Difference between revisions of "Talk:Poincaré conjecture"

Revision as of 03:49, 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)