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I've just read a proof that

If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that the obvious maps $M \rightarrow M \cup_f N$ and $N \rightarrow M \cup_f N$ are smooth imbeddings.

The proof I read uses collar neighborhoods of the two boundaries to identify a neighborhood of the common boundary in the new manifold with a product of the common boundary and an interval.

This left me wondering about the uniqueness of the smooth structure. At first I thought it must be unique and I tried to show that the identity map is smooth but I couldn't show smoothness at points on the common boundary. Then the thought of a decomposition of an exotic sphere into hemispheres made me think perhaps uniqueness isn't guaranteed. But then I wasn't sure whether the hemispheres were still $smooth$ submanifolds when you change to the exotic smooth structure. Can anyone help me out by telling me whether we always have uniqueness and if so is it easy to see that the identity map will be smooth at points in the common boundary of M and N? Thanks very much for your time.

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    If you don't mind my asking, where did you read it? My gut instinct is to pull out my copy of Kosinski's Differential Manifolds and look for the result in there. Unfortunately, my copy is still at school, so I won't be able to look until tomorrow.2012-12-20
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    @JasonDeVito: Hi Jason. I'm the OP (I used a temporary account because I couldn't remember my password but then I was signed out for some reason). I read the proof in the new edition of Lee's "Introduction to Smooth Manifolds."2012-12-20
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    Dear Braudel, quite off-topic: did you choose your nickname in honour of the historian Charles Braudel?2012-12-20
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    @GeorgesElencwajg : Actually the historian Fernand Braudel! Maybe he also went by Charles?2012-12-20
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    Dear @Tim: no,no, my memory failed me, Braudel's first name was indeed Fernand. I find it pleasantly unexpected that your nickname refers to him : when I asked you if it did, I was pretty convinced that your answer would be "no". I am glad I was wrong... And by the way: welcome to our site!2012-12-20
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    @GeorgesElencwajg : thanks! It's funny: many times I've confused Braudel's name with Baudelaire whose first name was Charles.2012-12-20

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