Parallels between Heegaard splittings and trisections of 4-manifolds

This dissertation develops the analogies between Heegaard splittings and trisections in two distinct directions. In one direction, we show that invariants of 3-manifolds defined using Heegaard splittings can be adapted in order to provide invariants of 4-manifolds. More precisely, given two smooth, oriented, closed 4-manifolds, 𝑀1 and 𝑀2, we adapt work of Johnson to construct two invariants, 𝐷𝑃(𝑀1,𝑀2) and 𝐷(𝑀1,𝑀2), coming from distances in the pants complex and the dual curve complex, respectively. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of "nearby" manifolds.

In another direction, we show that tools used to distinguish Heegaard splittings of a 3-manifold can be adapted to distinguish trisections of 4-manifolds. As a result, we exhibit the first examples of inequivalent trisections. We in fact show that, for every 𝑘 ≥ 2, there are infinitely many manifolds admitting 2𝑘−1 non-diffeomorphic (3𝑘, 𝑘)- trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections.