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?
Why abstract manifolds?
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differential-topology
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17If you insist that a manifold has to be embedded in $\mathbb R^n$, then you need to show that whatever property you are interested in is independent of that embedding. – 2011-03-12
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0By the way, PL manifolds DO come embedded in $\mathbb{R}^n$. You need an embedding on $\mathbb{R}^n$ to define a polyhedron. So this is a feature of the smooth and topological categories. – 2011-11-27