Re: 4 large dimensions

Posted by DickT on July 21, 19101 at 18:24:41:

In Reply to: Re: 4 large dimensions posted by phobos on July 21, 19101 at 10:07:25:

Here's a famous one from topology: The Poincare Conjectue.

It's about the three-sphere, and I will first explain n-spheres. We live on a (approximate) two-sphere, the surface of the Earth. It's two dimensional not three, because two numbers, Latitude and Longitude do the job of providing coordinates for every point (with a coordinate singularity in Longitude at the poles). In four dimensional space, the points equidistant from the origin would constitute a three sphere, and in higher dimensions the same construction would produce spheres of any dimension.

Now in the case of our familiar two-spheres we can draw any curve on a globe and then imagine a deformation of the curve that shrinks it down to a point. You can't draw a curve on a spherical surface that doesn't have this property.

Topologists describe this by saying the two-sphere is simply conected. All the higher n-spheres are simply connected too. Another way to say it is that the Fundamental Group (of addible loops) of the n-spheres vanishes.

Now for the Poincare Conjecture:

If a threee dimensional manifold without boundary has a vanishing Fundamental Group, is it homeomorphic to a three sphere?

Homeomorphic means there's a one-to-one mapping of the points of the manifold onto a three-sphere, continuous in both directions. Poincare conjectured the answer was yes.

The result is easy for the two sphere. Smale proved it for n_spheres, n >= 5. Freedman proved it for the four-sphere.
Nobody has yet proved it for Poincare's case of the three-sphere. The problem is now a century old.


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