀,z₁∈H, and S⁸ for z₀,z₁∈O. These are spheres because we complete them with a point at infinity. The entire spaces have double the dimension less one, namely S³, S⁷, and S¹⁵.

Please note that the isin operator being used here (is it an arrow, or subset?) is showing up on this system as a little square box. <math> may be better for the article. Septentrionalis PMAnderson 21:47, 14 November 2007 (UTC)

I think the essential fact for us is that these bundles (and none higher) exist. (Jakob thought it worth mentioning somewhere — perhaps in Applications? — that topology gives the existence results for an algebraic question; I objected that we use existence to compute homotopy groups of spheres, not the other way around.) I'll add a minor clarification of "corresponds", and cite Hatcher. --KSmrq^T 22:37, 10 November 2007 (UTC)

I've now done as Jrdodge suggested, fleshing out and tidying up Hopf fibration (although there is still work to be done there), and trimming unnecessary detail here. Geometry guy 15:04, 18 November 2007 (UTC)

[edit] Strange referencing style

I have reverted Gg's reversion of my fix. The referencing used the footnote system mixed with Harvard references in a chaotic way. Most of the footnote contents were pure Harvard refs! The remainder was chatty little comments, which I converted to parenthetical remarks, since that is what they are. (If they are not good enough to retain in the body, they don't deserve to be in a footnote either.) Please don't let's edit war over this. Really, go back and look; I fixed a mess. --KSmrq^T 21:34, 14 November 2007 (UTC)

PS: I also tried to fix some other things while I had the whole article open. Especially, the table of stable homotopy groups now presents the image of the J-homomorphism using underlining instead of bold italic, which I think works better. --KSmrq^T 21:37, 14 November 2007 (UTC)

PPS: I had wanted to work on something completely different: to amplify the discussion of π_n(Sⁿ) to include mention of the Hurewicz isomorphism and the exemplars by spinning a hemisphere (θ↦mθ). Along the way I found that the markup of the Hurewicz theorem article was a mess, and spent a lot of time cleaning that up. Then I come back to my original task, note the AWB cock-up, fix that, and note the chaotic referencing (which somehow had slipped under my radar). *Sigh.* So I still have accomplished none of the simple little task I hoped to accomplish! Ain't Wikipedia grand?! --KSmrq^T 21:46, 14 November 2007 (UTC)

I know you dislike footnotes, KSmrq, and I appreciate your point of view. However, the referencing here has long been of the form "Notes" and "References", which is a standard Wikipedia format (see e.g. the Georg Cantor FA). The links (using the citation template and Harvard macros) are a courtesy to people, such as yourself, to provide links to the references using the citation template and the Harv macros. This does not mean that the body of the text uses Harvard referencing. It does not, and a wholescale change is unjustified. Furthermore, there was no AWB cock up: only a spelling and whitespace was fixed. I'm sorry if you think this has distracted you, but you have been distracted by your own preferences, not by any deep flaws in the article. Geometry guy 22:12, 14 November 2007 (UTC)

Thank you for responding.

It's hard to believe you actually looked at the AWB change, as I did; I kept the spelling fix and the link fix, but the blank line removal between tables was indeed a cock-up.

I am familiar with the Notes/References format. I do not appreciate the ad hominem dismissal. I do not appreciate the multiple reverts without discussion. I stand by my objections to the way the footnotes were used.

I would appreciate it if you restored the work I put into the table, not expecting it would encounter an edit conflict because of a revert war. --KSmrq^T 02:19, 15 November 2007 (UTC)

Sincere apologies. The table does indeed look better with underlines, and was easy to restore. I found your edit summary confusing, and did not spot that the blank line made the tables too close together: I have added a comment so that the next time someone with AWB stops by, it does not happen again. As for those pesky footnotes, this is one of those "keep to the original style" issues that needs discussion before wholesale changes are made, which is why I concentrated on your actions rather than the issues. I'm sorry if that was rude or ad hominem. Anyway such discussion is now happily underway. Geometry guy 11:02, 15 November 2007 (UTC)

I agree with KSMrq that the footnote reference style is kind of senseless when it is just giving the link to a reference in the reference section. (It is a bit more useful if there are 3 references to the same paper). Comparing

Higher homotopy groups were first defined by Eduard Čech in 1932.^[5]

to

Higher homotopy groups were first defined by Eduard Čech (1932).

