Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $\Bbb R^n$

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.