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

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 $H_1(\mathrm{\Sigma };\mathbb{Z})$ generated by the three sets of curves and the intersection pairing on the diagram surface $\mathrm{\Sigma }$. 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.