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Let $M_1^m,M_2^m$ be two smooth closed connected manifolds and $N_1^n,N_2^n$ their submanifolds respectively. Choose embeddings $(D^m,D^n)\to(M_i,N_i),i=1,2$, we can use them to define $(M_1,N_1)\#(M_2,N_2)$. Is the diffeomorphism type depend on the choice of the embeddings?

Thanks.

Ivy
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  • Not if the $N_i$ are connected (well, you get up to four diffeomorphism types, depending on whether the map from $D^m$ and from $D^n$ are orientation preserving). The proof is the same. –  Oct 22 '16 at 03:38
  • Can I say that two orientation preserving embeddings of $(D^m,D^n)$ in $(M,N)$ are isotopic? I don't know how to keep $D^n$ stay in $N$ when I move $D^m$ in $M$. @MikeMiller – Ivy Oct 22 '16 at 08:26

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