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Thurston’s geometrization conjecture
Thurston’s geometrization conjecture, also known simply as the geometrization conjecture, states that compact 3manifolds can be decomposed into pieces with geometric structures. The geometrization conjecture is an analogue for 3manifolds of the uniformization theorem for surfaces. It was proposed by William Thurston in the late 1970s, and implies several other conjectures, such as the PoincarÃ© conjecture and Thurston’s elliptization conjecture.
Grigori Perelman sketched a proof of the geometrization conjecture in 2003 using Ricci flow with surgery, which (as of 2006) appears to be essentially correct.
1 The conjecture
Every closed 3manifold has a prime decomposition: this means it is the connected sum of an essentially unique collection of prime threemanifolds. This reduces much of the study of 3manifolds to the case of prime 3manifolds: those that cannot be written as a nontrivial connected sum.
Every prime closed 3manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions, described in the next section. Cutting a prime 3manifold along tori into pieces that are Seifert manifolds or atoroidal is called the JSJ decomposition: there is a minimal way of doing this, which is essentially unique.
There are similar statements for compact manifolds with boundary without $S^{2}$ boundary components.
2 The eight Thurston geometries
A model geometry is a simply connected smooth manifold $M$ acted on by a Lie group $G$, such that $G$ is maximal among groups acting smoothly and transitively on $M$ with compact stabilizers, and there is at least one compact manifold modeled on $M$.
A geometric structure on a manifold is an isomorphism of the manifold with $M\Gamma$ for some model geometry $M$ where $\Gamma$ is a discrete subgroup of $G$ acting freely on $M$.
$S^{2}$ is the round 2sphere and $\mathbb{H}^{2}$ is the hyperbolic plane.
Seven of the eight geometries (all except hyperbolic) are now clearly understood and known to correspond to Seifert manifolds and torus bundles. Using information about Seifert manifolds, we can restate the conjecture more tersely as:
Every prime, compact 3manifold falls into exactly one of the following categories:

It has a spherical geometry.

It has a hyperbolic geometry,

The fundamental group contains a free abelian group on two generators (the fundamental group of a torus).

