> I don't really get the whole surgery concept, if you can just cut the torus (and this could be anywhere, eg along the length of its inner or outer diameters) and glue the edges together aren't you just waving away the problem? By that logic I can slice open a sphere and call it a sheet, or conversely declare all shees to be spheres that haven't been glued together yet.
It's admittedly been a good 15 years since I cracked open a topology textbook, but the high-level, hand-wavey idea behind this sort of topological surgery isn't that you slice up manifolds and glue them together willy-nilly, but that you do so in very precise and controlled ways, which (and this is the very important bit) you've proven ahead of time preserve (or modify in a knowable fashion) some property of the manifold you care about. Rather than waving away the problem, you're decomposing it into a set of simpler ones which are ideally more tractable for some reason or another (perhaps you can compute a given property from first principles for a sphere, but not for a torus, for instance).
It's admittedly been a good 15 years since I cracked open a topology textbook, but the high-level, hand-wavey idea behind this sort of topological surgery isn't that you slice up manifolds and glue them together willy-nilly, but that you do so in very precise and controlled ways, which (and this is the very important bit) you've proven ahead of time preserve (or modify in a knowable fashion) some property of the manifold you care about. Rather than waving away the problem, you're decomposing it into a set of simpler ones which are ideally more tractable for some reason or another (perhaps you can compute a given property from first principles for a sphere, but not for a torus, for instance).
The wikipedia page on this stuff (https://en.wikipedia.org/wiki/Surgery_theory) is quite technical, but you might be able to squint at it and get a sense for how it works.