That's a complicated question you're asking. It belongs to differential geometry, which I specialised in 15 years ago. So, you're looking for interesting 4 dim

That's a complicated question you're asking. It belongs to differential geometry, which I specialised in 15 years ago. So, you're looking for interesting 4 dimensional manifolds. Problem is, there are many of those. Too many to count, iirc. Contrast this with the 3 dimensional manifolds, of which there are only eight (and simple compositions of these eight). Now, unfortunately, the Klein bottle is a 2 dimensional manifold (it's surface is the manifold, and that is locally isomorphic to a R^2). It's interesting because it does not have an embedding into 3d space, unlike most other 2d manifolds, so you need to embed it into R^4. I don't know if there is a list of these. I can look up more details and give more pointers if there is interest. Interesting side note: the Klein bottle was originally called "kleinsche Fläche" (Klein surface). When saying this German word with an English voice it'll sound like "Kleinsche Flasche" which translates to "Klein bottle".