It has $S^{2}\times\mathbb{R}$ geometry.
3 History
If Thurston’s conjecture is correct, then so is the PoincarÃ© conjecture (via Thurston’s elliptization conjecture). The Fields Medal was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds.
The case of 3manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes ”almost round” just before the collapse. He later developed a program to prove the Geometrization Conjecture by Ricci flow.
In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by extending Hamilton’s Ricci flow program to include surgery whenever the Ricci flow produces singularities. As of 2006 the consensus among those who have checked his work is that it is essentially correct, and the details have now been filled in. According to the Clay Mathematics Institute, he may be eligible for the Institute’s Millennium Prize Problems, although he has not submitted his work to a peerreviewed journal.
This entry was adapted from the Wikipedia article Geometrization conjecture as of November 10, 2006.
References
 1 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002
 2 G. Perelman, Ricci flow with surgery on threemanifolds, 2003
 3 B. Kleiner and J. Lott, Notes on Perelman’s Papers, 2006
 4 W. Thurston, Threedimensional geometry and topology, Vol. 1. Edited by S. Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.
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Comments
Copying from Wikipedia
Haven't we discussed in some previous posts about this issue? In order to keep our entries uniquely ours and to offer new perspectives if possible, and, to avoid potential plagirism, I think we need to revisit this issue. I still think it is best that entry authors contribute entries based on his/her own words and knowledge. And if I recall, someone suggested that we look at other internet sources, such as wikipedia and math world, only after we have written and submitted the entries.
I am not trying to be critical, I just want PlanetMath to thrive and offer unique and alternative sources to online mathematics for readers at large!
Re: Copying from Wikipedia
Probably. But notice that you're bringing it up because PrimeFan clearly labelled this as being based on a Wikipedia entry, because it is one that he has never worked on. PrimeFan has taken some articles of which he has been the starting and principal author on Wikipedia and copied them over here practically verbatim without comment (I noticed this because I also contribute to Wikipedia). But in this case, he's given us the ability to make this article take on a more PMflavor to it and we should avail ourselves to that ability.
Re: Copying from Wikipedia
Well, unless he's principal contributor for Thurston's geometrization conjection on Wikipedia, I have no problem (and in fact I apologize). But otherwise, I still don't think it is a proper thing to do. How would you like it if someone copies your stuff verbatim and publishes it somewhere else as his/her own?
I see your point of enhancing the Wikipedia entry by putting it here. That is wonderful. But we are already doing that, whether consciously or not. It is likely that a lot of entries on Planet Math can be found in Wikipedia or Math World. It is also likely that a lot of these entries on Planet Math are unique in that when you compare like entries on the difference online resouces, you will find a fair amount of differences in content and presentation.
Nevertheless, my point about not copying from other sources should also be stressed (and I am not pointing finger at any individual now). Planet Math should not (and will not) be a duplicate of other online math resources.
Re: Copying from Wikipedia
This is an interesting string of posts, as this is a topic that I have not thought much about.
If I am not too badly mistaken, PrimeFan added this entry to fill the requests made for such an entry. While I am glad that Thurston's geometrization conjecture has a reference here on PM, I also respect CWoo's and others viewpoints on this. I would hate to see PM get into trouble for copyright infringement or other matters that may come from indiscriminately drawing from sources. Also, I agree that PM should not try to become a carbon copy of some other math source on the web. I know from experience that PM and other similar sites are quite different. Some have resources about certain mathematical topics that others do not, and even when they do have something in common, they tend to have a different flavor to them. I feel that this diversity is important, mainly because people who use these resources to research a mathematical topic can see different expositions and get different things out of them.
One thing that I am very glad that PrimeFan did was make the entry in question world editable. Hopefully, those who are knowledgeable about this topic will edit the entry and give it its own PMesque flavor.
Finally, something that I would like to point out is that, in the collaboration "New User's Guide" it says:
"As a rule of thumb, if you cannot provide at least a sketch of a given topic without referring to a source, you are probably not yet qualified to write an entry about that topic. Not only is this policy prudent from the legal standpoint, it also makes sense from the point of view of mathematical content....A document written from your own understanding will be much more useful than one that purports to present facts that you yourself do not understand."
I feel that these are very useful guidelines. If you have not read this particular collaboration, I recommend that you do so, even if you are not new to PM. Actually, in my opinion, all of the collaborations make for a good read.
Re: Copying from Wikipedia
> I would hate to see PM get into trouble for copyright infringement
> or other matters that may come from indiscriminately drawing from
> sources.
In the case of Wilkipedia, this is not a problem because of the license.
Re: Copying from Wikipedia
I am in favor of copying works where the copyright situation is OK.
While initially not much might be changed, it is almost inevitable that copied content will acquire its own "PlanetMathflavor" after not too long.
As an example, early on I copied about 100 entries from the Data Analysis Briefbook, with permission to put them under the GNU FDL. Now most of them bear only a vague resembleance to their initial form. Considerable changes were made for clarity, rigour, removal of mistakes, and simple augmentations.
Also, consider that Wikipedia has copied a considerable amount from PlanetMath. It makes sense for us to "backport" what we are missing. In fact there is some work ongoing to methodically select the Wikipedia math entries, so that this can be done.
As Ray points out, the copyright situation for the latter could hardly be more optimal.
apk
Papers on the PoincarÃƒÂ© Conjecture
http://www.ims.cuhk.edu.hk/~ajm/vol10/10_2.pdf
http://arxiv.org/PS_cache/math/pdf/0605/0605667.pdf
Re: Copying from Wikipedia
Here's something that might be interesting. In the
next version of the FDL  or at least in its draft 
there is a 'Wiki relicensing'clause (Â§8b). This can
be found at
http://gplv3.fsf.org/fdldraft20060922.html
See also:
http://wikiangela.com/wiki/GNU_Wiki_License
The idea seems to be that if the FDL'd content
has been produced on a 'system for massive public
collaboration', then users may relicense it under
the GNU Wiki License. However, no draft on this
currently seems to be available.