Symplectic forms on trisected 4-manifolds

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.