I don’t think it would be too controversial to assert that the Kirby Theorem is an important theorem in low dimensional topology. Given a 3-manifold and an framed link (by “framed” let’s mean “integer framed”), let denote the 3-manifold obtained from by surgery around . The Kirby Theorem states that, given two framed links , the 3-manifolds and are homeomorphic if and only if and are related by a finite sequence of the following local moves:

- Kirby I
- Stabilization

- Kirby II
- Handle-slide

- Kirby III
- Circumcision

If then it is easy to show that Kirby III can be realized using Kirby I and Kirby II moves.

Kirby calculus is the art of manipulating framed links using Kirby moves. By the way, Akbulut has an interesting objection to the term “Kirby calculus”.

The Kirby Theorem’s importance in quantum topology is that it provides a way of proving that some map from surgery presentations of 3-manifolds to something simpler (integers, rational numbers, polynomials…) is a 3-manifold invariant. Typically, we know the value of for some basic 3-manifold (usually ), and we have some formula for how changed under surgery around a framed link (in terms of some variant of the linking matrix of , for example). We verify that for any other framed link related by Kirby moves, the values of on and on coincide. Such proofs appear in quantum topology papers over and over and over again…

The problem with the Kirby Theorem is that Kirby moves are quite violent, in that they make a mess of the link. Say you have a surgery presentation for a 3-manifold which has some nice property (such as: has linking matrix with unit determinant, belongs to the Johnson kernel of the mapping class group of some Heegard splitting, has unknotted components, or whatever), Kirby moves will lose these properties. If one wants to find a set of moves between surgery presentations with nice properly then, you have to do something extra. It’s also unknown how to determine whether or not two surgery presentations present the same 3-manifold.

I’ve been interested for a long time in understanding the proofs of Kirby Theorem in order to prove a relative version, where the nice properly I want to preserve is the existence of a representation of the fundamental group of the complement of in onto some fixed group. Here’s a micro-outline of the main proofs of the Kirby Theorem.

The Kirby Theorem was proved by Rob Kirby [1]. The idea of his proof was to view 3-manifolds as boundaries of 4-dimensional handlebodies (a 4-ball with 2-handles attached), and, interpreting Cerf theory using handles, to show how two Morse functions on the same manifold can be related to one another. The problem is that Cerf theory is a awful huge machine, and the proof is too difficult more most people to understand (certainly I can’t make head or fishtail of it, pardon the stupid joke).

Next, Fenn and Rourke (and later Roberts [2]) substantially simplified Kirby’s proof. Modulo a few Cerf theory “black boxes”, a useful insight which emerges from their proof is that the key point is that in their approach one needs a certain class in to vanish, where denotes the fundamental group of the cobordism between and . The class itself seems impossible to calculate directly. In every attempt to generalize the Kirby theorem which I have seen, itself vanishes, and the problem is side-stepped. In the setting I am interested in, I have no real control over

Anyway, pretend you didn’t hear that, and let’s move forward. The Kirby Theorem is a 3-dimensional statement and should have a 3-dimensional proof. This was provided by Ning Lu in 1990 [3](Matveev and Polyak give a similar proof [4]), who showed that your favourite finite presentation of the mapping class group implies the Kirby Theorem (he chose Wajnryb’s presentation [5], but Gervais’s presentation [6] – which didn’t exist at the time of course- works just as well if not better) . It would be wonderful if one could prove the opposite implication, but remember… Kirby moves are tremendously violent. Handle-slides in particular play such mayhem with generators of the mapping class group that I cannot imagine using them to find a presentation of the MCG. This proof is beautiful because each step in it is completely elementary (Wajnryb’s presentation is a deep result, but it’s completely elementary). I can’t see how to generalize it at all though, because I don’t know how to find finite presentations for quotients of the MCG.

How else could you imagine proving the Kirby Theorem?

Hey Daniel,

I guess I view surgery as something that you can unnaturally view as a 3-dimensional phenomenon. But the proof that “you can get N from surgeries on M” is equivalent to “M and N are cobordant” is too natural to ignore. So for me 3-dimensional surgery is just a face of the subject of cobordisms/handle attachments/morse functions for 4-manifolds.

As a side question, has anyone worked out a “Kirby calculus” for spin and spin^c 3-manifolds? The Kaplan algorithm shows you how to represent the spin structures in a combinatorial surgery-theoretic way. But has an analogue of “Kirby moves” been worked out? Has someone set up a similar spin^c formalism for surgery diagrams?

Cerf’s characterisation of 1-parameter families of functions on manifolds, that they’re generally Morse except for certain cubic singularities, I believe there are several modern and simpler proofs of this but they’re not well-advertised. Hmm, it’s not clear to me where they are. I believe the singularity theorist codewords for the machine from which this falls out of are things like Thom-Mather stratification. I’ll do a little digging as 3-manifold theory really deserves a textbook-style treatment of things like Cerf’s theorem that Diff(S^3) has two components, etc.

Comment by Ryan Budney — September 26, 2009 @ 10:23 pm |

Hi Ryan!

I’m gradually moving more and more towards agreeing that there is something natural about the Cerf theory approach, but I’m not there yet. It is clear that cobordism implies existence of a surgery presentation (handle decomposition with no 0-handles or 4-handles, then trade 1-handles for 2-handles, then slide to the top of the cobordism), and vice versa (take the bottom boundary times the unit interval, and attach 2-handles)- where it all goes sour is when one tries to show that two cobordisms between the same 3-manifolds (or two Morse functions of the same cobordism) must be related by births, deaths, and handle-slides. This isn’t even intuitively obvious (not to me anyway), and its proof relies on murky analysis of 1-parameter families of functions which I still don’t fully follow (because of lack of effort, no doubt). It’s not even clear to me why it should conceptually be true- although it is true, and proven, and it’s the Kirby Theorem.

