When someone starts learning topology, they’re given the rules that you cannot cut or glue, but you can distort continuously. Wikipedia details that surgery theory allows us to do just that, cut out regions of a manifold and replace it with another manifold that matches along the cut, but does this not change the manifold?
What *is* surgery theory and how does it work, informally, and *why* is it interesting?
In: 4
Yes, it changes the manifold or whatever other space you have. In what way depends on the type of surgery. The statements about “no cutting or glueing” are about the internal workings of topological spaces, but between different spaces there is often no such thing (well there are homotopy equivalences and more, but those are special things not fitting the desired generality).
Edit: **I’ve added an actual ELI5 down in the responses, including a tl;dr.**
It often is very useful to for example consider a point inside a larger space, or remove that point and look at the rest. Or even compare those three settings with each other (for example resulting in _exact sequences_, to throw advanced terminology). None of those are in any way continuous, in the naive sense spaces are discrete objects among each other.
One of the basic example is the sum of two manifolds of equal dimension: remove a full-dimensional open ball from each, then identify the boundary spheres the left behind. It is quite a paper-and-scissors kind of glueing if one draws [a picture](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Connected_sum.svg/1200px-Connected_sum.svg.png).
Surgery is kinda the same idea, but instead of connecting two things we just replace a part of a single thing with another. That’s ultimately the same procedure, before we replaced a ball by another manifold. The standard example replaces the boundary of the product space of ball times a sphere with that of a sphere times a ball, where the latter have other dimensions than the former ones (and the order is swapped for a reason).
The simplest non-trivial example is replacing the mantle of a cylinder (circle times a line segment) by two spherical “caps” (2D ball times the 2-pointed 0D sphere); both are bounded by two copies of a circle, where the surgery happens to detach and attach things. In medical terms: remove an artery and saw the now two open ends shut; or reversely transplant a new tube into someone’s damaged body.
Getting back to “why?”: because it works! Just like medical surgeries, it helps us “treating” the space, in particular figuring out how it behaves. Two often used examples are:
**A**: Is our given object, up to “deformation” (_homotopy type_), actually a manifold? Maybe even a _differentiable_ (or _smooth_, without hard folding edges, corners and such) one? So can we remove a couple of kinks and intersections in a continuous way? Replacing an already smooth(-ish) manifold-y part by another might not change this property. But it may allow us to deform the object in new ways we didn’t consider possible before.
It’s a bit like unknotting a knot, but instead of going through all the hassle of figuring it out, we just cut it open, straighten it out, and then glue the ends again. The Alexander the Great way of doing things. We just have to make sure that we are at that point only interested in the stuff where we do **not** cut!
**B**: Is a given deformation of one thing into another _differentiable_, or _smooth_? Or rather, can we change it in a continuous way to make it smooth(er)? Again this question might become simpler by replacing one part by another which is easier to deal with. Doing it step-by-step in some sense.
**C** (yes I said two two above): calculating the various _invariants_ such as homology, cohomology, homotopy, the ring structure, and whatever else one might desire. This is often only a means to an end, though. We find those because we want to answer the two questions above, or another problem, or just to invent homework problems for students.
Edit: added an example.
Latest Answers