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Archive 1 | Archive 2 |
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)
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)
S3 is a four dimensional sphere not three as was stated. Made a minor correction on 08/21/2008. —Preceding unsigned comment added by 147.153.247.34 ( talk) 02:36, 22 August 2008 (UTC)
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)
I've been thinking about the table, and I made a version that I like on a user subpage. It has the following features:
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)
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 πn+k rather then by πn: 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 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)
I deleted two sentences:
"See also Appendix 3 of Ravenel (2003). There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections.)"
Here's why:
One of the entries in Ravenel's table in his book was \pi_17 S^7 = \Z/24 + Z/4. On Oct. 25, 2009, I emailed Ravenel and said that this contradicted the EHP sequence and also disagreed with Toda's table. He emailed back and said (1) that I was right, (2) he would correct the online version of his book, and (3) the intention when he wrote the book was to copy Toda's table; he did not attempt corrections.
So at the very least the sentence "There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections." needs to be deleted. I have not checked whether the online version of Ravenel's book matches Toda's table now. Jfdavis ( talk) 19:57, 7 November 2009 (UTC)
Under the "General theory" section, can phrasing like
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 πn(Sm) 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)
The article has lovely pictures, and if you don't read it carefully, it may appear to be well written.
It is not.
Consider as a typical example this sentence [bracketed phrases are mine]:
"The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S^3 around the usual sphere S^2 in a non-trivial fashion, and so is not equivalent to a one-point mapping."
Let's see now, S^3 is wrapped around S^2 "in a non-trivial fashion" [whatever that means], and "so" is "not equivalent [which in English is called not homotopic]" to a "one-point mapping [which in English is called a constant mapping]".
Well, that certainly explains things. Not. Daqu ( talk) 05:54, 26 November 2009 (UTC)
For:
"Most of the groups are finite. The only unstable groups which are not are either on the main diagonal or immediately above the jagged line (highlighted in yellow)."
The second sentence isn't a complete sentence as far I as I can tell. It looks to me like <blah1> which are not <blah2> but doesn't actually say anything about them.
Additionally I don't see anything yellow. If there is yellow it is too close to white for me to tell on my computer. —Preceding unsigned comment added by 69.33.111.74 ( talk) 19:41, 15 February 2010 (UTC) #
Under "applications", note that the Kervaire invariant question has been resolved except for dimension 126. —Preceding unsigned comment added by 156.56.144.19 ( talk) 00:12, 20 May 2010 (UTC)
The article goes through simple examples giving trivial and Z groups, but can someone lead us through a simple example giving a finite group? -- JWB ( talk) 20:17, 4 January 2013 (UTC)
Criteria 2b asks that the article "provides in-line citations from reliable sources for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines" (bolding added by me). There are no inline citations in this article and some of the statements have been challenged for over five years now. I am hoping someone watching this can provide inline citations (not just to the statements marked but to all the statements that need them). Otherwise I am afraid that this article will have to undergo a reassessment. AIRcorn (talk) 23:07, 15 March 2013 (UTC)
While the keyring model will always hold a special place in my heart and in the history of this article, I think it is not as illustrative as other newer pictures. On the Hopf fibration page, it has been replaced with one that I made. There are other good pictures of the Hopf fibration online, and this article itself now has other pictures which more effectively communicate what it means to "wrap" one sphere around another. Let's not keep the keyring picture merely because we are used to it and like its historical significance. Is there an expository reason not to replace it?
For reference, here are the two pictures I'm referring to
— Preceding unsigned comment added by Nilesj ( talk • contribs) 22:05, 25 February 2019 (UTC)
It seems that oeis:A048648 gives the orders of the stable homotopy groups that differ from those given in the article for k = 23, 29..33. It seems that the OEIS numbers originate from the program referenced here, which ultimately references Ravenel's book, just like this article. So the question arises: which data is correct? -- colt_browning ( talk) 14:27, 4 March 2020 (UTC)
Never mind, the OEIS is now corrected. -- colt_browning ( talk) 07:59, 25 August 2020 (UTC)
The way the table of πn(Sk) changes to three different colors all of a sudden in the stable range is a) delightful and b) reminiscent of how the movie "The Wizard of Oz" morphs from black & white to color, once the protagonist has entered the Land of Oz.
