Talk:Homotopy groups of spheres/Archive 1

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Contents

Table?

Should there be a table accompanying this article giving the first X many homotopy groups of spheres? I'm tempted to put in the table of the first stable homotopy groups at least, just to educate the reader on their apparent lack of pattern. - Gauge 02:42, 13 May 2005 (UTC)

  • It seems like a reasonable thing to do: I thought of this as well, but didn't have a good idea for layout. - Dave Rosoff 01:46, May 14, 2005 (UTC)
    • Done, although took me numerous edits and playing with layouts. I've tried to give the co-efficients using the fewest number of terms and keepign teh terms as small as possible. I've kept powers of terms sepeate though (e.g. writing 3+22 rather than 6+2). Does anyone know where can obtain values for higher n and k? (If you can only cite articles in journals that's fine - I'm a student and there's a good chnace I have access to them via my university). Tompw 00:22, 16 December 2005 (UTC)
      • would be nice to have a discussion in the intro which is slightly less technical. i know this is difficult with such a technical topic, but would broaden the audience for this important topic. Covalent 19:39, 17 May 2006 (UTC)

Copied from Reference Desk

Where can I find the homotopy groups of order n+k of the n-dimenstional sphere [i.e. πn+k (Sn) ] for k>20 and n>19 ? It's for the article Homotopy groups of spheres. Tompw 00:36, 16 December 2005 (UTC)

I was hoping there might be something more in the updated (online) version of Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres [1], but I didn't spot anything. Keep in mind, one reason people are interested in these things is because they are notoriously irregular and difficult to calculate. It's not an area in which I play, and I've got no special insight. I just didn't want the question to be ignored. By the way, it might be helpful to add a link to Ravenel, and also one to some discussion by John Baez [2]. Good luck! --KSmrqT 11:01, 17 December 2005 (UTC)

Failed GA

Needs references --Jaranda wat's sup 21:41, 12 April 2006 (UTC)

Umm... that's what those three links at the bottom are... and in case it's not clear these are references, they main text gives them as referencesee. Tompw 22:47, 12 April 2006 (UTC)

That section should be renamed then --Jaranda wat's sup 22:51, 12 April 2006 (UTC)

This may surprise you, but Jaranda is absolutely correct. WP:CITE explains that "An ==External links== or ==Further reading== section is placed at the end of an article after the References section, and offers books, articles, and links to websites related to the topic that might be of interest to the reader, but which have not been used as sources for the article. Although this section has traditionally been called "external links," editors are increasingly calling it "further reading," because the references section may also contain external links, and the further-reading section may contain items that are not online." So, if they were used as sources, they ought to be given as "references" not "external links". If you could write it in reference format (including author, date of last update, and "last known good" date of URL access), that would be brilliant. A lot of maths articles currently are ignoring the referencing guidelines, which is a shame really! TheGrappler 04:45, 13 April 2006 (UTC)
Actually, my point was that references were given in the text, with links at the bottom. Anyway, this has now been sorted, so the point is moot. I agree that many maths articles do not give references... I think this is because many editors of math article see a reference as being something they consulted, rather than something to verify infomation given. As articles tend to get written out of the editor's own knowledge, then this means no refercnes get given. Anyway, it's something I am working on. Tompw 11:43, 13 April 2006 (UTC)
Good work! And it's great to see a "paper" reference too. And good point about the maths references in general. Is there any chance this article could be illustrated with the example of a 1-sphere and 2-sphere, or would it be hard to make a meaningful diagram? TheGrappler 14:08, 13 April 2006 (UTC)
Thanks :-) . Diagrams... tricky. I think I could "show" why pi1 of S1 and S2 are Z and 0 respecitvely, though I'd need someone else to actually do the diagrams. Wikipedia:WikiProject_Mathematics/Graphics Tompw 15:41, 13 April 2006 (UTC)
Yes sound good. I really think this article needs the simplest example possible. I might have a go at it or could could post a request on Wikipedia:WikiProject Mathematics/Graphics. --Salix alba (talk) 23:52, 19 April 2006 (UTC)
Tompw, if you can give me an idea of what you want in the way of diagrams, I might be able to come up with something. Put ideas on my user page. Thanks for the work you've done so far. Dave Rosoff 08:45, 18 May 2006 (UTC)
I've had a bash at creating an image for π1(S1) --Salix alba (talk) 08:57, 18 May 2006 (UTC)

