Lecture notes on trisections and cohomology
Peter Lambert-Cole

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 when is a trisected 4-manifold. The main insight is that, by analogy with Hodge theory and sheaf cohomology in algebraic geometry, classes in can be usefully interpeted as -classes. These notes formed the first half of an aborted seminar on symplectic trisections at the Max Planck Institute for Mathematics during Spring 2020.
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Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology
