Talk:Homotopy groups of spheres/Archive 1

message: 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)
thread_title: Table?

message: 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)
replies: text: I was hoping there might be something more in the updated (online) version of Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres , 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 . Good luck! --KSmrqT 11:01, 17 December 2005 (UTC)
thread_title:  Copied from Reference Desk 

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

text: 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)
replies: text:  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)
replies: text: 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)
replies: text:  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)
replies: text: 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)

text: 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)

text: I've had a bash at creating an image for \pi_1(S^1)  --Salix alba (talk) 08:57, 18 May 2006 (UTC)
thread_title:  Failed GA 

message: 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:
# no explanation of what homotopy theory is.
# 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?
# what's a homotopy group?
# what are n-spheres and i-spheres?
# what's a homotopy class?
# From the algebraic aspect, there is ample evidence that they involve substantial complexity of structure - don't understand what is meant.
# 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!
# 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)
replies: text: 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)
replies: text: I did a major rewrite of the intro in an attempt to address these problems. --agr 16:36, 19 May 2006 (UTC)
thread_title: GA review

message: 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)
thread_title:  Big improvement 

message: 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)
thread_title: This article makes my browser crash

message: 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)
replies: text: 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)
replies: text: 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)
replies: text:  Maybe I'm being thick, but does "inline citation" equal "footnotes"? Tompw 19:06, 26 September 2006 (UTC)
replies: text: Footnotes or Harvard referencing, either of which would be pointless and ugly here.  Septentrionalis 21:12, 19 November 2006 (UTC)

text: 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)
replies: text: 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)
thread_title:  GA Re-Review and In-line citations 

message: 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)
replies: text: No, no, no. Not recognisable. Charles Matthews 22:04, 18 October 2006 (UTC)
replies: text: 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)
thread_title:  Proposed move / re-name 

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

message: 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)
thread_title: Dimension

message: 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)
replies: text: 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)
replies: text: All better. Thanks.--agr 14:49, 24 June 2007 (UTC)
thread_title:  Page numbers 

message: 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)
thread_title:  References tidied, rating signed 

message: 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)
replies: text: 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)
thread_title:  Missing unstable groups 

message: Substituted at 06:33, 7 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
thread_title:  Consistent notation 

message: 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, OhanaUnitedTalk page 21:45, 8 September 2007 (UTC)
replies: text: 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)
replies: text:  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)
replies: text: 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)
thread_title:  GA on hold 

message: 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=\Delta^1 / boundary, the other one is the multiplicative group Gm.
This can be found in a paper of Morel and Voevodsky . 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)
replies: text: 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)
replies: text: 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)
thread_title:  Homotopy theory of schemes 

message: So I don't forget, and for others to contemplate, topology can use perhaps a half-dozen different definitions of n-sphere.
# Implicit surface, x02+⋯+xn2 = 1
# Parametric surface, such as (cos φ sin θ,sin φ sin θ,cos θ)
# Suspension of equator, Sn = ΣSn−1
# Interval mod boundary, In/∂In
# Disk mod boundary sphere, Dn/Sn−1
# Spin of semi-equator, (x0,…,xn−2,xn−1 cos θ,xn−1 sin θ), xn−1 ≥ 0
Each of these has its uses and benefits for our topic. For example, the spin version is helpful for explaining πn(Sn) = Z. --KSmrqT 06:05, 4 November 2007 (UTC)
replies: text: 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)
replies: text:  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 πn(Sn) = Z. --KSmrqT 20:46, 4 November 2007 (UTC)
thread_title:  Spheres 

message: 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)
replies: text: 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)
replies: text:  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)
thread_title: A-Class Review?