Uniqueness of 4-manifolds described as sequences of 3-d handlebodies

Abstract

Given a description of a manifold, two natural questions arise: existence and uniqueness. For example, closed, orientable 3-manifolds can be described as surgeries on a link in S 3 . The existence problem for this description was addressed by Lickorish and Wallace [21] [29] who showed that any closed, orientable 3-manifold is given by surgery on a link in S 3 . In [15], Kirby proved a uniqueness theorem for these descriptions, showing that any two link-surgeries yielding the same 3-manifold are related by handle slides and blowups. Similarly, Moise [24] showed that every closed, orientable 3-manifold can be described as a Heegaard splitting, and Reidemeister and Singer [26] [25] showed that any two Heegaard splittings giving the same 3-manifold are related by a stabilization operation.Moving up a dimension, there has been a variety of new descriptions of smooth, orientable, closed 4-manifolds. In [8], Gay showed that any such 4manifold can be described as a loop of Morse functions on a surface, and used this fact to give a novel proof of that the smooth, oriented, 4-dimensional cobordism group is isomorphic to Z. In [17], Klug and the author showed that these 4-manifolds can be represented as a loop in the pants complex, and this was used to provide a combinatorial calculation of the aforementioned cobordism group as well as to give invariants of loops in the pants complex. Kirby and Thompson [16] have also showed that any such 4-manifold can be described as a loop in the cut complex, and they used this to define an invariant of 4-manifolds which detects the 4-sphere among homotopy spheres. Finally, in [13] Naylor and the author introduced multisections of 4-manifolds and used them to describe 4manifold operations such as cork twisting and log transforms diagrammatically on a surface.As these descriptions all stem from trisections [7], there are inherent similarities between them. Most obviously, all of the information in these decompositions can be given on a surface. Nevertheless it was not clear how to pass between these descriptions. The first half of this paper is focused on constructing explicit correspondences between certain quotients of the aforementioned sets. These quotients still retain all of the information needed to construct a unique smooth 4-manifold, together with a multisection structure.The second half of the paper is dedicated to showing how any two multi-1