Talk:Geometrization conjecture

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[edit] New layout

I like the fact that the geometries are all given their own section, but I want to ask a few questions.

1. How do we avoid bloat? Instead of allowing long lists of examples, we should give each of the geometries their own page, and put most of the information there.
2. Do we have to use bold face? It looks bad in my browser. Let's return to blackboard bold, if we can fix the alignment problems. Sam nead 20:07, 10 November 2006 (UTC)
1 Some geometries already have their own page, and others will probably get one if they grow too much.
2 Unfortunately the alignment problem makes blackboard bold look even worse than boldface. At the moment, the standard seems to be to use unicode inline and latex for displays. The least worst fix might be to wait until blackboard bold unicode becomes standard on browsers, and then switch. R.e.b. 20:48, 10 November 2006 (UTC)

[edit] Two-level??

This may be partly because the standard two-level decomposition article isn't written yet, but I find the description of this phenomenon given here to be ambiguous.

Does it mean that each compact 3-manifold can be decomposed into prime 3-manifolds, each of which can be further decomposed via the Jaco-Shalen-Johannson torus decomposition?

Yes, that's right. See irreducible for a description of the prime decomposition. The JSJ decomposition is applied to each prime manifold except for certain exceptions (since the JSJ decomposition requires irreducibility and every prime manifold, except for these exceptions are irreducible).
Yeah, someone should write something on the JSJ decomposition, and the prime decomposition also, although I've somewhat abated the problem by writing some stuff under irreducible.

Does it mean that each compact 3-manifold can be decomposed into prime 3-manifolds OR via the the Jaco-Shalen-Johannson torus decomposition?

Does it mean that each compact 3-manifold can be decompsed in two quite different ways, into prime 3-manifolds AND via the Jaco-Shalen-Johannson torus decomposition?

It means, first decompose along certain essential spheres to get prime pieces and cap off sphere boundaries with three-balls. THEN decompose along certain essential tori to get the JSJ decomposition. By the way -- I've never heard anyone use the term "two-level" decomposition, so I am getting rid of it. Sam nead 23:50, 5 September 2006 (UTC)

[edit] Sol Geometry

The description of sol geometry given here differs from that on MathWorld.

Yeah, it doesn't make any sense. Anyway, I removed the error, but at some point I will try and write an article. I've added some references for now, in case anybody wants to look at them or write some stuff. --C S (Talk) 08:33, 11 June 2006 (UTC)

[edit] Thurston conjecture?

I finally got sick and tired of seeing the page called "Thurston conjecture" when not a single 3-manifold topologist calls it that. Most of the time, it's just "geometrization conjecture" or sometimes "Thurston's geometrization conjecture"; usually "Thurston" is dropped because he's so famous and his name ubiquitous. I thought I better move it now before there's a zillion pages linked to "Thurston conjecture". --C S 11:57, Dec 3, 2004 (UTC)

[edit] Various issues

It's probably a good idea to not blow the Perelman result out of proportion. As far as I know, his work looks good for the case of manifolds with finite fundamental groups but there's still significant hurdles to deal with in the case of manifolds with non-trivial JSJ-decompositions, large seifert fibred manifolds in the JSJ-decomposition, etc.

Actually if the JSJ decomposition is non-trivial, then the manifold is Haken, so Thurston's result covers this case. The difficult case as I understand it is for irreducible atoroidal non-Haken 3-manifolds with infinite fundamental group. Quasicharacter 18:31, 22 July 2005 (UTC)
Indeed there's no concensus yet on Perelman's work outside of the Poincaré conjecture. The difficult case is as Quasicharacter stated. I made a small edit to the article to reflect this. --Dylan Thurston 16:49, 5 September 2006 (UTC)
The articles in Wikipedia around the proof of the Geometrization conjecture (that is, those on Grisha Perelman and the Poincaré Conjecture) make conflicting comments on the status of the proof. The introduction to Perelman's biographical article says quite clearly that he has proved Geometrization, whereas other parts of those articles seem more cautious. I would like to know whether there have been any advances on verification since the above notes by Dylan Thurston and Quasicharacter were posted. As far I know, both the Cao/Zhu and the Kleiner/Lott teams have pronounced themselves firmly in favour of the correctness of the proof, after having canvassed them. The Morgan/Tian team focused on the "easy" part, that is, the Poncaré conjecture, whose proof by Perelman has received the ultimate endorsement in the form of a book. But they have mentioned in the introduction to that book that they are going to deal with Geometrization in a forthcoming article. My question is: can Wikipedians follow the lead of the world's most renowned topologists and state that Geometrization has been proved? Quasicharacter says that a case remains to be settled. That does not seem to be the opinion of the two teams. I know mathematical proofs may take years to be fully recognised, but Perelman's proof has survived five years of intense scrutiny. Could Quasicharacter provide any disclaim by a mainstream topologist that Geometrization has been settled? Where does he get the information about the remaining case? (I know he is interested in Mathematics, but surely he has other sources than his own assessment of the status of the proof.) Has anyone publicly declared to differ from the verification teams on the assessment of Perelman's coverage of Geometrization? I mean, Thurston posted his comments almost a year ago and nothing has been heard from him or Quasicharacter since. I may also remark that the Cao/Zhu paper has been published in a peer-reviewed journal (although I concede that, as the New Yorker has revealed, circumstances around the publication seem a little fishy; but they pubished an ameded version on December 2006 where they sustain their claim that Perelman solved Geometrization) and that ought to suffice for Wikipedia... unless a serious professional can be shown to question the validity of the proof, in which case it would be reasonable to state clearly what the nature of the rebuttal is. If these concerns of mine are not shown to be unreasonable in a few weeks, I shall take it upon myself to modify all conflicting parts of the related Wikipedia articles so that they state Geometrization has been proved. —The preceding unsigned comment was added by 200.167.222.212 (talk) 02:02, August 23, 2007 (UTC)

