Comparing $4$–manifolds in the pants complex via trisections

Abstract

Given two smooth, oriented, closed $4$–manifolds, $M_1$ and $M_2$, we construct two invariants, $D^P\left(M_1,M_2\right)$ and $D\left(M_1,M_2\right)$, coming from distances in the pants complex and the dual curve complex, respectively. To do this, we adapt work of Johnson on Heegaard splittings of $3$–manifolds to the trisections of $4$–manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of “nearby” manifolds. This naturally leads to various graphs of $4$–manifolds coming from unbalanced trisections, and we briefly explore their properties.