# Torsions and intersection forms of 4-manifolds from trisection diagrams

## TL;DR

This paper explores how trisection diagrams of 4-manifolds encode algebraic invariants like twisted homology, Reidemeister torsion, and intersection forms, providing explicit formulas relating surface data to 4-manifold topology.

## Contribution

It introduces formulas to compute twisted homology, Reidemeister torsion, and intersection forms of 4-manifolds directly from their trisection diagrams.

## Key findings

- Expressed twisted homology in terms of surface data.
- Derived formulas for Reidemeister torsion from diagrams.
- Connected intersection forms of 4-manifolds to surface intersection forms.

## Abstract

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $\Sigma$, guiding the gluing of the handlebodies. Any morphism $\varphi$ from $\pi_1(X)$ to a finitely generated free abelian group induces a morphism on $\pi_1(\Sigma)$. We express the twisted homology and Reidemeister torsion of $(X;\varphi)$ in terms of the first homology of $(\Sigma;\varphi)$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi)$ in terms of the intersection form of $(\Sigma;\varphi)$.