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
- Kirby II
- Kirby III
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 . 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 ) 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 (Matveev and Polyak give a similar proof ), who showed that your favourite finite presentation of the mapping class group implies the Kirby Theorem (he chose Wajnryb’s presentation , but Gervais’s presentation  – 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?