| See Milnor, "On manifolds homeomorphic to the 7-sphere", in Annals of
Mathematics (2) vol 64, (1956), pp 399-405 and Kerviare & Milnor,
"Groups of homotopy spheres: I", Annals of Math vol 77 (1963), pp 504-537.
These are the citations I've seen, but I don't have easy access to
a good math library and have not read the papers myself.
Milnor proves that the 7-sphere supports 28 non-homeomorphic
differentiable structures - a counterexample to a conjecture
that smooth manifolds always had only one such structure!
Kerviare and Milnor show that there are many examples, for example
S15 has 16245 structures. And it is *not* known if S3 has one or many!
Even worse, every Euclidean manifold has only 1 obvious structure except
for R4, which has many!! There's a book on this now.
Some exotic stuff, but fun. Even though the subject is somewhat advanced,
I've read some of Milnor's work and he writes very clearly so it should
not be so bad.
- Jim
|
| > Milnor proves that the 7-sphere supports 28 non-homeomorphic
> differentiable structures - a counterexample to a conjecture
> Even worse, every Euclidean manifold has only 1 obvious structure except
> for R4, which has many!! There's a book on this now.
Yes, fake R4 theorem was done by Donaldson(sp?).
One of the most significant thing in his theory is that
the proof is using the gauge theory.
After his work, the fact that there are uncountable number of differentiable
structures on one R4 prooved.
|