= Generalized Poincaré conjecture
{tag=Classification (mathematics)}
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
|| Smooth types
| 1
| 1
| 2
| 1
| 3
| 1
| 4
| ?
| 5
| 1
| 6
| 1
| 7
| 28
| 8
| 2
| 9
| 8
| 10
| 6
| 11
| 992
| 12
| 1
| 13
| 3
| 14
| 2
| 15
| 16256
| 16
| 2
| 17
| 16
| 18
| 16
| 19
| 523264
| 20
| 24
$n = 4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
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