The first reads less smoothly to me. Or look at a footnote à la

This material is also in many textbooks on algebraic topology, such as Hatcher 2002.

If we want the reader to know this, we should write it where the footnote takes place. Otherwise we can also discard it and link to the reference directly. Jakob.scholbach 22:25, 14 November 2007 (UTC)

I'm guess this reference was a fix to get through the GA system requiring inline references. Probably not needed now the article is much better referenced. --Salix alba (talk) 23:06, 14 November 2007 (UTC)

You're guessing?! You added the first footnotes a year ago in response to a GAR challenge. :-)

But since then, as you say, the article has changed considerably. In particular, three references have become … well, more than I care to count! As well, we now have {{harv}} templates with autolinking. And no longer is there any question of needing footnotes (instead of Harvard style) for GA. Am I to understand that you would be willing to join with Jakob and me in abandoning footnotes? --KSmrq^T 02:19, 15 November 2007 (UTC)

Yep, the footnote was a hack (technology). --Salix alba (talk) 08:29, 15 November 2007 (UTC)

I much prefer KSmrq's version of the references, and suggest that we should revert back to it. The "footnotes" style introduces a pointless double redirect every time one wants to look up a reference. R.e.b. 02:28, 15 November 2007 (UTC)

Okay, I'd better defend the footnotes. I agree with Jakob that in the case of Cech, where both the author and year are interesting and relevant to many readers, there is no advantage in the footnote system. However, in the second case, we are providing meta-information. The footnote in full reads:

"The mathematics described in the first two sections and the beginning of the fourth section can be found in many textbooks on algebraic topology, such as Hatcher 2002 or May 1999b. For the homotopy groups of spheres, see Hatcher 2002 p=339ff."

I inserted this footnote in several places to meet the scientific citation guidelines. Making the same parenthetical remark in several places would be distracting, and is not relevant to the article content. Indeed, often the source is not relevant. In some cases the author is relevant (as it was the person who did something), but the year is not particularly interesting. The Harvard referencing format is a bit of a straight jacket here: it forces one to cite primary and secondary sources in the same way.

So I disagree with the point of view that any statement should either be stated in the text, or removed. In particular, there is no place for meta-information in the body of the article.

As for GA/FA, I know that the Harvard system is accepted as one consistent method for formatting references, but it is outside the comfort zone of many reviewers. For an example see the recent good article review of Marsileaceae. I want to give an FAC the best chance possible so that this experiment is a useful one. We are certainly going to be asked to add cites, and I don't want the FAC to degenerate into an argument over citations, as I am more interested in discussion about advanced content. For this reason, I strongly prefer that we follow the much more widely used Notes/References format. Geometry guy 11:23, 15 November 2007 (UTC)

I've never participated in a FA or GA review. But it seems weird to me that one format will not be allowed by reviewers. After all, it is mainly a cosmetic difference. I believe it is more important to arrive at a coherent style. For example, footnotes like

O'Connor, John J; Edmund F. Robertson "Camille Jordan". MacTutor History of Mathematics archive

seem to me just to belong to the references section.

Also, the "meta information" in

^ Kochman 1990. Actually there remains a small ambiguity in three of these 2-components.

is redundant as this is clearly exposed in the "Table of stable h.g." section.

^ This is described in Ravenel 2003.

could just be a harvard link to the Ravenel paper.

Footnotes which are explaining something which is covered in subarticles like

^ Adams 1966. Adams also introduced certain order 2 elements μn of πnS for n = 1 or 2 mod 8, and these are also considered to be "well understood".

could be deferred to the subpage, in this case J-homomorphism.

I could imagine that structuring the "references" section may render unnecessary footnotes like "the math. described here and there can be found here and there.". If we had a subsection "General theory" in the refs section, we can omit the footnote and a harvard style link for such general references at all. Also, writing a very brief description of the references (if it is not clear by the title) can be helpful. (Not everybody has MathSciNet access). Jakob.scholbach 13:05, 15 November 2007 (UTC)

Then you have no idea of the amount of sheer stupidity and crankishness that FA (and especially GA) involves. It depends to some extent on which reviewers happen to notice your article, which is random; but some reviewers will oppose articles on

• The number of footnotes
• The usage of an en-dash vs. a hyphen (– vs. -)
• Whether footnotes use the {{cite book}} template and its sisters (this last is rare, I am glad to say, since, unlike the rest of these follies, it is unsupported by the letter of guidelines.)
• Whether it complies with every jot and tittle of WP:MOS, which is itself written and supported by a handful of provincial and ignorant cranks.)
  • I regret to say that all of these are references to actual current discussions.

