Comparing 4-Manifolds in the Pants Complex via Trisections

TL;DR

This paper introduces invariants for 4-manifolds based on distances in the pants and dual curve complexes, adapting 3-manifold techniques to 4-manifold trisections, and explores their properties and implications.

Contribution

It develops new invariants for 4-manifolds using complexes from 4-dimensional trisections, extending Johnson's 3-manifold work to 4D.

Findings

Invariants are independent of choices made in their construction.

Interpretations of 'nearby' 4-manifolds are provided.

Exploration of graphs of 4-manifolds from unbalanced trisections.

Abstract

Given two smooth, oriented, closed 4-manifolds $M_1$ and $M_2$, we construct two invariants, $D^P(M_1, M_2)$ and $D(M_1, M_2)$, 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.