Difference between revisions of "Talk:Poincaré conjecture"

Revision as of 05:35, December 23, 2009

The smooth Poincare conjecture

The page states that "the h-cobordism theorem actually demonstrates that a diffeomorphism exists for n >= 5. The only open case is the four dimensional one". Perhaps I'm misunderstanding what this is supposed to mean, but I think it's false, the counterexamples being provided by the so-called exotic spheres. These are known not to exist for n=1,2,3,5,6, but there are 28 distinct smooth manifolds which are homeomorphic to the 7-sphere but not diffeomorphic to it (Milnor). For general larger n the conjecture is false, though there are a few cases (n=12 if memory serves) where it's still true. Generally the set of smooth structures on the n-sphere can be assembled into a finite abelian group. It's a tricky matter, and as noted in the article, remains open in 4 dimensions (though it's generally thought to be false). --JimR 21:03, 22 December 2009 (EST)

You're absolutely right, I misstated something. I'll fix it now. JacobB 21:19, 22 December 2009 (EST)

I don't know why I wrote that, I've seen a few exotic S^7s. In my defense, this was written pretty late at night (heh). JacobB 21:26, 22 December 2009 (EST)

I surely understand, and great work on this page! Would you mind if I add a link to fundamental group, which looks much better than homotopy group and related pages? --JimR 21:33, 22 December 2009 (EST)

It's a wiki, Jim! With your edit history, you hardly have to ask before contributing to a math article! JacobB 21:45, 22 December 2009 (EST)

Layman's statement

I made a couple tweaks to make this more accurate, and I hope I haven't compromised the accessibility too much:

1. Changed "orange" to "surface of an orange", to emphasize that S^n should not be thought of as solid.
2. Not sure what a "covering" means here, so I made it a statement about loops. But this is probably less clear. Any suggestions?
3. The page made it sound like part of the conjecture is that S^3 is simply connected, but this is an easy fact. The hard part is that it's the only simply connected thing.
4. The page made reference to "manifold in four-dimensional space". What we're really interested in is three-manifolds, and some of them don't even fit in four dimensional space, so this needs a tweak! A somewhat analogous example is the Klein bottle, a 2-dimensional manifold which doesn't fit in 3-dimensional space (without self-intersections). Thus I wrote "three dimensional space", with a link to manifold.

I think my changes are not optimal, please continue to improve! --JimR 22:24, 22 December 2009 (EST)

Your improvements sound great! I'm learning as I go here, which is part of the benefit in contributing. Researching more to improve more ....--Andy Schlafly 22:41, 22 December 2009 (EST)

While Andy's addition of "compact" is certainly a necessary correction, since I didn't even think to add this necessary term, I feel that the term itself isn't that useful in the laymans definition. I'm going to change it to give an explanation in the same way "manifold" is explained. My explanation isn't exact, but the only possibility I've ignored is that a closed set is removed from a compact manifold, which isn't really an object of study, ever. JacobB 22:46, 22 December 2009 (EST)

Great improvement, as the layman's definition should not be overloaded with jargon.

I need to understand the entropy angle better on this next.--Andy Schlafly 00:06, 23 December 2009 (EST)

Ricci Flow and Entropy: I think I can help explain the entropy relation here. It won't be too technical, since this is very high-level stuff here, but I can give the basics. There'll be some calculus:

First of all, every three-dimensional manifold (three manifold) is said to "admit a smooth structure," which basically just means we can do differential calculus in them.

Now, Ricci flow on a manifold is a way of deforming the manifold over time. A great way to visualize Ricci flow on two-dimensional manifolds is to imagine gas inside: for example, picture a manifold that looks two balloons, one full and one empty, which are attatched at their bases. Now, over time, gas will go from the large, inflated part (which has "positive curvature") to the deflated part (the curvature of which might vary, but will definitely be more curved than the inflated part). By the time the gas has distributed itself throughout the interior of the two balloons, the part which was originally inflated will have shrunk, and the part which had high curvature will have expanded. This is very similar to Ricci flow - the details of the manifold's geometry disappear, but the essential structure remains - two balloons connected.

Of course, dissipating gas is an example of entropy, as is the disappearance of details over time. So it shouldn't surprise us that the effects of Ricci flow have a good deal in common with entropy in physics.

For example, look at the heat diffusion equation . This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take , where j is any number less than the dimension of the manifold and no summation is implied on that index.

I'm going to expand on this explanation for possible incorporation into the article. JacobB 00:30, 23 December 2009 (EST)