Minimality and fiber sum decompositions of Lefschetz fibrations

TL;DR

This paper provides a new proof of a conjecture on the minimality of fiber sums in Lefschetz fibrations and constructs examples of genus g > 1 Lefschetz fibrations with unique fiber sum decompositions on minimal symplectic 4-manifolds.

Contribution

It offers a simplified proof of Stipsicz's conjecture and presents the first examples of such fibrations with unique decompositions.

Findings

Short proof of Stipsicz's conjecture on minimality

First examples of genus g > 1 Lefschetz fibrations with unique decompositions

Fibrations constructed on minimal symplectic 4-manifolds

Abstract

We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.