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)
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)
I made a couple tweaks to make this more accurate, and I hope I haven't compromised the accessibility too much:
- Changed "orange" to "surface of an orange", to emphasize that S^n should not be thought of as solid.
- Not sure what a "covering" means here, so I made it a statement about loops. But this is probably less clear. Any suggestions?
- 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.
- 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, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).
Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion, Perelman formulated the concept of "Perelman entropy."
Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincare conjecture. First, and this is the lesser of the insights, it was realized that no matter how complicated a manifold might be, the simple knowledge that a region of a three manifold could be bounded by a loop (or sphere) that could be shrunk to a point allows us to know for CERTAIN that Ricci flow on that region will either make it look like a patch of a three sphere after some finite "run time" for the flow, or "pinch" it beyond recognition (a singularity). This insight is due to Hamilton.
The second of these insights, which is the result of Perelman, is whenever the flow pinched the manifold, one could perform surgery on the manifold which would remove the pinch but leave the structure of the manifold unchanged otherwise.
I'm going to expand on this explanation for possible incorporation into the article. JacobB 00:30, 23 December 2009 (EST)
- I have to ponder this further. Thanks for your explanation and I look forward to your additional insights.--Andy Schlafly 00:51, 23 December 2009 (EST)
- Here is as precise a definition of Perelman entropy as exists for a non-geometer: Ricci flow on two manifolds which are basically the same (ie, diffeomorphic) can still be different. Perelman discovered a similar flow on manifolds which depends on not only the geometry of the manifold, but some other function (it can be just about any function). While we can create many different flows of this sort by altering that other function, there is a diffeomorphism of the manifold for which any flow you can think of will be identical to the Ricci flow. Perelman further modified this flow to be scale invariant, and called this discovery "Perelman entropy."
- JacobB 01:00, 23 December 2009 (EST)