= Generalized Poincaré conjecture
There are two cases:
* (topological) manifolds
* differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
* for topological manifolds: this is a generalization of the <Poincaré conjecture>.
Original problem posed, $n = 3$ for topological manifolds.
<Millennium Prize Problems>.
Last to be proven, only the 4-differential manifold case missing as of 2013.
Even the truth for all $n > 4$ was proven in the 60's!
Why is low dimension harder than high dimension?? Surprise!
AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
For dimension two, we know there are infinitely many: <classification of closed surfaces>
* for differential manifolds:
Not true in general. First counter example is $n = 7$. Surprise: what is special about the number 7!?
Counter examples are called <exotic spheres>.
Totally unpredictable count table:
| Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Smooth types | 1 | 1 | 1 | ? | 1 | 1 | 28 | 2 | 8 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 | 523264 | 24 |
$n = 4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??