Hi, this is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when i say minimal i think something like adding a finite number of points or at least cells of dimension less than than the dimension of manifold) for a smooth non compact manifold, in this sense: if the one point compactification of the manifold is smooth and the embedding is smooth ok we are done, but what if the one point compactification is singular? can i embed the manifold in a "minimal" compact manifold of the same dimension?

This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite number of points or at least cells of dimension less than than the dimension of manifold) for a smooth non-compact manifold, in this sense: if the one point compactification of the manifold is smooth and the embedding is smooth, we are done; but what if the one point compactification is singular? Can I embed the manifold in a "minimal" compact manifold of the same dimension?