Given a trisection diagram we can uniquely construct a smooth, connected, closed 4-manifold as follows. We first begin by constructing 3-dimensional handlebodies, by attaching 3-dimensional 2-handles to along the -tuples and respectively. We then attach to along (where we view as a subset of ), and likewise for After smoothing corners, this gives us a smooth 4-manifold with boundary. A schematic of this intermediate step is given below.
By definition, we require that any two of give a Heegaard diagram for Thus we have constructed a 4-manifold with three disjoint boundary components, each of which is diffeomorphic to Finally, we make use of a theorem of Laudenbach and Poénaru which tells us that there is a unique way to fill in this boundary components. This associates to the given diagram a unique, closed 4-manifold as desired.