Symplectic forms on trisected 4-manifolds

TL;DR

This paper provides explicit criteria for when a trisection of a smooth 4-manifold can be used to construct a symplectic structure, offering a new characterization of symplectic 4-manifolds and linking 3D contact geometry to 4D symplectic topology.

Contribution

It establishes the converse of previous work, giving explicit conditions on trisections to produce symplectic structures, thus characterizing symplectic 4-manifolds.

Findings

Criteria for constructing symplectic structures from trisections

New characterization of symplectic 4-manifolds

Connections to contact geometry and Thurston norm

Abstract

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the converse direction and give explicit criteria on a trisection of a closed, smooth 4-manifold $X$ that allows one to construct a symplectic structure on $X$. Combined, these give a new characterization of 4-manifolds that admit symplectic structures. This construction motivates several problems on taut foliations, the Thurston norm and contact geometry in 3-dimensions by connecting them to questions about the existence, classification and uniqueness of symplectic structures on 4-manifolds.