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??

Classification of closed surfaces

0 0

So simple!! You can either:

cut two holes and glue a handle. This is easy to visualize as it can be embedded in $R^{3}$ : you just get a Torus, then a double torus, and so on
cut a single hole and glue aMöbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
- 1: real projective plane
- 2: Klein bottle

A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.