Confused about A. Kosinski's description about surgery in his book "differential manifolds"

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

• The purpose of Kosinski's approach is that the classical handle attachment language puts you in a category of manifolds with corners, and he wants to avoid that. I would suggest reading Kosinski's description of handle attachment first. Once you are comfortable with that, the surgery description follows. Commented Jan 16, 2020 at 20:51
• thank you @RyanBudney. I will do so.
– Zest
Commented Jan 17, 2020 at 19:11