Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well defined.
Say, we have two smooth embeddings $f:M\rightarrow{\mathbb R}^n$ and $g:N\rightarrow{\mathbb R}^n$. I need to construct a smooth embedding $f\#g:M\# N\rightarrow {\mathbb R}^n$.
Usually you just connect $f(M)$ and $g(N)$ with a tube along an arc and then smooth the result. I know why the result does not depend on the choice of an arc in codimension at least 3. But why does it not depend on the smoothing procedure?