There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- Original problem posed, for topological manifolds.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 differential manifolds:Counter examples are called exotic spheres.Totally unpredictable count table:is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
So simple!! You can either:
- cut two holes and glue a handle. This is easy to visualize as it can be embedded in : you just get a Torus, then a double torus, and so on
- cut a single hole and glue a Möbius 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:
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.