Question About Exotic Smooth Structures
Posted by John Baez
Here’s a question that came up as Aaron Lauda and I have been writing A prehistory of $n$-categorical physics. It’s about whether you can prove the existence of an exotic $\mathbb{R}^4$ with the help of Khovanov homology.
I don’t understand this stuff at all, but here is the story I’ve heard. I’d like to know if this story contains errors. Even if it doesn’t, I’m sure it can be improved! I don’t really know who did what…
Apparently work of Freedman combined with Kronheimer and Mrowka’s work on gauge theory (Donaldson theory) proved there are knots that are ‘topologically but not smoothly slice’:
- Michael H. Freedman, A surgery sequence in dimension four; the relations with knot concordance, Invent. Math. 68 (1982), 195–226.
- Peter B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces I, Topology 32 (1993), 773–826
Think of $\mathbb{R}^3$ as a subspace of $\mathbb{R}^4$ in the obvious way, and give both these guys their ordinary smooth structure. A knot — a plain old smooth knot in $\mathbb{R}^3$ — is said to be smoothly slice if it bounds a smoothly embedded disc in $\mathbb{R}^4$. It’s said to be topologically slice if it bounds a flat topologically embedded disc in $\mathbb{R}^4$.
‘Flat’? Yes, this fine print is important: a topologically embedded disk in $\mathbb{R}^4$ is said to be flat if its embedding extends to an embedding of a thickened version of that disk: that is, the product of that disc and another disc. If we drop this fine print, all heck breaks loose: every knot bounds a topologically embedded disk in $\mathbb{R}^4$.
Apparently Gompf or somebody (?) used the existence of topologically but not smoothly slice knots to give a proof of the existence of exotic smooth structures on $\mathbb{R}^4$.
More recently Jacob Rasmussen used Khovanov homology to construct an invariant that can prove certain knots aren’t smoothly slice:
- Jacob Rasmussen, Khovanov homology and the slice genus, to appear in Invent. Math., available as arXiv:math/0402131.
And in fact, we can apply this idea to a specific knot that’s known to be topologically slice, and prove it’s not smoothly slice, and thereby conclude the existence of an exotic $\mathbb{R}^4$.
Is this right?
If so, this would be cool because it would set up the beginnings of a connection between categorified quantum groups (which underlie Khovanov homology) and exotic smooth structures in 4 dimensions. This was an old dream of Crane and Frenkel.
Re: Question About Exotic Smooth Structures
I just got some more information on this, from Aaron himself! He pointed out page 13 of Rasmussen’s paper Knot polynomials and knot homologies, which says, in part: