Uniqueness of 4-Manifolds Described as Sequences of 3-d Handlebodies

Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we prove a corresponding uniqueness theorem for each of these descriptions so that, for example, any two loops of Morse functions on a surface yielding diffeomorphic 4-manifolds are related by a given set of moves.