Calculating the homology and intersection form of a 4-manifold from a trisection diagram

TL;DR

This paper provides explicit formulas and an algorithm to compute the homology and intersection form of a 4-manifold directly from its trisection diagram, enhancing understanding of 4-manifold topology.

Contribution

It introduces a method to determine homology and intersection forms from trisection diagrams, including explicit formulas and an algorithm, and shows all certain trisections admit algebraically trivial diagrams.

Findings

Explicit formulas for homology groups of 4-manifolds from diagrams

An algorithm to compute the intersection form

All (g;k,0,0)-trisections admit algebraically trivial diagrams

Abstract

Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(\Sigma;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing on the diagram surface $\Sigma$. This includes explicit formulas for the second and third homology groups of $X$ as well an algorithm to compute the intersection form. Moreover, we show that all $(g;k,0,0)$-trisections admit "algebraically trivial" diagrams.