Talk:Poincaré conjecture/archive1

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A call to arms

It seems likely (given the strengths of the verified breakthroughs) that Perelman's proof of PC w ill be found correct, and it behooves the Wikipedia community to get working on cleaning up this page.

IMHO, there is a good chance that within the year, major figures in the mathematical community will announce their confidence in Perelman's proof of PC. When this happens, there will be major media attention, like when the news about Perelman first broke. Recall that when the media first got interested, several journalists actually referenced and quoted the Wikipedia article on PC.

I think everyone should put more effort into this page and work to make it at the level of a featured article. If we can get a lot of people looking at this page, journalists included, then this can only further Wikipedia's cause.


Recent progress in solving the problem

Grigori Perelman may have solved another problem called Thurston's conjecture.

Poincaré's conjecture is a special case of Thurston's conjecture, so a proof of Thurston's conjecture immediately proves the Poincaré Conjecture.

Consequently Perelman may be in line for the Clay Millennium Problem million dollar prize.

Re the proof, Tosha would probably know if there were a problem. Charles Matthews 11:18, 18 May 2004 (UTC)

Also, There is as-yet no problem with the proof, and it is widely accepted; maybe the page should change to reflect this? Gartogg, Sep. 2, 04

No problem with the proof so far does not imply "widely accepted"! No respected mathematician has come forth saying s/he finds it correct. Currently, Kleiner, Lott, et al, are not confident of all the details and assertions of the second Perelman paper.


I was wondering what the word "closed" in the conjecture means. Is it "compact and wouthout boundary"? If so, we should probably clarify, since "closed" is used in other meanings as well. AxelBoldt

Yes, it means "compact and without boundary". This is explained in the manifold article, but it might well be best to clarify it anyway. --Zundark, 2002 Feb 19

Could someone merge the information from Poincaré Conjecture (note capital C) into this article and then make the other one redirect here? -- Timwi 00:14 29 Jun 2003 (UTC)

I don't think there's any useful extra information there, but I've moved the contents here (below), just in case anyone disagrees. --Zundark 13:27 2 Jul 2003 (UTC)

The Poincaré Conjecture is a topological problem. It is one of the most well known problems in mathematics, after Fermat's Last Theorem. The problem was formulated by Henri Poincaré in 1904.

Here is the problem, taken from the official problem description (an Adobe Acrobat PDF)

  • Consider a compact 3-dimensional manifold V without a boundary.
  • Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

The problem is the subject of a one million dollar prize at the Clay Mathematics Institute.

Popular descriptions of this conjecture are surprisingly dependent upon international culinary definitions. Most descriptions of the conjecture involve comparing the result of constricting a spherical object with the effects of performing a similar constriction exercise upon a doughnut-shaped object.

Unfortunately, there are two types of doughnut, one of which is a torus or ring (a 'ring doughnut') and the other variety is approximately spherical (called a 'jam doughnut' in the UK because it is typically filled with what the British call jam, and called a jelly doughnut in the USA because they call the same filling jelly) and this doughnut dichotomy has the potential to render the standard explanation of the Poincaré conjecture unintelligibly confusing.

The (admittedly mischevious, not to mention potentially frivolous) suspicion is that the two different types of doughnut divide Europe from the United States and this may be in part responsible for the longevity of the unsolved mystery of the problem.

Edit of User:C S

I do not like this edit. The article should be reasonably short, such things a sphere, homomorphism and so on one can learn elsewhere. Tosha 05:48, 6 Sep 2004 (UTC)

Length is a problem, as always; I agree that there shouldn't be too much explanation.
The current version is unsatisfactory since it does not explain the history or motivation. The content basically consists of comments about the higher dimensional conjectures, a short comment about the statement of the conjecture, and several paragraphs about Perelman and his work. This does not do justice to such a venerable conjecture as the Poincare Conjecture.
I wonder how carefully you read the edit. I do not attempt to explain "homeomorphism", at least not explicitly. There is other material besides that on spheres. I think my edit gives the entry room to grow, unlike the current revision, and so your (basically) complete revert is a step backwards. So what I will do is keep some of the stuff about motivation, e.g. homology spheres, and some history, but cut out the the long explanation. I think we should move to having some more explanation though.

