One point compactifications of diffeomorphic $X$ and $Y$

Question

I know that if two topological spaces $X$ and $Y$ are homeomorphic then so are their one point compactifications $X^*$ and $Y^*$. If $X$ and $Y$ (say both are smooth manifolds) are diffeomorphic what do we know about $X^*$ and $Y^*$? Are they diffeomorphic too or only homeomorphic? If the latter is true, are there further condition under which $X^*$ and $Y^*$ will be diffeomorphic?

Answer 1

One point compactification of a manifold does not always have to be a manifold. For more details read this:

One-point compactification of manifold

But even when it is, I don't think it has to be unique. Please correct me if I'm wrong (manifolds is not my field of study).

For example take $M$ to be any exotic sphere of dimension $n\neq 4$. Then when you remove a point you obtain $\mathbb{R}^n$ as a topological space but since $n\neq 4$ then it is $\mathbb{R}^n$ as a manifold (the only exotic Euclidean space is $\mathbb{R}^4$).

Reversing, you have standard $\mathbb{R}^n$ which has at least two, non-diffeomorphic compactifications: the standard and exotic sphere.

Answer 2

You can't put a differentiable structure on the one-point compatification, so your question is not well defined. For example, consider the union of several open segments $(0,1)$, the one-point compactification will be a wedge of circle which is not a smooth manifold. I also think that one point compactification of $\mathbb R^2 \backslash D$, where $D$ is a union of disks, can't be a smooth manifold.