Torsions and intersection forms of 4-manifolds from trisection diagrams

Abstract

Gay and Kirby introduced trisections which describe any closed oriented smooth 4manifold 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 Σ, guiding the gluing of the handlebodies. Any morphism ϕ from π 1 (X) to a finitely generated free abelian group induces a morphism on π 1 (Σ). We express the twisted homology and Reidemeister torsion of (X; ϕ) in terms of the first homology of (Σ; ϕ) and the three subspaces generated by the collections of curves. We also express the intersection form of (X; ϕ) in terms of the intersection form of (Σ; ϕ).

Trisecting a 4-dimensional book into three chapters

“…It would be interesting to see if one can determine if a given trisection is induced by an open book. Here it is worth pointing out that we can compute the signature [11,39] and the intersection form with coefficients in the group ring [12] from any trisection diagram.…”

The algebraic topology of 4-manifolds multisections

“…A trisection diagram determines a smooth 4manifold up to diffeomorphism, so that one should be able to read topological invariants of the manifold on the diagram. In the setting of closed 4-manifolds, Feller, Klug, Schirmer and Zemke [FKSZ18] provided a computation of the homology and intersection form of the manifold from a trisection diagram, and Florens and Moussard [FM19] derived the twisted homology and torsion, and the twisted intersection form. Following these papers, Tanimoto [Tan21] computed the homology of 4-manifolds with connected boundary.…”

“…We keep in this section the assumption that ∂X = ∅. The intersection forms are formally identical to the closed case treated in [FM19]. The upshot is that the intersections between various cycles in X can all be made to coincide with intersections in Σ.…”

“…Similarly, let Z ∂ 2 be the relative 2-skeleton of Lemma 4.2. As observed in [FM19], we may push each Z ∂ 2 ∩ C i slightly into its collar so that it is pushed into ∪ 1≤j≤i X j . This being done, the intersections between 2-chains in C 2 (Z 2 ) and C 2 (Z ∂ 2 ) will coincide with intersections between the boundaries of the sub-chains lying just in Z 2 ∩ C i with the boundaries of the sub-chains lying just in Z ∂ 2 ∩ (∪ i<j≤n C j ), see the left-hand side of Figure 3.…”