Let $M$ be a closed smooth manifold. If for some point $p$ on $M$ we can find a diffeomorphism between $M-\{p\}$ and $\mathbb R^n$, then is $M$ diffeomorphic to $S^n$(with the standard differential structure)?
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Nice question ! – Georges Elencwajg Sep 26 '14 at 18:40
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In other words, is true that if $M-{ p}$ is diffeomorphic to $S^n-{ p}$ then $M$ is diffeomorphic to $S^n$ for a closed smooth manifold $M$? – Semsem Sep 26 '14 at 18:47
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No!
For example, take one of the 27 exotic differential structures $X$ on $S^7$ and delete a point $p\in X$ : you will obtain a differentable structure on the manifold $X\setminus \{p\}$, homeomorphic to $\mathbb R^7$.
But there is only one differential structure (up to diffeomorphism) on $\mathbb R^7$ (or for that matter on any $\mathbb R^n$with $n\neq 4$ : a result of Donaldson and Freedman).
So we may take $\mathbb R^7=X\setminus \{p\}$ and adding the point $p$ obtain a manifold $X$ not diffeomorphic to the standard differential manifold $S^7$.
Georges Elencwajg
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