Certifying a compact topological \(4\)-manifold (Q2001001)

The authors show that every compact topological \(4\)-manifold \(M\) can be presented by a finite amount of data. The authors call this data a certificate for \(M\). This is obviously true for PL manifolds. As in dimension four every PL manifold admits a unique smooth structure, it is also true in the smooth category. In the topological category, the authors use that every connected manifold admits a PL structure away from a point. They then show that it is possible to give a triangulation of a large enough compact subspace containing a topologically flat embedded \(S^3\) such that cutting along this sphere and gluing in \(B^4\) recovers the original topological manifold.