The root cause of all this is plain: it is much easier to do these things than actually to read and review the content of articles; this will, of course, be doubly true of this article. So far, common sense has occasionally prevailed over all this; but it can't be relied on. In short, don't worry about it; GA is absolutely worthless, and FA is almost worthless. It would be nice to see this on the front page, but if it doesn't happen, oh well. Septentrionalis PMAnderson 20:34, 15 November 2007 (UTC)

I think the "in-use" and "close this debate" was unnecessarily proceduralistic, and in any case it is somewhat premature to close a discussion which has an unanswered comment less than 24 hours old. Thank you though, to R.e.b., for taking care to preserve e.g. Turgidson's edits when updating the article. I had promised to do this.

By and large (as far as I am aware) those editing here are professional mathematicians. We are used to Harvard references and citing sources in the text. General readers are not. It simply isn't the tradition in encyclopedias, which make content more accessible by footnoting technicalities and sources. Further, as I have indicated already, Harvard referencing confuses the role of verification (to secondary sources) with citation (to primary sources).

I think it is a big mistake to switch to Harvard. If others don't care about the FA experiment, that is fine: I will forget it too. But if we do, then we are just throwing a spanner into the works by dropping the footnotes. As Pmanderson points out, common sense cannot be relied upon at FAC, and this exposes a potential FAC to additional randomness, making the whole endeavour less worthwhile. Geometry guy 08:14, 16 November 2007 (UTC)

The use of Harvard is more likely to escape WP:FAC because it is supported by the letter of WP:CITE (we cannot guarantee that reviewers will know this; but quoting it may suppress them). But what we want to do for the general reader is more important.

It should be possible to have a variant of {{Harvtxt}} which would link from the author's name alone, which would be less forbidding to readers not used to it. (We have only three cases where we cite multiple papers by the same author, and all three appear to be joint citations of related papers, so we don't really need years very much.) Septentrionalis PMAnderson 15:34, 16 November 2007 (UTC)

Isn't this possible with Template:Harvard_citations? Jakob.scholbach 15:45, 16 November 2007 (UTC)

So it is; good. I was thinking for a second that {{harvtxt}} didn;t need {{citation}}, but I see I was overoptimistic. Septentrionalis PMAnderson 16:25, 16 November 2007 (UTC)

If necessary, we can switch back to footnotes. It's not that much work, I guess. Jakob.scholbach 08:53, 16 November 2007 (UTC)

Of Wikipedia:Featured article candidates/Battle of Red Cliffs‎ is promoted, mentioning it may help. Septentrionalis PMAnderson 03:25, 17 November 2007 (UTC)

I encourage editors here follow the link to that FAC, which contains exactly the sort of unedifying discussion of citation that I wish to avoid. Geometry guy 12:51, 17 November 2007 (UTC)

I read what's opined there. Except for one person (Hadseys), who did not (yet) care to give an argument, everybody else up there does not object. Apparently, lots of editors don't know the harvard style (yet), but this should be no reason for us to avoid it (rather it should be a reason to promote it when it is sensible). Jakob.scholbach (talk) 13:28, 17 November 2007 (UTC)

