## motivation of surgery

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an $n$-surgery on m dim manifold M is to cut out $S^n\times D^{m-n}$and replace it by $D^{n+1}\times S^{m-n-1}$. 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 $D^{n+1}\times D^{m-n-1}$.Now the boundary of $D^n\times D^{m-n}$ is $S^n\times D^{m-n}\cup D^{n+1}\times S^{m-n-1}$,i think there must be some close relation between the special form of $n$-surgery and handle.can someone help make this clear?

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Some help may be gained by looking at Milnor's "Lectures on the h-cobordism theorem," in the Princeton Yellow series. Also look at Fenn Rourke and Sanderson's book --- sorry I forgot the name. Finally, try to compare what is going on using surfaces with boundary --- disks with handles attached --- and consider $3$-manifolds, their Morse functions, and their non-critical levels. Favorite examples include lens spaces. One has to be meticulous about the parametrizations. FRS introduce the belt sphere, the attaching sphere, the core disk and the co-core disk. All of these ideas help! – Scott Carter Mar 16 2010 at 13:22

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Please improve punctuation and capitalization. – Theo Johnson-Freyd Mar 16 2010 at 16:04

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@Theo, to whom is your comment directed? In my comment, "Yellow" should be lower case. The phrase "sorry I forgot the name" should be enclosed in parenthesis. The phrase "handles attached --- and consider [...]" should be "handles attached. Consider [...]" I think you have enough points to edit the original question:-) – Scott Carter Mar 16 2010 at 21:18
I just read another paper of Milnor and Kervaire, "groups of homotopy spheres." In it they talk about surgery, only they call it "spherical modification", so I think it is early in the discussion. It is also highly enlightening because it shows the best set of algebraic-topological arguments about surgery that allowed one to compute cobordism groups. I'm not sure what the question is though. OP, I second Theo's comment: could you please articulate what you are asking and what you have been thinking in separate sentences? – Elizabeth S. Q. Goodman Oct 2 2011 at 6:40

## 3 Answers

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

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That is quite an impressive article you've (mostly) written! – S. Carnahan♦ Apr 17 2010 at 23:38

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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, $S^n\times D^{m-n}$ 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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Or "surgery is what happens to the boundary during a handle attachment" – Ryan Budney Mar 16 2010 at 18:51
Well said, Ryan! – Petya Mar 16 2010 at 23:19

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 a-priori construction which somebody pulled from a hat- it is rather an operation which stems naturally and inevitably from Morse theory.

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