On the Fukaya-Seidel categories of surface Lefschetz fibrations

TL;DR

This paper establishes that positive allowable Lefschetz fibrations (PALFs) can be given an exact structure, and shows their Fukaya-Seidel categories are invariant under symplectic changes, providing new insights into their algebraic and geometric properties.

Contribution

It proves PALFs admit an exact Lefschetz fibration structure and demonstrates the invariance of their Fukaya-Seidel categories, revealing deeper algebraic structures.

Findings

Fukaya-Seidel categories are independent of symplectic structure choices.

Derived Fukaya-Seidel categories contain more information than Milnor lattices.

PALFs can be equipped with an exact Lefschetz fibration structure.

Abstract

We prove that a positive allowable Lefschetz fibration, PALF in short, admits a structure of exact Lefschetz fibration in the sense of Seidel \cite{Se08}. If the two-fold first Chern class of the total space is zero, we obtain the Fukaya-Seidel category. We prove that the derived Fukaya-Seidel category of PALF is independent of the choice of the symplectic structure. At the end of this paper, we study examples and show that derived Fukaya-Seidel categories have more information than the Milnor lattices of PALFs.