Trisections and symplectic structures on smooth 4-manifolds & Higher dimensional generalizations – SyTriQ

This project concerns geometric topology, more precisely the study of smooth manifolds of dimension 4 and higher. Building on the recent theory of trisections of smooth 4--manifolds introduced by Gay and Kirby, the project investigates the relationship between this theory and symplectic geometry, and the possible generalization to higher dimensional manifolds.

A Heegaard splitting is a decomposition of a compact 3-manifold into two handlebodies glued along a surface, a key notion in the study of 3-manifolds. Gay and Kirby developed an analogous construction for smooth compact 4-manifolds: they define a trisection of such a manifold as a decomposition into three 4-dimensional 1-handlebodies, with conditions on the gluing of the pieces. This project aims at exploring two aspects of trisections. The first objective is to study the relation between trisections and symplectic structures. This is motivated by the induced structures on the boundary of the manifold: a trisection induces an open book decomposition, a symplectic structure induces a contact structure, and the Giroux correspondence establishes a strong relation between open book decompositions and contact structures. The second objective is to generalize the theory of trisections to higher dimensions, studying first a notion of quadrisections for smooth compact 5-manifolds.