[edit] In defence of footnotes

Sorry for being a latecomer to the party, but I also feel that it was premature to close the discussion on the best mode of referencing the article. Regardless of your general opinion on the relative merits of comments/notes/footnotes and Harvard referencing, I urge everyone concerned about the readability of this article to examine the two versions. My conclusion is that the notes worked better. Far from all the material formerly present therein, which generally included comments as well as the citations, can be captured by the citation template, and I am especially alarmed by proliferation of citations such as (Scorpan 2005), which is not, to the best of my knowledge, a standard reference for homotopy groups of spheres. (On the other hand, Ravenel's book is such a standard reference, yet it would seem strange to cite it almost everywhere, just because most of the quoted results can be found in it.) An explanation for some of the reasons why footnotes worked better was given by Geometry Guy above: the fundamental distinction between primary and secondary sources (one has to be quite careful even with regards to primary sources; it may not be the best course of action to quote Hopf's original paper by a way of introducing the Hopf fibration, especially in an article on a different subject). I would like to add that Harvard referencing also tends to conflate the scholarly sources such as textbooks of Hatcher and May and Ravenel's monograph with popular expositions such as MacTutor archive, and I think it is quite a mistake to create an impression that their authoritativeness, accuracy, and level of presentation are comparable (they are not). Unfortunately, there is no way around it within the current scheme. I would also caution against basing a strategic decision about the best way of presenting material in an article on minor implementation details, such as the "double redirect" mentioned by R.E.B. above. Arcfrk (talk) 10:06, 17 November 2007 (UTC)

Eloquently argued, Arcfrk. In addition to these and my previous comments, I would like to mention the scientific citation guidelines, in particular, the following sentence:

"In sections or articles that present well-known and uncontroversial information – information that is readily available in most common and obvious books on the subject – it is acceptable to give an inline citation for one or two authoritative sources (and possibly a more accessible source, if one is available) in such a way as to indicate that these sources can be checked to verify statements for which no other in-line citation is provided."

It is much easier to indicate this using a footnote, and the current version is not so clear on this point. Additionally, although some information, such as page numbers, can be provided in a Harvard reference, this tends to add unnecessary clutter to the article text. Geometry guy 12:44, 17 November 2007 (UTC)

I tried to structure the references section in order to make the nature of the references clearer: (more or less) current research papers, papers with a historical perspective, general introductions to topology. Does that help? Jakob.scholbach (talk) 13:28, 17 November 2007 (UTC)

My turn.

I did not set out to eliminate footnotes. To begin, I saw that we had accumulated numerous footnotes whose sole purpose was to contain a name-date (Harvard) citation. As almost everyone agrees, that makes no sense. So I went though and fixed those to be straightforward Harvard style. That left a handful of footnotes that contained auxiliary information. These were now easier to consider. I began to look at them one by one, and each time concluded that either the comments added no information, or that the information should be in the article proper (at best). I was startled to see that no footnotes were left. I eliminated the now-empty Notes section.

This experience troubles me, as do Arcfrk's comments. In fact, my interpretation of Arcfrk's complaints is radically different. The footnotes hid problems. Now he doesn't like seeing the problems. So fix the problems, don't hide them under more footnotes! As for discriminating sources, popular versus scholarly, that's bunkum, for several reasons. Do we really think readers are so stupid they won't notice the difference between a MacTutor history and a textbook?! The footnotes citing MacTutor said nothing about the difference. And if we feel it essential to say something, we can either say it in the body or annotate the references.

In a context or a field where heavy citing is forced by tradition or pragmatic needs, footnotes may be the system of choice. Within Wikipedia they are often badly misused. Editors go for sheer quantity. No one — neither the "reviewers" nor the readers — tries to wade through them. When we see a handful of name-date items, we think about them; when we see tiny superscript numbers, we don't.

Confusion within Wikipedia notwithstanding, citations — even thousands of them — do not make reliable articles. Reliability comes from writers who understand the topic and reviewers who understand and do the work to check the contents. The article is for the readers; the talk page is for the editors and reviewers. I have written a number of large articles without collaborators, and never added a single citation to satisfy WP:V; yet I claim that those articles have more sources and better sources than the vast majority of mathematics articles, because I care about the readers.

The scientific citation guidelines neither require nor even recommend footnotes. They were drafted, in part, as a rebellion against the flood of footnote madness! We don't need a chatty little footnote just because we choose to cite only one or two sources out of numerous possibilities. Reader's (especially of this article) are not idiots. Choose well, cite, and get on with it.

The proposal that we should deviate from Wikipedia-approved citation style defensively, rather than do what we think best, is the worst idea yet. Look at what happens when the press censors itself out of fear; it's usually far more restrictive than the legal censors! Geometry guy cites the Good Article Review of Marsileaceae, yet I see nothing there to suggest name-date style had any real impact; reviewer ignorance about plant families was a factor. Geometry guy and Septentrionalis point out that many reviewers are unfamiliar and uncomfortable with name-date citations, which is true. Frankly, if that is the quality of review we get, we don't stand a chance. Wikipedia has explicitly chosen to treat different styles as equally valid, just like different spellings — which are also unfamiliar and uncomfortable. A reviewer who would reject the article on such invalid and superficial grounds is probably incapable of understanding so much as the introduction.

