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-pt. can be triangulated and explicitly give such a triangulation for a large initial segment X of M-pt., and an even larger initial segment Y of M-pt., together with a certificate that  Y-X contains a topologically flat embedded S^3 cutting off the end of M-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 ask 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.