Gay and Kirby introduced trisections, which describe any closed, oriented, smooth 4-manifold X as a union of three 4-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 $\varphi $ 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 $(X;\varphi )$ in terms of the first homology of $(\Sigma ;\varphi )$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi )$ in terms of the intersection form of $(\Sigma ;\varphi )$ .

Gay 和 Kirby 引入了三等分法，将任何封闭的、定向的、光滑的 4 流形X描述为三个 4 维手柄的结合。三等分被编码在图表中，即封闭定向表面 $\Sigma $ 中的三个曲线集合，指导手柄的粘合。从 $\pi _1(X)$ 到有限生成的自由阿贝尔群的任何态射 $\varphi $ 都会在 $\pi _1(\Sigma )$ 上引发一个态射。 我们用$(\Sigma ;\varphi )$ 的第一同调来表示 $(X;\varphi )$ 的扭曲同调和 Reidemeister 扭转以及由曲线集合生成的三个子空间。我们还根据 $(\Sigma ;\varphi )$ 的交集形式来表达 $(X;\varphi )$ 的交集形式。