Septentrionalis mentioned hyphens versus dashes. As it happens, I tend to be sensitive to such typographic details, and routinely substitute en dashes, em dashes, and minus signs where appropriate. (For example, I fixed this in the new Berrick reference.) So, not to worry. --KSmrq^T 23:31, 17 November 2007 (UTC)

We are clearly not going to agree on this. We do need a footnote because we cite only one or two secondary sources out of many possibilities (and I fail to see in what way the original footnote was "chatty"). For one thing, Hatcher did not invent this material, and mentioning his name multiple times in the text is inappropriate, although I'm sure he'd appreciate the advertising for his book. I agree with the comment below that we should cite a classic source like Spanier also. Footnotes allow for the citing of multiple sources without disrupting the flow of the text. Most readers do not care about verifiability, they just want to learn or check something.

I initiated this exercise with the idea of working with the rest of Wikipedia, and giving it the best possible chance to support and evaluate technical content, not to work against it and expose how idiotic FAC and reviewers can be. The attitude that the rest of Wikipedia is confused, wrong, incompetant, etc., whether it is true or not, is unhelpful.

The footnotes had some problems, but the solution is to fix the problems, not remove the footnotes. Footnotes do not deviate from a Wikipedia-approved citation style. Instead, a wholesale change has been made to the original citation style that I supported and developed since my first edit on this article in June. Anyway, the case has been made both for and against, and there isn't much more to say: I don't own the article, and I've already said that I am happy to abandon this experiment. Thanks to the effort of many editors, this is a really impressive article now, and that is what really matters. Geometry guy 13:32, 18 November 2007 (UTC)

I'm puzzled by your response. How much more clear can I be? I did not set out to eliminate all footnotes, yet you seem to want to tar me with that brush. Do you see me or anyone else on this talk page saying "Ban all footnotes from this article, now and forever!"? I don't. And sorry for my diatribe about Wikimadness, but notice it begins by accepting a potential role for footnotes!

I did not add mentions of Hatcher, I merely exposed them. Now that you see them, you care; when they hid in footnotes, you (apparently) did not. I guess I'm just really stupid, because for the life of me I can't see how that is an argument that the footnote style was better!

Thousands of technical papers and print encyclopedia articles and textbooks seem satisfied to cite a handful of sources without fear of implying that there are no others. Do you think if we don't write a footnote to explicitly point out we haven't listed every possible source the bewildered readers and editors will suffer tragically by inevitably concluding that, say, only two textbooks on Algebraic Topology exist? Again, readers are not idiots. Or do you assume reviewers are idiots?

I did fix problems with the footnotes. When I was done fixing, there were no more footnotes. Oh well. The result is still a Wikipedia-approved citation style! That was not politics, it was just editing. The editor who added the first footnotes prefers the result. So does almost everyone else. If you're so convinced the FAC reviewers won't know a good thing when they see it, why would you want to ask their opinion? But I think you're insulting them just like you're insulting me and your other colleagues here.

I'm not trying to pick a fight with you; I'm not trying to pick a fight with anyone! Yet you attack my motives and my judgment. I don't know why you've done so, but it helps neither the article nor my equanimity. I would appreciate an apology and an end to the attitude. --KSmrq^T 23:42, 18 November 2007 (UTC)

That was an ill-judged response KSmrq: I know I have been guilty of the same, but I have no further reason to apologize. I have insulted no one in my last comment. I respect your point of view as well, including your point of view on footnotes. I will respond only to the substantive additional point you make about Hatcher: the advantage of footnotes is that they allow multiple secondary sources to be cited easily for the same fact. The original footnotes did not do this as well as they could have done, but eliminating them was not the best solution, in my view. I accept that you disagree. Geometry guy 00:05, 19 November 2007 (UTC)

PS: I am delighted to see a flurry of new activity, with many additional editors contributing to the article.

