The wild frontier, in Brisbane

Towards the end of June the University of Brisbane is hosting a conference on a subject which as the quote above suggests, is something of a barren landscape scattered with unsavory characters.

In my mind the motivation for this conference comes from two sources, one is entirely structuralist, and another comes from a remarkable theorem. The remarkable theorem is Rubinstein's 3-sphere recognition algorithm. Let me state it:

Theorem: There is an algorithm to determine if a compact triangulated 3-manifold is homeomorphic to the 3-sphere.

The really remarkable aspect of this theorem is the statement of the algorithm. The algorithm (as currently implemented in Ben Burton's software Regina) has an exponential run-time in the number of simplices in the initial triangulation. It can also be fairly memory intensive.

Algorithm: Start with your initial triangulation and enumerate all normal 2-spheres. Crush the normal 2-spheres until you have no more normal 2-spheres available to crush. This converts your original 3-manifold into a wedge of 3-manifolds. For each one of these wedge summands, check to see if there is an almost-normal 2-sphere. If one of the wedge summands fails to have an almost-normal 2-sphere, your original manifold (and that wedge summand) is not the 3-sphere. If every wedge summand has an almost-normal 2-sphere, they're all 3-spheres and your original manifold is the 3-sphere.

If the terminology normal and almost-normal is unfamiliar, the basic idea is that normal surfaces in a triangulated manifold are the ones that "look linear" inside of a simplex. So enumeration of these turns into an integer linear-programming problem. Almost-normal surfaces are an invention of Rubinstein's -- the idea is that normal surfaces are nice, but if you want to know isotopy relations, or embedded surgery relations between surfaces, they do not suffice. Almost normal surfaces are the "critical" surfaces in isotopies or embedded surguries between normal surfaces. There is a direct analogy from "Morse Theory to Cerf Theory" as with "Normal Surface Theory to Almost Normal Surface Theory."

So the idea is this: if 3-sphere recognition is so beautiful and algorithmic, shouldn't there be some aspect of this technology that should work for 4-manifolds? We want to know.

And that gets to the structuralist aspect of the purpose of this conference. Triangulations of 4-manifolds have not been used much at all in the field. So they introduce a new set of biases and prejudices, rather different from, say, the surgery approach. That's enough for me.