GA review

I'm afraid I've removed the article from the good article nomination page - I don't believe it meets the criterion which requires technical terms to be explained. I have a doctorate in astronomy but still can't understand what this article is telling me. A few examples:

  1. no explanation of what homotopy theory is.
  2. the ways in which spheres of higher dimension can wrap around spheres of lower dimension - I have no idea what this means. Can an example be provided?
  3. what's a homotopy group?
  4. what are n-spheres and i-spheres?
  5. what's a homotopy class?
  6. From the algebraic aspect, there is ample evidence that they involve substantial complexity of structure - don't understand what is meant.
  7. The cases i < n for n > 0 are very simple: πi (Sn) = 0 - this is likely to scare off any non-mathematical readers - doesn't look simple to me!
  8. The case i = n is always the infinite cyclic group of the integers Z by a theorem of Heinz Hopf, with mappings classified by their degree. - again, I don't understand at all what this means.

That's just from the first few paras. I appreciate it must be very difficult to explain a complicated idea like this to a layman, but at the moment I don't think more than a very small fraction of our readers would come away from this article with an idea of what it's about. I have a PhD in astronomy but that didn't help me at all! The article should be written so that an educated but ignorant reader can understand it. Worldtraveller 21:14, 24 April 2006 (UTC)

OK, those comments are fair enough, and I've made some attempt to address them. Have a look and say what you think. Tompw 23:44, 24 April 2006 (UTC)
I did a major rewrite of the intro in an attempt to address these problems. --agr 16:36, 19 May 2006 (UTC)

I've not convinced this is quite up to GA status yet, compare Riemann hypothesis. My main concern is that its lacking in the history of the subject. When did people start studing it, who were the major people involved. --Salix alba (talk) 08:42, 3 September 2006 (UTC)

Big improvement

This rewrite is extraordinary. A massive improvement.... I think there is still substantial room for improvement, but this has come a long way and must be one of the best resources there is on the subject, including as an introduction for novices. Many thanks to the editors involved! TheGrappler 17:42, 26 May 2006 (UTC)

This article makes my browser crash

When I try to edit the first table, my mozilla browser window disappears? What's going on? Michael Hardy 23:22, 12 August 2006 (UTC)

GA Re-Review and In-line citations

Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:54, 26 September 2006 (UTC)

OK, I'm confused by what is meant by "in line citations". Do you mean Footnotes? If you do, then I do not think fotonotes are a suitable way of providing citations for this article, which is why a list of references are given at the end. Tompw 11:38, 26 September 2006 (UTC)
Quite a few eyebrows have been raised over this, virtually all the maths GA and 100's of other articles have had the same message and theres some discussion at Wikipedia talk:WikiProject Good articles. The new criteria have not been settled upon and whether cites need to be inline is being debated. --Salix alba (talk) 13:44, 26 September 2006 (UTC)
Maybe I'm being thick, but does "inline citation" equal "footnotes"? Tompw 19:06, 26 September 2006 (UTC)
Footnotes or Harvard referencing, either of which would be pointless and ugly here. Septentrionalis 21:12, 19 November 2006 (UTC)
That said I still feel this article needs more work on the history side, and this information could use some good cites. Whats the history of these groups, when were they first introduced, when were some of the groups calculated, by who and using what methods? --Salix alba (talk) 13:44, 26 September 2006 (UTC)
I quite agree this article needs more work, especially on the history / motivation side. I'm not sure if the methods of calculation can be explained, because they are so varied and so complicated... I learnt about these groups in a post-grad level class, and we only covered the computation of the simplest. Tompw 19:06, 26 September 2006 (UTC)

Proposed move / re-name

I have long felt "Homotopy groups of spheres" is a cumbersome name, which is hard to actually use in a sentence. (e.g. "The homotopy groups of spheres describe the different ways..."). I have held off commenting on this because I couldn't think of anything better... until now.

