I am looking for specific examples of abstract manifolds. Let me be more clear.
By "specific" I mean the manifold should not be a manifold in general, such as "a loop in the plane" or a "sphere", etc.
By "important" I mean that the manifold should be nontrivial and of some importance in some research field, or appears in some real-world applications.
By "abstract" I mean the manifold should violate our geometric intuition and should exist beyond the physical world, like, if someone sees it for the first time, they will think "I didn't know this could be a manifold until you told me!"
So far I think I know two of such examples:
Teichmuller space: the space of all complex structures on a surface. The space can be made into a complex manifold.
Manifolds of data: I know this from machine learning. For example, one can view an image as a vector in a high-dimensional space by viewing each pixel as a component. Then we can assume that certain images lie on a submanifold of this high-dimensional space, and we can try to study this manifold to study these images.
Are there other such examples?
p.s. I am asking because there was once the idea that we only need to study manifolds embedded in the 3d space while the more abstract ones are essentially useless. But these examples certainly backs up the opposite argument, so I got curious.