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.

I don’t have a copy to hand, but from memory Hempel works entirely in the PL category.

Comment by Henry Wilton — August 2, 2011 @ 12:21 pm |

All smooth manifolds of any dimension can be given a unique PL structure. This is basically due to Whitehead and is not hard to prove. It’s the other direction that’s true in 3 dimensions and false in higher dimensions : all PL 3-manifolds can be given unique smooth structures. This is actually not a particularly difficult theorem (a proof is sketched in Thurston’s book). What’s much deeper is that a topological 3-manifold can be given a unique PL structure; this is due to Moise.

And there are some facts about 3-manifolds which have nothing to do with smooth vs pl but are much easier to prove in one category instead of the other. For instance, I don’t know how to give a smooth proof of Dehn’s lemma or the loop/sphere theorem — the standard proof using towers is definitely a PL proof (there’s an alternate proof due to Johannson using hierarchies that might work in the smooth setting, but I’ve never worked through the details of it). Similarly, I don’t know how to do normal surface theory in the smooth category, so I don’t know a smooth proof of the fact that every 3-manifold has a unique prime decomposition.

This is basically why Hempel’s book works in the PL category exclusively. I don’t think it even mentions smooth structures (as Henry noted).

However, none of the above disturbs me that much. Mixing different styles of math is good for a subject, not bad!

Comment by Andy P — August 2, 2011 @ 3:19 pm |

Dehn’s lemma and the loop/sphere theorem were among the primary examples I had in mind when writing this post- these are not inherently PL results, but I don’t know smooth proofs for them.

In my opinion at this moment (which has changed before, and may change again), something is lost when routinely switching from smooth to PL. I’m not sure I know how to formulate what that is with any sort of precision, but somehow when I switch from smooth to PL I feel I’ve traded my beautiful smooth manifold for an ugly jumble of tetrahedra- and some measure of conceptual transparency is being lost through the cracks.

So I’m sticking by the Chovot Halevavot philosophy- “a little purity is a great deal”. Of course there are other philosophies, and different people are comfortable with different levels of conceptual mixing!

Comment by dmoskovich — August 2, 2011 @ 6:01 pm |

I can’t say I get your point (criminally overused?), but as far as a smooth proof of Dehn’s lemma/loop theorem, isn’t that what Meeks-Yau did?

Comment by me — August 3, 2011 @ 3:51 pm

If everyone agreed that the theorem were criminally overused, then textbooks would mostly be either “pure smooth” or “pure PL”- so clearly this is a minority opinion. The reason I would characterize the theorem as “criminally overused” is that, rather than having two clear models of 3-manifold topology, instead 3-manifold topology looks to me like a bit of an incoherent jumble of techniques taken from all over (at least, if you want a full-detail black-box-free version of the theory), with no really clear transparent picture of its fundamentals (this is a bit of a pet peeve of mine). And it seems everyone is happy with this situation! At least from the aesthetic perspective, I really wish I could (for instance) teach two introductory graduate courses- “PL 3-manifold topology” and “smooth 3-manifold topology”, both clear, complete, and self-contained, and with no technical overlap.

Comment by dmoskovich — August 3, 2011 @ 6:50 pm |

I’m pretty sure there’s a straightforward adaptation of the loop and sphere theorems to the smooth world. Basically there’s just a lot of “smoothing the corners” going on, being careful that doesn’t cause trouble. In some sense Whitehead’s proof that smooth manifolds admit essentially unique compatible triangulations, this involves a similar variety of fussy details you have to mess around with to get smooth category proofs of the loop and sphere theorems off the ground. With normal surface theory you’re pretty much forced into a PL-smooth world because you have to address the triangulation, so maybe it’s splitting hairs to say any setup of normal surface theory is purely “smooth”. You’re going to be addressing piecewise-smooth issues at every step. But likely most of it can be buried in a few technical lemmas.

