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.