--C S 07:40, Sep 6, 2004 (UTC)

Keep in mind it is not natural to treat this conjecture identically as one would any other. PC is special, being tractable, as a statement, to non-mathematicians, and yet profoundly difficult to mathematicians. It's an intriguing combination. In my arrogant opinion, it is incumbant on the wiki (and mathematics) community to bolster that intrigue, i.e., broad accessibility is good here. MotherFunctor

Poincare's invention of the fundamental group

Regarding:

He then applied another tool, called the fundamental group. He wondered if a 3-manifold with trivial fundamental group had to be a 3-sphere.

which was changed from the statement, "He then came up with..."

The point I was trying to make was that his invention of the fundamental group was motivated by desire to get a finer invariant of the 3-sphere than the homology groups (in particular the first homology group). The revised statement gives the impression he just had another tool handy, instead of emphasizing that in this case, as usual, "necessity was the mother of invention".

There's a lot of mathematics invented by Poincare just to form the conjecture, and then of course, even more was invented to try and prove it. We should try to emphasize this to give the conjecture its due. --C S 15:19, Sep 8, 2004 (UTC)

It seems he introduced the fundamental group in 1894, though. Charles Matthews 16:13, 8 Sep 2004 (UTC)
Milnor gives the approximate date as 1900-1904 in the external links articles by him. But I just noticed that in the published versioin of "Towards the Poincare Conjecture..." he changes it to 1895! I'm guessing the editor caught the mistake. Thanks.

--C S 00:20, Sep 9, 2004 (UTC)


I do not know history of the question at all, is it true that fundumental group was envented only after homology groups? Tosha

Let me clarify my response to Charles Matthews and answer your question. Milnor's article gives the date of publication for the concept of fundamental group as 1895, and I corroborated this by checking the date of the document in question, which I found, for example, by checking the University of St. Andrews biographies website.
However, I decided to dig through some history books, mainly Klein's 3-volume history. The papers he wrote after 1895, which includes the one (1904) in which he asked if fundamental group distinguishes the 3-sphere, were supplements to the 1895 paper. In the 1895 paper he not only defines the fundamental group, but talks about the Betti numbers and proves, in fact, Poincare Duality. So at this point, it looks like the fundamental group was invented more or less during the time Poincare was thinking about homology.
One thing to note though, is that Poincare thought of homology as Betti numbers and torsion coefficients. I think the idea of thinking of homology groups came along later.
In terms of what to fix in the article, we should mention that Poincare's work on homology was based on earlier work by Enrico Betti. Also, it should be mentioned that Poincare was able to distinguish the Poincare sphere from the 3-sphere using the fundamental group (1904). Earlier, in 1900 he had claimed that homology (i.e. Betti numbers and torsion coefficients) would distinguish the 3-sphere. So he was led naturally to add the simple-connectivity condition. I'll straighten this out in the article.--C S 05:47, Sep 9, 2004 (UTC)

I think the history needs to be added with a little care. Please note that there are historical remarks already in the Poincaré duality article. I feel it is clear that PD was more significant an influence on others than PC, up to 1930 (say). I am also fairly clear that the definition of the fundamental group was available during the 1890s, just because so much had been done on monodromy. There are also historical remarks to make about Stokes' theorem (1899) and de Rham cohomology, and torsion and the formulation of homology as abelian groups (Emmy Noether); and other matters such as Poincaré polynomials and the Kunneth theorem. I really think this is diffucult all to hang on the Poincaré conjecture peg.

A timeline of homology theory might therefore be better.

