Talk:Poincaré conjecture
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)
