Torsions and intersection forms of 4-manifolds from trisection diagrams

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 $\phi$ 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 $(X;\phi )$ in terms of the first homology of $(\Sigma ;\phi )$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\phi )$ in terms of the intersection form of $(\Sigma ;\phi )$.

The homological computations have been slightly modified, switching from Mayer-Vietoris sequences to long exact sequences of pairs. Examples have been added