I believe Poincare’s switch to PL came earlier. He was trying to write down a good proof of what we now call Poincare duality. At some point he realized his notion of cycle wasn’t ideal — he seemed to be initially thinking in terms of singular bordism of smooth manifolds. But then simplicial homology struck him and he saw the dual polyhedral decomposition and was much happier. As far as I know, the proof of Poincare duality is the initial reason for simplicial homology. That’s the impression I get from Dieudonne’s history, anyhow.

Comment by Ryan Budney — August 2, 2011 @ 8:48 pm |

I agree that you can imitate a PL proof in the smooth world, but somehow it isn’t “naturally” a smooth proof. Ideally, maybe I’d want something that works with a space of functions, in some bigger space of functions, and I’d show by some sort of dimension count that a connected component of a smooth immersion of a disc with “nice” singularities bounding a homotopically trivial loop contains an embedding, or something like that (somehow analogous to the proof that a knot has an essential quadrisecant). I haven’t thought about this at all, but I wonder what the obstruction to that type of naive approach might be.

Comment by dmoskovich — August 3, 2011 @ 6:41 pm |

But a lot of purely smooth proofs do look quite a lot like PL proofs. To get Morse theory off the ground you really need things like Sard’s theorem and transversality. But transversality arguments have much of the same flavour as basic PL topology, because you get stratifications. Even the proof of Sard’s theorem has an inductive study of various families of degenerate points in the function’s domain. Further development of Morse theory along the lines of Cerf require the study of further stratifications on function spaces. So I think the distinction can get fairly artificial, depending on how you look at it. I think many people consider the smooth/PL divide to be a flavour inherited by your choice of techniques, rather than a major divide between areas. Or perhaps a more smooth-centric perspective on this would be to say PL-topology turns something that would require an analytic argument into one that’s just linear algebra. Smooth topology keeps the analysis around — the stratifications and fussy analytic arguments remain, they haven’t been preconverted into combinatorial/linear algebra problems.

Comment by Ryan Budney — August 3, 2011 @ 10:58 pm

Somehow, I see stratifications as part of the smooth machine… somehow, I feel there’s a qualitatively different understanding one obtains from an argument concerning stratification of a space of functions, as opposed to an argument involving gluing tetrahedra in prescribed ways. Maybe that’s the issue I’m groping at- if the point of proofs is to furnish understanding, then “smooth 3-manifold topology” contains a coherent conceptual picture which is worthwhile to understand, and “PL 3-manifold topology” contains a different coherent conceptual picture, and I feel I’m just beginning to separate them out from one another, because none of the standard references which I’ve used separated things out for me like that.

Comment by dmoskovich — August 4, 2011 @ 5:43 pm |

I agree that there is something aesthetically appealing about working only in one category. But for a text book, the goal is to convey the ideas to the newcomer as painlessly as possible, and sometimes one category is better than the other in terms of minimal overhead. My favorite example is the proof of the 3D Schoenflies lemma in Hatcher’s notes on 3-manifolds. To do it carefully, I think PL is the best category, but then there are a lot of messy details. In the smooth category, Hatcher is able to sweep a lot of details under the rug and give a very intuitive and understandable proof. Now switching back and forth in papers is another matter…

Comment by Jesse Johnson — August 4, 2011 @ 4:32 pm |

The Alexander Theorem proof in Hatcher’s notes is lovely, isn’t it! There are loads of details swept under the rug, but they’re standard details which there is conceptual value in working out.

The purpose of a lot of 3-manifold textbooks is to present qualitative results about 3-manifolds, so they take a bee-line to those results. My lament is that, if the purpose of proof is to understand, then there’s a “smooth category” understanding of 3-manifold topology, and a different “PL category” understanding of 3-manifold topology, and having one or the other, or both, is much better than having an incoherent mix of the two. I always had that “incoherent mix” understanding, and I’ve been working a bit on separating things out inside my mind, in order to tidy up my basic 3-manifold topology understanding.

Comment by dmoskovich — August 4, 2011 @ 5:50 pm |

The beauty of being a Professor is that you can write a book. Why not write two on 3-manifolds, one for PL, and one for smooth?

Comment by Mayer A. Landau — August 5, 2011 @ 12:09 am

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