I nominate the following for the most criminally overused theorem in Low Dimensional Topology.
Theorem: Every smooth 3-manifold has a triangulation, unique up to common subdivision.
This theorem implies that the smooth and the PL categories coincide as far as 3-dimensional topology is concerned, so we can switch from one to the other at will. We can, but I wish we wouldn’t quite so often. Smooth and PL worlds have quite different flavours you see. Thanks, in part, to discussions with my co-blogger Ryan Budney, I find conceptual value in keeping the two worlds separate.
Smooth 3-manifold topology is the world of Morse functions. The basic building blocks of smooth 3-manifolds are handles, from handles come surgery presentations, and surgery presentations of the same smooth 3-manifolds are related by Kirby moves. There’s a big analytic machine in the background, which isn’t really as fully developed as we might want it to be (perhaps framed functions are really the “right” tool), but there’s something natural and intrinsic about smooth 3-manifold topology- no wonder that Poincaré considered smooth manifolds first. This is the world which is more closely tied in with the rest of mathematics.
PL 3-manifold topology is the world of triangulations. The basic building blocks of PL 3-manifolds are 3-simplices (tetrahedra), and two simplicial decompositions of the same PL 3-manifold are related by Pachner moves. Compact PL 3-manifolds are finite combinatorial objects, so they can be handled by computers, and PL proofs are often easy finite combinatorics. No wonder Poincaré changed over to thinking about PL 3-manifolds.
Ideally, I think we really do want both worlds. But if we could choose only one per theorem, then smooth facts (e.g. existence of surgery presentations) should have smooth proofs, PL facts (e.g. existence of special spines) should have PL proofs, and both-world facts (e.g. existence of Heegaard splittings) should have one proof in each world. Right now, I feel people tend to sort-of mix and match techniques from both worlds inside the same proof, and I don’t like that.
My greatest lamentation on this topic is that none of the textbooks seem to be pure (smooth or PL). Somewhere along the line, every 3-manifold textbook which I know cheats. For example, I don’t think there’s ever been a single textbook which proved the existence of a surgery presentation (a smooth fact) using only smooth tools. It’s not at all difficult, but for some reason nobody seems to have considered it worth doing.