TITLE: Why abstract manifolds?
QUESTION [27 upvotes]: 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?

REPLY [6 votes]: If you restricted the definition of manifolds to be "submanifolds of Euclidean space", you'd run into the problem that sometimes you want to do calculus on an object -- natural objects like homogeneous spaces, Grassmann manifolds, cut-and-paste objects like the "pac man" torus and such.  But the formal restrictions on usage of manifolds would mean you would have to embed the manifold in Euclidean space before you could apply the machinery of calculus to it.  In that sense the submanifold restiction is like a type of formal red-tape that gets in the way of understanding.  Abstract manifolds dispense with that.