Poincare Conjecture proven

It is so embarassing that the first I head of this was on Paul Harvey's radio news report (although it is cool that Harvey chose to mention it). The Poincare Conjecture is essentially this: any three dimensional manifold (just think of an abstract lump of clay and you won't be far off) that is simply connected (a loop of string embedded in it can be shrunk to a point without leaving the lump) is homeomorphic (can be deformed without tearing) to the three sphere. That's it. As easy to state as Fermat's famous theorem. And as hard to prove. The proof is under review, but it's short and sweet, and so far it seems good. This will earn M.J. Dunwoody $1 million. To put this prize in perspective, however, Joseph Naccio, CEO of the telcommunications corporation Qwest earns 100 times that much every year to run it into the ground.

John Starrett

John,

I think I understood the conjecture except for what
does it mean to deform:

> to the three sphere

Does that mean to mold the manifold until it is lying
flat upon a 2-D space defined as teh surface of a 3D
sphere? Or ??

- J

--- In metatuning@y..., "jdstarrett" <jstarret@c...> wrote:
> It is so embarassing that the first I head of this was on Paul Harvey's radio news report (although it is cool that Harvey chose to mention it). The Poincare Conjecture is essentially this: any three dimensional manifold (just think of an abstract lump of clay and you won't be far off) that is simply connected (a loop of string embedded in it can be shrunk to a point without leaving the lump) is homeomorphic (can be deformed without tearing) to the three sphere. That's it. As easy to state as Fermat's famous theorem. And as hard to prove. The proof is under review, but it's short and sweet, and so far it seems good. This will earn M.J. Dunwoody $1 million. To put this prize in perspective, however, Joseph Naccio, CEO of the telcommunications corporation Qwest earns 100 times that much every year to run it into the ground.
>
> John Starrett

Starrett, you're nuts. This guy Dunwoody, although a serious mathematician, has been working on this for years and this is his seventh official draft. Maybe it's been proven, maybe not. Why don't you wait until all the reviews are finished before shooting off your mouth?

John Starrett

--- In metatuning@y..., "X. J .Scott" <xjscott@e...> wrote:
> John,
>
> I think I understood the conjecture except for what
> does it mean to deform:
>
> > to the three sphere
>
> Does that mean to mold the manifold until it is lying
> flat upon a 2-D space defined as teh surface of a 3D
> sphere? Or ??
>
> - J

By the phrase "deform to a three sphere" I mean you can continuously transform, or morph, the object in question, to the three sphere. Imagine a *solid* sphere with its infinitesimally thin skin removed. This is called an open 3-ball. This can be continuously deformed into ordinary three space by expanding it to infinity, whereas you could not, for instance, continuously deform a donut into a ball, since you'd have to poke a hole, or roll it out and connect the ends.

As for the three sphere itself, it is fourth in a sequence of "n-spheres", the 2-sphere being the one we are most familiar with. Each of them separates n+1 dimensional euclidean space into three sets, the points inside the sphere, those on the sphere and those outside. The zero sphere is two points on a line; it separates the line (euclidean 1-space) into those points between the two points, the two points and those not between them. The 1-sphere is a circle, and it separates the plane (2 dimensional euclidean space) into the points inside the circle, those on the circle and those outside. The two-sphere is the one we are most familiar with. It is the mathematical equivalent of a balloon, separating three space into the points inside, on, and the points outside the sphere. The three sphere is *almost* euclidean three space, except you have to complete the sphere by adding a "point at infinity", that is, connect all the points "at" infinity to this one other point.

Another way to think about this sequence is that in each size of space, we take a space one dimension lower and connect its ends. This image is not that good in one dimension, but in two, for instance, take the line (it is one dimension lower and infinite in length) and connect its ends: voila! the circle. In three space, take the plane, curl it up and connect all its points at infinity, and you have the sphere. In four space, you take three space and connect all its points at infinity, and you have the three sphere.

John Starrett