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

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 H1(F; Z) generated by the three sets of curves and the intersection pairing on H1(F; Z). 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.