TITLE: How to specify a compact topological 4-manifold with a finite amount of data
QUESTION [36 upvotes]: Finite data for 4-manifolds: I have a somewhat hazy memory of seeing a Ph.D. thesis around 2005 which showed that any compact topological 4-manifold $M$ can be specified by a finite amount of data. The very rough idea is to exploit the fact that $M \setminus pt$ ($M$ minus a point) can be triangulated and explicitly give such a triangulation for a large initial segment $X$ of $M \setminus pt$, and an even larger initial segment $Y$ of $M \setminus pt$, together with a certificate that  $Y \setminus X$ contains a topologically flat embedded S^3 cutting off the end of $M \setminus pt$. This data allows you to build the closed topological manifold $M$ and argue that the result is unique. The important detail is how to give the required certificate. Over the years several people have asked me about this “finite data for 4-manifold” question. I would like to locate the reference, or failing that find time to write up a proof.

REPLY [4 votes]: In my understanding, I guess that the following strategy could be be attempted. Let $M$ be a closed connected orientable 4-manifold (while nonorientable 4-manifolds are doubly covered by orientable ones). Surgerying $M$ along finitely many embedded tame circles produces a simply connected 4-manifold $M'$. Then, $M$ can be recovered from a certain simply connected manifold by surgerying finitely many pairwise disjoint locally flat 2-spheres with trivial normal bundle. Now, simply connected 4-manifolds are classified by Mike Freedman's celebrated theorem in terms of the intersection form and the Kirby-Siebenmann invariant (which amounts to finite data). It remains to understand how much data we need to determine the embedded spheres to be surgered in a simply connected 4-manifold. Of course, we need more than homotopy (= homology since $\pi_1 = 0$) classes.