However: Has anyone proven that three colors suffice for coloring the stable range of πnSk ??? — Preceding unsigned comment added by 2601:200:C000:1A0:DD44:41FE:AA4:A892 ( talk) 23:19, 28 November 2022 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
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)
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)
S3 is a four dimensional sphere not three as was stated. Made a minor correction on 08/21/2008. —Preceding unsigned comment added by 147.153.247.34 ( talk) 02:36, 22 August 2008 (UTC)
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)
I've been thinking about the table, and I made a version that I like on a user subpage. It has the following features:
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)
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 πn+k rather then by πn: 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 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)
I deleted two sentences:
"See also Appendix 3 of Ravenel (2003). There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections.)"
Here's why:
One of the entries in Ravenel's table in his book was \pi_17 S^7 = \Z/24 + Z/4. On Oct. 25, 2009, I emailed Ravenel and said that this contradicted the EHP sequence and also disagreed with Toda's table. He emailed back and said (1) that I was right, (2) he would correct the online version of his book, and (3) the intention when he wrote the book was to copy Toda's table; he did not attempt corrections.
So at the very least the sentence "There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections." needs to be deleted. I have not checked whether the online version of Ravenel's book matches Toda's table now. Jfdavis ( talk) 19:57, 7 November 2009 (UTC)
Under the "General theory" section, can phrasing like
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 πn(Sm) 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)
The article has lovely pictures, and if you don't read it carefully, it may appear to be well written.
It is not.
Consider as a typical example this sentence [bracketed phrases are mine]:
"The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S^3 around the usual sphere S^2 in a non-trivial fashion, and so is not equivalent to a one-point mapping."
Let's see now, S^3 is wrapped around S^2 "in a non-trivial fashion" [whatever that means], and "so" is "not equivalent [which in English is called not homotopic]" to a "one-point mapping [which in English is called a constant mapping]".
Well, that certainly explains things. Not. Daqu ( talk) 05:54, 26 November 2009 (UTC)
For:
"Most of the groups are finite. The only unstable groups which are not are either on the main diagonal or immediately above the jagged line (highlighted in yellow)."
The second sentence isn't a complete sentence as far I as I can tell. It looks to me like <blah1> which are not <blah2> but doesn't actually say anything about them.
Additionally I don't see anything yellow. If there is yellow it is too close to white for me to tell on my computer. —Preceding unsigned comment added by 69.33.111.74 ( talk) 19:41, 15 February 2010 (UTC) #
Under "applications", note that the Kervaire invariant question has been resolved except for dimension 126. —Preceding unsigned comment added by 156.56.144.19 ( talk) 00:12, 20 May 2010 (UTC)
The article goes through simple examples giving trivial and Z groups, but can someone lead us through a simple example giving a finite group? -- JWB ( talk) 20:17, 4 January 2013 (UTC)
Criteria 2b asks that the article "provides in-line citations from reliable sources for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines" (bolding added by me). There are no inline citations in this article and some of the statements have been challenged for over five years now. I am hoping someone watching this can provide inline citations (not just to the statements marked but to all the statements that need them). Otherwise I am afraid that this article will have to undergo a reassessment. AIRcorn (talk) 23:07, 15 March 2013 (UTC)
While the keyring model will always hold a special place in my heart and in the history of this article, I think it is not as illustrative as other newer pictures. On the Hopf fibration page, it has been replaced with one that I made. There are other good pictures of the Hopf fibration online, and this article itself now has other pictures which more effectively communicate what it means to "wrap" one sphere around another. Let's not keep the keyring picture merely because we are used to it and like its historical significance. Is there an expository reason not to replace it?
For reference, here are the two pictures I'm referring to
— Preceding unsigned comment added by Nilesj ( talk • contribs) 22:05, 25 February 2019 (UTC)
It seems that oeis:A048648 gives the orders of the stable homotopy groups that differ from those given in the article for k = 23, 29..33. It seems that the OEIS numbers originate from the program referenced here, which ultimately references Ravenel's book, just like this article. So the question arises: which data is correct? -- colt_browning ( talk) 14:27, 4 March 2020 (UTC)
Never mind, the OEIS is now corrected. -- colt_browning ( talk) 07:59, 25 August 2020 (UTC)
The way the table of πn(Sk) changes to three different colors all of a sudden in the stable range is a) delightful and b) reminiscent of how the movie "The Wizard of Oz" morphs from black & white to color, once the protagonist has entered the Land of Oz.
However: Has anyone proven that three colors suffice for coloring the stable range of πnSk ??? — Preceding unsigned comment added by 2601:200:C000:1A0:DD44:41FE:AA4:A892 ( talk) 23:19, 28 November 2022 (UTC)