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 $H_1(\Sigma ;\mathbb{Z})$ 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.

14 pages, 2 figures. Submitted to appear as part of the Proceedings of the National Academy of Sciences collection on trisections of 4-manifolds