I am reading Milnor's Lectures on h-cobordism theorem, and I am stuck on Milnor's definition on surgery of type $(\lambda,n-\lambda)$ on manifold, where the definition following can be found on page 31 of the book. Milnor uses this to describe the elementary cobordism corresponding to an non-degenerate critical point of index $\lambda$.
$D^k$ is the unit open disk of dimension $k$, and $S^k$ unit sphere, and $1\le \lambda\le n-1$.
Definition: Given a manifold $V$ of dimension $n-1$ and an embedding $\varphi: S^{\lambda-1} \times D^{n-\lambda}\to V$. Let $\chi(V,\varphi)$ denote the quotient manifold obtained from the disjoint sum $(V-\varphi(S^{\lambda-1}\times\{0\}))+(D^{\lambda}\times S^{n-\lambda-1})$ by identifying $\varphi(u,\theta v)$ with $(\theta u,v)$ for each $u\in S^{\lambda-1}, v\in S^{n-\lambda-1},0<\theta<1$. If $V'$ denotes any manifold diffeomorphic to $\chi(V,\varphi)$, then we will say that $V'$ can be obtained from $V$ by surgery of type $(\lambda,n-\lambda)$
It seems that this surgery is very much different from the common one. I don't really understand what the surgery construction, though I computed some examples in low dimensions. Milnor says that This surgery on (n-1)manifold has an effect of removing sphere of dimension $\lambda -1$ and replacing it by an embedded sphere of dimension $n-\lambda-1$. Could someone try to explain it?
Actually, I know that given a morse function on a manifold $f: V\to \mathbb R$, and $p\in V$ be a critical point of $f$ with index $\lambda$, then $f^{-1}(-\infty, f(p)+\epsilon]$ has the same homotopy type with $f^{-1}(-\infty, f(p)-\epsilon]$ added by a $\lambda$-cell. How adding a $\lambda$-cell is related to this surgery?
Any help would be appreciated!