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I want to ask how to show $\mathbb{R}^{n}+\mathbb{S}^{n}\cong \mathbb{R}^{n}$ as connected sums where the isomorphism is a differeomorphism between $\mathbb{R}^{n}$ and $\mathbb{R}^{n}$. The proof in Kosinski's book is not readable. Sorry if this problem is too trivial.

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It would be helpful for answerers if you would let us know where your confusion is. – Rasmus Oct 13 '12 at 9:02
well. clearly the original statement is wrong. – Bombyx mori Oct 14 '12 at 21:13

The addition happens like this: You remove a disc from $\Bbb R^n$, and one from $\Bbb S^n$, and glue the two of them together along the edge of the hole you just opened. But if you remove a disc from $\Bbb S^n$, what's left is diffeomorphic to a disc, so what you are really doing to your $\Bbb R^n$ is removing a disc, and then glueing a new disc onto the hole. All in all, you're back to $\Bbb R^n$

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But we can use an exotic sphere, which has a different differential structure. I am not sure if the new $\mathbb{R}^{n}$ produced is differeomorphic to the original one. – Bombyx mori Oct 12 '12 at 17:57

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