So i was trying to get my head around it, but i still haven't managed to do so. I am currently reading A. Kosinski's Differential Manifolds. On p.112 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 $$S^m =\partial D^{m+1} = \partial (D^\lambda\times D^{m-\lambda+1}) = S^{\lambda-1}\times D^{m-\lambda+1} \cup D^\lambda \times S^{m-\lambda}$$
but I'm still missing the link between Kosinski's definition of surgery and the common definition (as of Ranicki) i've stated.
Can anyone help me understanding how they're related or what i might not seeing here?
thank you very much
The key observation here is that for the sphere $$S^m = \partial ( D^\lambda \times D^{m-\lambda+1})$$ it holds that
$$\partial\left(D^\lambda\times D^{m-\lambda+1}\right) = \left(S^{\lambda-1}\times D^{m-\lambda+1}\right)\cup \left(D^\lambda\times S^{m-\lambda}\right)$$
Therefore, surgery on $M$ here means that one removes $S^{\lambda-1}\times D^{m-\lambda+1}$ and attaches $D^{\lambda}\times S^{m-\lambda}$ to the boundary $$\partial(S^{\lambda-1}\times D^{m-\lambda}) = S^{\lambda-1}\times S^{m-\lambda-1}$$ of the cut. One can consult Ranicki's Surgery Theory for the general idea.