Compact Connected Manifolds of dimension $n$

Question

I started studying topology and seeing some differential geometry and i found it interesting when the book i was reading said that the only one-dimensional compact connected manifold is the circle, but they don't say anything about other dimensions. So just out of curiosity i wanted to know if we can have this results for more dimensions, say for an arbitary $n$-dimensional compact connected manifold we know what it is.

Thanks.

Answer 1

You are asking about one of the most fundamental questions in topology, still very far from being resolved in full.

For 2-manifolds see here for a complete classification.

Then it gets complicated.

For 3-manifolds see here for an overview. The question of whether there is a "complete" classification might be regarded as answered, but it might also be regarded as still open since no-one has really quite described the complete list yet.

Then it gets impossible.

For manifolds of dimension $\ge 4$, there is a lot of deep theory, and many partial results, but a complete classification seems still far beyond our present capabilities. See here for an overview of 4-manifolds, which has many special features. See here for a very brief description of manifolds of dimension $\ge 5$, which has several kinds of coherent theories independent of dimension, although the actual answers provided by those theories do depend on dimension.

Answer 2

For two-dimensional manifolds (i.e. surfaces), the situation is well understood; any connected, closed manifold (closed meaning compact without boundary) is either a sphere, a connected sum of tori, or a connected sum of real projective planes. This can be proved in an intro topology class with some work. (https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces).

For three-dimensional manifolds, the answer is much more complicated but has somewhat of an answer in what is called the Geometrization Conjecture, which has only been proven quite recently. (https://en.wikipedia.org/wiki/Geometrization_conjecture)

In dimensions 4 and greater, not only is there no classification known, it is in fact not possible to give an algorithm that can always determine whether or not two manifolds are isomorphic. (https://en.wikipedia.org/wiki/Classification_of_manifolds#Computability)