Consequently, I proposed this article be re-named Sperical homotopy groups. Tompw 21:36, 18 October 2006 (UTC)

No, no, no. Not recognisable. Charles Matthews 22:04, 18 October 2006 (UTC)
Agreed. "Homotopy groups of spheres" is the standard name, see the references section. Using 'Sperical homotopy groups" would be coining a neologism, something that is prohibited on Wikipedia.--agr 22:43, 18 October 2006 (UTC)

History

Trying to write something on the history of the topic. Some snipits --Salix alba (talk) 19:29, 19 November 2006 (UTC)

From http://www-history.mcs.st-and.ac.uk/~history/Biographies/Jordan.html Jordan introduced the notion of homotopy of paths looking at the deformation of paths one into the other. He defined a homotopy group of a surface without explicitly using group terminology.

From J. P. MAY, STABLE ALGEBRAIC TOPOLOGY, 1945–1966 http://www.math.uchicago.edu/~may/PAPERS/history.pdf

  • Hurewicz’s introduction of homotopy groups - 1935

[Hur35] W. Hurewicz. Beitr¨age zur Topologie der Deformationen. Nederl. Akad. Wetensch. Proc. Ser. A 38(1935), 112-119, 521-528; 39(1936), 117-126, 213-224.

The first manifestation of stability in algebraic topology appeared in Freudenthal’s extraordinarily prescient 1937 paper [Fr37, Est], in which he proved that the homotopy groups of spheres are stable in a range of dimensions.

It was shown by G.W. Whitehead [Wh53] that there is a metastable range for the homotopy groups of spheres.

The power and limitations of such direct homotopical methods of calculation are well illustrated in Toda’s series of papers [To58a, To58b, To58c, To59] and monograph [To62b]; while cohomology operations, spectral sequences, and the method of killing homotopy groups are used extensively, most of the work in these calculations of the groups ¼n+k(Sn) for small k consists of direct elementwise inductive arguments in the EHP sequence.

Although the credit for the invention of spectral sequences belongs to Leray [Le49, Mc], for algebraic topology the decisive introduction of spectral sequences is due to Serre [Se51].

Serre’s introduction of class theory [Se53a], and his use of the spectral sequence to prove the finiteness of the homotopy groups of spheres, save for ¼n(Sn) and ¼4n¡1(S2n), were to change the way people thought about algebraic topology.

The method of killing homotopy groups introduced by Cartan and Serre [CS52a, CS52b] was also profoundly influential. It provided the first systematic route to the computations of homotopy groups.

Milnor’s results just cited are also among the many implications of Adams’ celebrated theorem that ¼2n¡1(Sn) contains an element of Hopf invariant one if and only if n is 1, 2, 4, or 8 [Ad60]. This result was announced in [Ad58b], which was submitted in April, 1958.

Using the structure theory for mod p Hopf algebras of Milnor and Moore and Milnor’s analysis of the Steenrod algebra, I later developed tools in homological algebra that allowed the use of the Adams spectral sequence for explicit computation of the stable homotopy groups of spheres in a range of dimensions considerably greater than had been known previously [May65, May65, May66].

As we have already mentioned, the starting point of modern differential topology was Milnor’s discovery [Mil56b] of exotic differentiable structures on S7. Kervaire and Milnor classify the differentiable structures on spheres in terms of the stable homotopy groups of spheres and the J-homomorphism. In 1950, Pontryagin [Pon50] showed that the stable homotopy groups of spheres, in low dimension at least, are isomorphic to the framed cobordism groups of smooth manifolds. His motivation was to obtain methods for the computation of stable homotopy groups, and he used this technique to prove that ¼n+2(Sn) »= Z=2Z, thus correcting an earlier mistake of his.

Dimension

One of the things we are (quite sensibly) sweeping under the rug is what dimension means here. Spheres have homotopy groups because they are topological spaces, but the confused reader will not be helped by being sent to Lebesgue covering dimension, or other dimensions for spaces in general. I addressed this by saying manifold, but not linking; if someone has a less fragile dodge, please put it in. Septentrionalis 21:49, 19 November 2006 (UTC)

Initial questions

  1. What's a homotopy?
  2. What is a manifold?
  3. Does this topic have any bearing on other, larger questions?
  4. What do the 1 in 1-sphere and 2 in 2-sphere refer to? Isee something resembling a possible explanation in the 2nd paragraph, but...
  5. What does "one can be continuously deformed into another" mean?
  6. How can you assign the points on one sphere to a single point on the second?
  7. What is a "metastable range," and why is it important?
  8. What are "fundamental invariants"?