Charles Matthews 08:52, 9 Sep 2004 (UTC)

I agree that things shouldn't get too overloaded with historical detail. But I wanted to motivate why Poincare would settle upon simple-connectivity as the fundamental condition. I think that's been done, more or less, without making the section on the statement unwieldy; do you agree? With my last change, I just wanted to straighten out the historical order. I'm not planning on adding more to that section.
If you're worried I'm going to transfer all the stuff from the talk page I've written to the article, don't worry. I was just overexplaining things. I agree that stuff like Poincare Duality and facts about homology should have their own pages, but in the case of homology spheres, I felt it should be mentioned.
Perhaps I shouldn't have added the part on Betti, but I wanted to credit Poincare in the article without being unfair.--C S 09:14, Sep 9, 2004 (UTC)

The current section on the statement seems to be more or less OK. Charles Matthews 09:29, 9 Sep 2004 (UTC)

Applications

I would remove this subsection, it is bit misleading and does not add much to the article. Tosha

Yes, it would be better on the Thurston conjecture page. Charles Matthews 18:59, 9 Sep 2004 (UTC)

I'm not sure any part of it fits on the Thurston conjecture page. But anyway, I moved the section here, in case anyone wants to salvage some part of it. --C S 01:38, Sep 10, 2004 (UTC)

The three-dimensional Poincaré Conjecture is related to the problem of classifying 3-manifolds. A "classification of 3-manifolds" is generally accepted to mean that one can generate a list of all 3-manifolds up to homeomorphism with no repetitions. Such a classification is equivalent to a recognition algorithm, which would be able to check if two 3-manifolds were homeomorphic or not.

Thurston's geometrization conjecture, in addition to containing the Poincare Conjecture as a special case, is known to imply a classification of 3-manifolds.

It is not true that any two 3-manifolds can be distinguished by their fundamental groups. In fact, J.W. Alexander showed in 1919 that there are lens spaces with isomorphic fundamental groups but are not homeomorphic. On the other hand, these kinds of examples are seen as exceptions, and it is generally believed that most 3-manifolds are determined by their fundamental groups. The Poincaré Conjecture is therefore an important test case of this heuristic.

Homotopy spheres

The statement given in terms of homotopy spheres could be a lot clearer. The product of a sphere with the real line is a counterexample to what is currently said. Charles Matthews 14:58, 13 Apr 2005 (UTC)

Fixed. The dimension should be specified and now is. --C S 07:15, Jun 24, 2005 (UTC)

what does the final step mean?

see the lastest link in the introduction page.

[1]

Is it obvious from Thurston's geometrization conjecture to Poincaré conjecture or there really still need a 300-pages proof? just want know the importance of the china scientist work.—Preceding unsigned comment added by mathematic (talk • contribs)

The paper isn't about going from geometrization to Poincare: that, as you suspect, is trivial and requires at most a short paragraph (see elliptization conjecture for short explanation). I haven't looked at the article, but I read the abstract. It sounds like they have basically a complete write up of Perelman's work, with probably details that he didn't fully explain, and probably cast and reformulated somewhat differently. The Xinhua article is very poorly written, and probably the headline and focus of the story gives a misleading impression of what the two mathematicians did. --C S (Talk) 08:59, 4 June 2006 (UTC)

Interesting link

I found a very interesting link related to the Poincare's conjecture: [2]. However I don't know if it is a hoax or not. Can someone verify their proof? --Matikkapoika 19:59, 4 June 2006 (UTC)

This is the same thing I responded to in the previous section. No hoax. Just, well, a very interesting point of view that I think would not be universally agreed to. It's an even worse article than the previous one mentioned...the quotes are kind of misleading, and I wonder if the journalist did a very selective job of quoting. Both articles do not do a good job of explaining the different contributions (and kind of contributions). One particularly bad thing is that this article's emphasis on Cao and Zhu's contributions comes at the expense of lessening Perelman's contributions; however, it is clear that is is really Perelman's work that is significant, not theirs. Of course, they have done something important here, which is to clarify Perelman's work...but other groups of people have and are doing this. This team was apparently just one of the first to complete the task. Perelman will probably win a Fields medal for what he did. --C S (Talk) 22:19, 4 June 2006 (UTC)