Trisections and Lefschetz fibrations with $(-n)$-sections

TL;DR

This paper extends the construction of trisections from Lefschetz fibrations to those with $(-n)$-sections, providing explicit diagrams for complex 4-manifolds and their fiber sums, generalizing previous work on $(-1)$-sections.

Contribution

It introduces methods to construct explicit trisection diagrams for 4-manifolds with $(-n)$-sections and their fiber sums, broadening the applicability of Lefschetz fibration techniques.

Findings

Constructed trisections for manifolds with $(-n)$-sections.

Explicit trisection diagrams from monodromies of Lefschetz fibrations.

Extended previous $(-1)$-section results to general $(-n)$-sections.

Abstract

Castro and Ozbagci constructed a trisection of a closed 4-manifold admitting a Lefschetz fibration with a $(-1)$-section such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. In this paper, for a closed 4-manifold $X$ admitting an achiral Lefschetz fibration with a $(-n)$-section, we construct a trisection of $X \# n\mathbb{C}P^2$ if $n$ is positive and $X \# (-n)\overline{\mathbb{C}P^2}$ if $n$ is negative such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. We also construct a trisection of the fiber sum of two achiral Lefschetz fibrations with $n$- and $(-n)$-sections such that the corresponding trisection diagram can be explicitly constructed from monodromies of the Lefschetz fibrations.