In the 3-manifold world, surgery along a single unknot is the action of an element of the mapping class group of a Heegaard splitting of the 3-manifold by Dehn twisting. This also looks pretty natural to me. Now you have to show, given 2 manifolds related by different actions of elements of the mapping class group of the same Heegaard splitting (you can assume the Heegaard splitting is the same by stabilizing), that the product of those elements in the mapping class group coincides. And you already know a finite presentation of the mapping class group.

So this is why I cannot yet see the Cerf theory approach as being more natural. I wish I understood it better. In the setting I care about (manifolds equipped with a colouring onto a group G), you’d have a pair of Morse functions, one for the manifold and one for an Eilenberg-Maclane space of G, with some kind of map between them, and you would want to do some kind of funky Cerf theory on such an object…

I’m wondering whether there is yet another approach to be found… say via hyperbolic geometry, or indeed perhaps by a technique which uses spin^c structures, and whether that might make the proof of the Kirby theorem more transparent (at least in the presence of extra structure).

Habiro is working on problems closely related to the problem you mention, and there is a chance he would know the answer to that question (at least for homology spheres)- although he is looking at clasper surgery descriptions (of integral homology spheres) rather than Dehn surgery descriptions.

I agree that there needs to be a textbook-style treatment of things like Cerf’s theorem…

Comment by Daniel Moskovich — September 29, 2009 @ 2:13 am |

The vague idea of why something like Cerf theory should be true is this. Maybe it’s obvious to you but I think it’s worth mentioning.

Consider the space of all real-valued smooth functions on a manifold M, call it Map(M,R).

Some basic transversality (using things like Sard’s theorem) states that Morse functions form an open dense subset of Map(M,R).

So you start to think of this as part of a stratification of Map(M,R), where the Morse functions are the co-dimension zero part of the stratification.

Map(M,R) is a contractible space (since R is contractible) so if you have any two Morse functions on M, there is a path between them (path in Map(M,R)). And since Morse functions are open and dense, this means generically you’d expect that path of in Map(M,R) to be Morse at all but finitely many times.

The point of Cerf theory is that you can give a local description to something that is effectively the co-dimension 1 part of this stratification — these are the functions that have a cubic local model at a single point, linear or quadratic local models elsewhere — functions of the form x^3-tx provide such nice local models, and elaborations on that idea provide birth/death families where various quadratic critical points can cancel.

Anyhow, I’ve been meaning to determine what the simplest textbook level proof of Cerf theory is nowadays. I’ve put in some interlibrary loan requests. Hopefully I’ll have an answer soon…

Comment by Ryan Budney — September 29, 2009 @ 8:37 pm |

Hi Ryan,

Thanks for this! I understood some but not all of this (I still don’t really understand all of it). In particular, Map(M,R) looks monstrous, and I can’t see “wall-crossing” arguments in it so well (or even how the “cubic in 1 point” functions divide it into cells).

On a related subject, I wondered this morning whether there might be a “combinatorial group theory” approach to the Kirby Theorem for aspherical 3-manifolds. So say you have two groups $\latex G_1$ and $\latex G_2$ (the “manifolds”), and two ways of relating them by “Dehn surgery” (adding generators and relations corresponding to integral Dehn surgery on a corresponding aspherical manifold). Are these related by “combinatorial Kirby moves”? The answer is yes- it’s a special case of the Kirby theorem. But can you prove this directly, combinatorially? Such a proof should be modifyable to work in a “coloured” setting (and should give a new angle on the Kirby theorem).

Comment by Daniel Moskovich — September 29, 2009 @ 9:16 pm |

If you consider only C^k-smooth functions, Map(M,R) is a Hilbert space, if you demand C^\infity-smooth functions, it’s called a Frechet space. These are manageable topological vector spaces, but a tad more on the analytic end of things. The basic analytic details are in Hirsch’s Differential Topology text.

The characterisation of the walls is Cerf’s theorem. The theorem is analogous to the Morse lemma in spirit. Neither is obvious. But it’d be nice to have a proof of Cerf’s theorem as cute and presentable as Milnor’s textbook proof of the Morse lemma.

Cerf theory is philosophically related to things you know pretty well. My understanding is Cerf theory was the main inspiration for almost-normal surface theory, in 3-manifold triangulations. The idea being that normal surfaces are sort of generic surfaces in a 3-manifold triangulation (modulo embedded surgery issues). The thing normal surface theory is after is 1-parameter families of surfaces, like if you want to know if two normal surfaces are isotopic (in the non-normal sense). If the surfaces aren’t normally isotopic, then your isotopy has to fail to be normal at some times — almost normal surfaces are what you need to fill in the gaps.

Comment by Ryan Budney — September 30, 2009 @ 1:13 am |

Incidentally, has Cerf theory been worked out for circle-valued functions, or in the setting of Novikov rings?

Comment by Daniel Moskovich — September 30, 2009 @ 10:40 pm |

Cerf’s proof for real-valued functions works just as well for circle-valued functions. Well, there’d be a caveat — the analogous theorem would only apply to homotopic functions to the circle.

I’m kind of insensitive to your 2nd question, but if there’s analogous arguments to the Whitney trick / h-cobordism for Novikov-type Morse functions, then it seems unlikely there’s much standing in the way of a Cerf-type theory for such functions. I’m completely insensitive to the details though so I might be way off.

Comment by Ryan Budney — October 1, 2009 @ 12:34 pm |

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