More tomorrow... --Ling.Nut 22:46, 19 November 2006 (UTC)

Lots of questions:

  1. Good point, a brief explantion homotopy should be given, or linked to in article.
  2. Manifold. It's now linked, which I think is sufficient.
  3. (Didn't answer this one first time). Umm.... probably. The trouble is that not enough is known about the subject matter to form meaningful conclusion in many cases. The field is still at the stage of prodding around and seeing what happens. Give it 10-20 years and I'll be able to give a better answer :-). Tompw 00:15, 22 November 2006 (UTC)
  4. Now added: "More generally, the n-sphere is an n-dimensional object."
  5. Ummm.... which bit don't you understand? Is it the "continuously deformed" bit? Sorry, I'm not quite with you.
  6. Again, which bit don't you understand?
  7. No idea on this one. Anyone else know?
  8. Re-wrote as "From a geometric point of view homotopy groups are invariants of the n-sphere under any homeomorphism."

Hope this helps :-) Tompw 23:21, 19 November 2006 (UTC)

I've rewritten for Ling Nut's 5 and 6. "Metastable ranges" is part of Salix's history; I hope he knows what it means, but I think this whole section may be too detailed and technical here. On the other hand, mathematical articles are expected to vary in technical requirements as they go on; if GA is going to demand this change, so much the worse for them.
For the mathematicians: the equivalence of all one-point maps depends of course on the connectedness of spheres. I would like to slide this under the rug too, but feel I have to go add "path-connected". This has the same stylistic problem as linking manifold: too much accuracy can be confusing.
For Ling Nut; this article does, and must, assume that puzzled readers click on the links. Septentrionalis 00:16, 20 November 2006 (UTC)
We define a mapping as a rule that assigns each point in the first space to some point in the second.
Is assigns clear to the non-mathematician? Would takes be clearer? Septentrionalis 16:16, 20 November 2006 (UTC)

History

I am afraid that I don't quite like the "history" section. Such sections are often useful to put subject in a wider context, but I feel it adds little in this instance. Furthermore, a "history" section this early in the article will necessarily include many technical terms which cannot be explained. If we do decide to keep this section, it should probably be moved further down, but I'd rather just distribute its contents over the rest of the article.
I agree with Septentrionalis that we do not want to talk about connectedness here. -- Jitse Niesen (talk) 08:13, 20 November 2006 (UTC)
Well it is the first iteration, feel free to be WP:BOLD. Yes it could be spread about the article or placed lower, I wasn't sure of the correct place when I added it.
There is some questions as to who first devised homotopy groups. The St-Andrews site credits both Jordan and Poncari, but the May article credits Hurewicz. Poincaré may just have developed the fundamental group and not the higher dimensional extensions. I'm begining to doubt the reliability of the St Andrews site on this question, and its worth investigating funther as Jordan claim also appears on several other pages, which are all wowfully lacking in any history of the topic. Alas I don't have good access to a library so its hard for me to investigate this further. --Salix alba (talk) 13:40, 20 November 2006 (UTC)
A better source would be Dieudonne's history of topology. --C S (Talk) 21:59, 28 December 2006 (UTC)
I moved the history lower. I think having it in one section is useful if we can get it right.--agr 14:35, 20 November 2006 (UTC)

Page numbers

The recent reformatting of the references deleted the page number citations. I don't thing that is a good thing to do.--agr 13:39, 24 June 2007 (UTC)

I agree: I think you caught me in the middle of a group of edits. Are all the page numbers you want there now? If not, please restore them. Geometry guy 14:17, 24 June 2007 (UTC)
All better. Thanks.--agr 14:49, 24 June 2007 (UTC)

References tidied, rating signed

I've tidied and added some citations/references to meet the scientific citation guidelines and hence ensure that the article still meets the good article criteria. It could still use a reference for the paragraph on unstable homotopy groups. Anyone? Geometry guy 14:14, 24 June 2007 (UTC)

Missing unstable groups

Does anybody know why many unstable groups are left out of the table? For instance, all the groups pi_{n+1}(S^n) before the stable range are missing. They are certainly all there in Toda's table. Katzmik 15:25, 20 August 2007 (UTC)

Yes. They are stable. The column heading of the stable entries says this, but you do have to decode the heading to figure this out. It's done like this (I assume) to make the stable text justify into a column. Can anyone think of a clearer way of doing this? Adam1729 20:46, 25 August 2007 (UTC)