I'd like to ask one of you to take on a thankless task: skim through the article with the mindset of someone who really wants to come to grips with the topic, and make a note anywhere you would like to see a citation that would help you dig deeper. New eyes, not so close to the writing or the topic, could be a great help in this regard.

I'd also like to ask anyone who feels up to it to look over existing references, to help us be sure that the sources we cite are the most helpful ones available. These don't have to be one-size-fits-all, but do let us know your taste: formal, comprehensive, tutorial, seminal, whatever. Online references like Hatcher are great, and he seems to write well; but that doesn't mean we should not also cite, say Spanier, Algebraic Topology (ISBN 978-0-387-94426-5). Sometimes the first paper on a topic is still one of the best, but not always; newer understanding and better expositors can make more recent works a better place for a beginner to start.

I hope we all agree that what we cite is far more important than how we cite it! For example, the new Berrick et al. reference is a welcome addition, taking us into modern territory we had not previously seen. --KSmrq^T 01:27, 18 November 2007 (UTC)

Agreed. Carl(CBM) made some helpful comments at the A Class review. It would be good if other editors would comment there too. Geometry guy 13:32, 18 November 2007 (UTC)

I'm a bit lost with all the arguments about how to deal with citations, but I will try to familiarize myself with some of the subtleties, and will comment if I have something to add. In the meantime, let me second the idea of adding Spanier's book here, I think it's still a basic reference for this topic (eg, for some of the material on spectral sequences), even though it's a bit dated. While at it, why not also put George W. Whitehead's book as a general reference for homotopy theory, or perhaps more specifically for the J-homomorphism? (I just added a ref to that book on GWW's article.) Finally, I'm also a bit baffled as to why Scorpan's book is used as a reference for, say, Rokhlin theorem on the signature of smooth, spin 4-manifolds -- why not refer to the original paper on the subject? I'm not quite sure what's best, and I think it's a good idea to hash these things out as a model for other math articles, so I'd be interested to hear more opinions, if anyone is willing to elaborate on this. Turgidson (talk) 00:21, 19 November 2007 (UTC)

I agree with Spanier and Whitehead, and your comments about Scorpan. In my view we need a secondary source which explains Rokhlin's theorem: the original article (the primary source) is probably best cited only in Rokhlin's theorem itself, and Scorpan is bordering on being a tertiary source. I learnt the result from Lawson and Michelsohn, but there may be other secondary sources more suited to this article. Geometry guy 00:32, 19 November 2007 (UTC)

I forget where I first learned about Rokhlin's theorem (perhaps from the Kervaire-Milnor paper?), but there are many strands and stories about it. Despite its apparent simplicity (just a factor of 2, eh?), it's rather amazing how influential this result has been. Now that you mention it, that article also could use some expansion. I'll give it a shot at some point. Turgidson (talk) 00:43, 19 November 2007 (UTC)

[edit] Algorithm to compute homotopy groups of spheres

Is there an algorithm to compute all the homotopy groups of a sphere? The article mentions the Cartan--Serre "killing homotopy groups" method, but is vague on whether this is a complete method for doing so, albeit unwieldy. The Math Review calls it a "partial method" where one needs additional info to make the computations work. It's unclear to me whether the additional info (about "Eilenberg groups") is known for the case of spheres. In any case, if there is such a algorithm (even if useless in practice), it should be made clearer. --C S (talk) 04:13, 17 November 2007 (UTC)

There is now an algorithm — at least in principle — to compute the homotopy groups of S^2 (and, thereby, of S^3), using suitable braid groups. For higher-dimensional spheres, one still needs spectral sequences and the like, far as I know. Turgidson (talk) 06:29, 17 November 2007 (UTC)

So the question becomes: can the (e.g. Serre) spectral sequence be made computationally explicit? And the answer may actually be "yes": that's what Computational Algebraic Topology packages like Kenzo claim to achieve. Well, I have never took the time to check the precise extent of what Kenzo claims to be able to compute, but I think this includes arbitrary homotopy groups of spheres (at least in principle).

Anyway, I just stumbled on this page and was really impressed by its quality. After reading some of the comments above, I can see why it might indeed use a few further adjustments in structure here and there, but I'll still take my hat off to you guys. Great work! Bikasuishin (talk) 00:10, 24 January 2008 (UTC)