[edit] Non-uniqueness of geometric structure, compact manifolds with boundary

Currently there's one problem with the statement of geometrization. It talks about doing the sphere+torus decomposition to a COMPACT manifold, then the existance of a unique geometric structure. Technically this is not geometrization unless the original manifold was CLOSED. If the original manifold is compact with boundary, say a trefoil knot complement then that manifold admits no geometric structure in the standard sense since it has boundary and yet its sphere + JSJ decomposition is trivial. If you remove the boundary torus the resulting manifold admits TWO geometric structures, a hyperbolic x real line geometry and a PSL structure (you get this by viewing is a a seifert bundle over a disc with 3 singular fibres. It would be a sad thing if we "fixed" the statment by switching "compact" to "closed". How about we give a detailed account of how the geometry can be non-unique for the interiors of compact manifolds with boundary with trivial sphere+JSJ decompositions?

Oops, I guess I introduced this blunder about the (supposed) uniqueness. Anyway, the statement of geometrization is always supposed to be understood to be on the interiors of the pieces. Another thing that keeps being omitted is that the spherical boundary components should be capped off. I'll fix this part.
I think the best thing to do is keep things simple and only mention in a technical note (not in the main statement) the possible non-uniqueness (for manifolds with boundary). I'm not familiar with the exact statement of the non-uniqueness so you'll have to do that part. --C S 00:09, Jan 22, 2005 (UTC)

I think Walter Neumann worked this stuff out years ago, but I'm not sure which of his papers its in. If nobody else writes it up, I'll get to it eventually.

[edit] geometrization theorem vs. hyperbolization theorem

I wanted to make a few clarifying remarks about why the article mentions both these terms. They are both common, although I've never considered which would be more common. My initial revert was due to the fact that in my experience, when the hyp. theorem is explained, usually one starts "Let M be an atoroidal, blah blah" and ends with "M is hyperbolic", whereas the geometrization theorem's statement utilizes all 8 geometries. Of course, it's simple to go from one to the other, which is presumably why the names appear to be basically synonymous (at least from my cursory check).

Issues of common usage aside (this can be explained however others feel is ok in the article, names can be reordered if needed, etc.), both names should be mentioned as geometrization theorem redirects here. A common source of confusion after Perelman's work on GC has been some non-specialists and laymen calling GC the geometrization theorem, which as far as I know, is not at all that common to do so. "Geometrization theorem" has a long history behind it, starting with Thurston himself, so the name, unfortunately, is taken. I suppose people were never really concerned with this as it seemed highly unlikely GC would be proven in our lifetimes. One day "geometrization theorem" may have its own article (possibly under the name "hyperbolization theorem"), but at the moment, it doesn't. So the redirect and mention in this article is the workaround.

Also, I changed "hyperbolisation" to "hyperbolization" for style consistencies. I hope I haven't offended those in the Commonwealth. ^_^ --C S (Talk) 21:28, 21 April 2007 (UTC)

I agree with your basic motivation but the specific "geometrization theorem" might be confusing. I'm curious where you see geometrization theorem occur in the literature. All the books on the topic tend to call it the hyperbolization theorem: Otal, Kapovich, Hubbard.. etc. On top of that, wasn't it Peter Scott that spelt out all the details regarding geometrization of Seifert fibred-spaces? This is basically the only difference between 'hyperbolisation' and 'geometrization' in the Haken case anyhow, except for Sol manifolds. I think it's clear Thurston knew these details extremely well, but did he write them up? I haven't looked at his BAMS (1982) article in a while so the details are foggy. The only other article of Thurston's that touches the topic is his notes that got turned into his book, but it doesn't actually go into any Seifert-fibred details. Rybu 09:42, 22 April 2007 (UTC)
Where? Well, you can try Google Scholar and MathSciNet; you'll immediately get a number of relevant hits. --C S (Talk) 10:08, 22 April 2007 (UTC)

[edit] Sol or Solv?

A basic search in the Front for the arXiv produce the numbers of articles:

  • Solv geometry: 622, and solv manifold: 210

vs

  • Sol geometry: 55, and sol manifold: 31

going on with the majority it should be used Solv. Thruston-self uses the term solvegeometry.--kiddo 22:26, 16 July 2007 (UTC)

[edit] ???

what is a "point stabilizer"? —Preceding unsigned comment added by 74.170.68.234 (talk) 13:05, 17 March 2008 (UTC)

It is defined in the group action section [1] Rybu (talk) 17:22, 17 March 2008 (UTC)