If we can use Whitney embedding to smoothly embed every manifold into Euclidean space, then why do we bother studying abstract manifolds, instead of their embeddings in $\mathbb{R}^n$? A vague explanation I have heard is that from this abstract viewpoint, we gain understanding into the intrinsic behavior of the manifold, without knowing anything about the ambient space in which it can be embedded. Can anyone give some examples of this, or any other reason why abstraction is necessary?
[Math] Why abstract manifolds
differential-topology
Best Answer
The same reason we study abstract groups instead of their embeddings into symmetry groups. If we know some properties of a manifold $M$ which are intrinsic and don't depend on embeddings, and we later encounter that manifold $M$ somewhere else in a different embedding than the one we're used to, then we automatically know a list of things we can say about this new copy of $M$. The point is that the study of things like groups and manifolds naturally separates into the study of their abstract structure and the study of their concrete representations, and failing to make this separation is unnecessarily confusing.
For example, topological and smooth manifolds don't have a notion of length of paths or volume: these things are dependent on an embedding into $\mathbb{R}^n$, so if I meet a manifold in a different embedding I can't assume that things like length and volume are the same, and I will only get confused if I do assume this. I can assume, however, that things like the number of connected components, the homotopy groups, the cohomology, etc. are the same, so whenever I meet a manifold whose cohomology groups I know I can always apply that information.
I wouldn't say that the abstraction is "necessary." Nothing in mathematics is necessary. We make the definitions we make so we can more easily understand certain mathematical phenomena, and the phenomena we are interested in with regard to manifolds are easier to understand if we define abstract manifolds first and then worry about their embeddings later.