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.
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