Please excuse me, if MO is not the proper place for this question. I aksed the same question on M.SE
but i am not sure, whether MO is not the better place to adress this. I am really struggling to understand Kosinski's description of surgery on a manifold.
On p.112 in Kosinski's "Differential Manifolds" he introduces Surgery on a $(\lambda-1)$-Sphere in a manifold $M^m$. He says
Surgery on a $(\lambda-1)$-Sphere in a manifold $M^m$ is a special case of pasting. We paste $M$ and $S^m$ along $S$ and $S^{\lambda-1}$. The resulting manifold will be denoted $\chi(M,S)$. (...) it can be described as follows:
Let $T' = \{x \in S^m \mid x_\lambda^2 > 0\};$ we view $T'$ as a tubular neighborhood of $S^{\lambda-1}$ in $S^m$. Let $h: T' \to M$ be a diffeomorphism, $h(S^{\lambda-1}) = S$. Then $$\chi(M,S) = (M\setminus S) \cup_{h\alpha} (S^m \setminus S^{\lambda-1})$$
remark: $\alpha$ is the composition of the diffeomorphism $D^m \setminus S^{\lambda-1} \to \mathring{D}^\lambda \times D^{m-\lambda}$ and the involution on $(\mathring{D}^\lambda \setminus \boldsymbol{0})\times D^{m-\lambda}$
He then continues:
Note that the operation of attaching a $\lambda$-handle along $S$ becomes, when restricted to the boundaries, precisely surgery on $S$. This can be conveniently stated as follows. Consider $h$ as an embedding of $T'$ in $M \times \{1\} \subset M \times I$ and attach a $\lambda$-handle to $M\times I$ along $S$. Let $W = (M\times I)\cup H^\lambda;$ $W$ is called the trace of the surgery.
My question: whenever i read about surgery on a $m$-Manifold $M$, it's always described as cutting out $S^n\times D^{m-n}$ and gluing in $D^{n+1}\times S^{m-n-1}$ (see Ranicki's Surgery Theory) or any other source about surgery theory.
I simply can't get behind the way Kosinski describes this procedure. Where exactly do we remove $S^n\times D^{m-n}$ and glue in $D^{n+1}\times S^{m-n-1}$ ?
The way I understand Kosinski's approach is that we remove the embedded $(\lambda-1)$-sphere from $M$ and $S^m$ simultaneously and past them along the tubular neighborhoods of the embedded sphere $S^{\lambda-1}$... whilst i do recognize $S^m$ being $$S^m =\partial D^{m+1} = \partial (D^\lambda\times D^{m-\lambda+1}) = S^{\lambda-1}\times D^{m-\lambda+1} \cup D^\lambda \times S^{m-\lambda}$$
i'm still failing to see the link between Kosinski's definition of surgery and the common definition (as of Ranicki) i've stated.
On p.142 Kosinski even mentions himself:
"Surgery is informarlly described as "taking out $S^k\times D^{n+1}$ and gluing in $D^{k+1}\times S^n$."
But i fail to see how this relates to his definition i've stated.
Can anyone help me understanding how they're related or what i might not seeing here?
thank you very much
