Talk:Poincaré 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)

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

1. Not sure what a "covering" means here, so I made it a statement about loops. But this is probably less clear. Any suggestions?
2. 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.
3. 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)