Lecture notes on trisections and cohomology

TL;DR

This paper explores geometric interpretations of the second cohomology group in trisected 4-manifolds, drawing analogies with Hodge theory and algebraic geometry to deepen understanding of these structures.

Contribution

It introduces new geometric interpretations of $H^2(X)$ in the context of trisected 4-manifolds, linking cohomology classes to $(1,1)$-classes similar to Hodge theory.

Findings

Classes in $H^2(X)$ can be interpreted as $(1,1)$-classes.

Provides geometric insights into the structure of 4-manifolds with trisections.

Lays groundwork for further research in symplectic trisections.

Abstract

In these notes, we describe several geometric interpretations of $H^2(X)$ when $X$ is a trisected 4-manifold. The main insight is that, by analogy with Hodge theory and sheaf cohomology in algebraic geometry, classes in $H^2(X)$ can be usefully interpeted as $"(1,1)"$-classes. These notes formed the first half of an aborted seminar on symplectic trisections at the Max Planck Institute for Mathematics during Spring 2020.