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.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 |

- Homotopy | 0, 526, 7
- Topology | 72, 2k, 54
- Calculus | 17, 8k, 203
- Mathematics | 17, 28k, 633
- Ciro Santilli's Homepage | 262, 218k, 4k

- Classification | 107
- The beauty of mathematics | 391, 832, 8