# What are Kirby diagrams of candidate exotic 4-manifolds?

It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle decompositions of 4-manifolds can be written down as Kirby diagrams: Dotted circles represent 1-handles, undotted, numbered links represent 2-handles.

Is there a reference that lists candidate exotic manifolds, expressed in Kirby diagrams?

Edit: I am mainly interested in diagrams for manifolds that are not known, but conjectured to be exotic.

-

## 1 Answer

There is no comprehensive list in the format you ask about; you will probably want to look at the original papers. Searching for "exotic" and "4-manifold" on Mathscinet gives > 100 responses, and probably there are other phrases you could use for such a search. There are a number of examples listed in the book of Gompf and Stipsicz, but the subject has progressed quite a bit since that book was published.

The use of handle pictures in describing exotic 4-manifolds is not quite what you are suggesting, though. Typically, an exotic manifold would be described by some other construction (log transforms, fiber sums, knot surgery, rational blowdown, etc.) which enables you to compute a Seiberg-Witten or other gauge-theory invariant. The hard part is typically doing the right sequence of such operations in order to get the right topological type; even computing the fundamental group may be hard. It is usually possible, although often requires considerable effort, to draw a handle picture afterwards. This might be used to uncover some interesting property of the exotic manifold. But I don't know of any examples (at least any simply-connected ones) where one starts with an interesting handle diagram and then deduces exoticity. The point is that one still doesn't know how to effectively compute gauge-theory invariants from the handle picture.

-
I've seen quite a few articles with candidate exotic manifolds based on surgery methods, but none with a plain Kirby diagram. But thanks for the hint in Gompf and Stipsicz, I'll look at that later and accept. – Turion Jan 26 '14 at 14:03
I'm not exactly sure what you mean by surgery methods, but most of the techniques can be found somewhere. Non-spin complex surfaces are exotic copies of decomposable manifolds, and you can find lots of pictures of those. Gompf-Stipsicz Exercise 8.5.17 (solution in the appendix) shows a complicated rational blowdown. If you want to see an example of how to draw a handle diagram of the result of knot surgery on an elliptic surface, look in Akbulut's notes. – Danny Ruberman Jan 26 '14 at 14:49
The "problem" is maybe I'm looking for manifolds that are conjectured to be exotic, like current candidates for exotic 4-spheres, not for ones that are already known to be exotic. Akbulut's notes are a great reference, thanks! – Turion Jan 26 '14 at 17:47
@Turion: I don't think there are large numbers of such pairs sitting around for someone to prove them exotic. Here is one (but it's hard!) The Horikawa surfaces H(r) and H'(r) are complex surfaces of general type whose moduli space is disconnected; it's conjectured that they're not diffeomorphic but they have equal SW invariants. They are built as branched covers or by rational blowdown so (for what it's worth) handle diagrams can be drawn. – Danny Ruberman Jan 26 '14 at 21:05