an $n$surgery on m dim manifold M is to cut out $S^n\times D^{mn}$and replace it by $D^{n+1}\times S^{mn1}$. I want to know how this is invented? I do know that the effect of passing a critical point of index $n$ in $m$manifold is equivalent to attach an $n+1$handle $D^{n+1}\times D^{mn1}$.Now the boundary of $D^n\times D^{mn}$ is $S^n\times D^{mn}\cup D^{n+1}\times S^{mn1}$,i think there must be some close relation between the special form of $n$surgery and handle.can someone help make this clear?

The Wikipedia article on surgery theory explains this! In addition, the Edinburgh Surgery Theory Study Group provides all kinds of surgeryrelated materials, including YouTube videos (not for the squeamish). There is all kinds of surgery bricabrac on Surgery Bits and Pieces and also here. 


The best answer is to cite Tim Perutz answer to my question 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, $S^n\times D^{mn}$ lies in $M$, $D^{n+1}\times S^{mn1}$ lies in $N$ and spheres $S^n$ and $S^{mn1}$ 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 $S^n\times D^{mn} \subset M=M\times 1$, smooth the resulted cobordism and take a component of a boundary. 


This question has already been answered, but there's a tiny piece of intuition which I'd like to make explicit: If you're thinking about a manifold in the PL world, surgery might look a bit arbitrary why cut out and glue in those pieces and not others? Surgery's natural setting is the smooth world, where you're equipping a manifold with a Morse function $f\colon\, M\to \mathbb{R}$, and using information about critical points of $f$ to encode $M$. It's actually a bit more involved than you might think it might be, but when you pass a critical point of $f$ you add a handle to $M$, and the boundary changes by surgery. For details, see answers to this question. So really, surgery isn't an apriori construction which somebody pulled from a hat it is rather an operation which stems naturally and inevitably from Morse theory. 

