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The best answer is to cite Tim Perutz answer to my question #15641 (Surgery and homology: a reference request):

"To say that a smooth, closed manifold N is obtained by surgery along a (framed) sphere in M is to say that there is a cobordism P from M to N and a Morse function $f\colon P\to [0,1]$, with $f^{−1}(0)=M$, $f^{−1}(1)=N$, and exactly one critical point c."

Critical point corresponds to a handle, $S^n\times D^{m-n}$ lies in $M$, $D^{n+1}\times S^{m-n-1}$ lies in $N$ and spheres $S^n$ and $S^{m-n-1}$ are intersections of stable and unstable manifolds for $c$ with corresponding level sets.

In other words: to obtain $N$ from $M$ by a surgery one can consider $M\times [0,1]$, glue a handle using $S^n\times D^{m-n} \subset M=M\times 1$, smooth the resulted cobordism and take a component of a boundary.

2 added 217 characters in body

The best answer is to cite Tim Perutz answer to my question #15641 (Surgery and homology: a reference request)

"To say that a smooth, closed manifold N is obtained by surgery along a (framed) sphere in M is to say that there is a cobordism P from M to N and a Morse function $f\colon P\to [0,1]$, with $f^{−1}(0)=M$, $f^{−1}(1)=N$, and exactly one critical point c."

Critical point corresponds to a handle, $S^n\times D^{m-n}$ lies in $M$, $D^{n+1}\times S^{m-n-1}$ lies in $N$ and spheres $S^n$ and $S^{m-n-1}$ are intersections of stable and unstable manifolds for $c$ with corresponding level sets.

In other words: to obtain $N$ from $M$ by a surgery one can consider $M\times [0,1]$, glue a handle using $S^n\times D^{m-n} \subset M=M\times 1$, smooth the resulted cobordism and take a component of a boundary.

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The best answer is to cite Tim Perutz answer to my question #15641 (Surgery and homology: a reference request)

"To say that a smooth, closed manifold N is obtained by surgery along a (framed) sphere in M is to say that there is a cobordism P from M to N and a Morse function $f\colon P\to [0,1]$, with $f^{−1}(0)=M$, $f^{−1}(1)=N$, and exactly one critical point c."

Critical point corresponds to a handle, $S^n\times D^{m-n}$ lies in $M$, $D^{n+1}\times S^{m-n-1}$ lies in $N$ and spheres $S^n$ and $S^{m-n-1}$ are intersections of stable and unstable manifolds for $c$ with corresponding level sets.