Unique fiber sum decomposability of genus 2 Lefschetz fibrations

TL;DR

This paper explores the fiber sum decomposability of genus 2 Lefschetz fibrations, constructing exotic symplectic 4-manifolds via lantern relation substitutions and identifying a unique decomposing fibration.

Contribution

It introduces a novel method using lantern relation substitutions to construct exotic symplectic 4-manifolds and proves the uniqueness of a certain decomposing genus 2 Lefschetz fibration.

Findings

Construction of exotic symplectic 4-manifolds via lantern relation substitutions.

Identification of a uniquely decomposing genus 2 Lefschetz fibration.

Proof that the constructed manifolds are symplectically minimal.

Abstract

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariant of the resulting symplectic 4-manifolds, and show that they are symplectically minimal. We also investigate how these exotic smooth 4-manifolds constructed via lantern relation substitution method are fiber sum decomposable. Furthermore by considering all the possible decompositions for each of our decomposable exotic examples, we will find out that there is a uniquely decomposing genus 2 Lefschetz fibration which is not a self sum of the same fibration up to diffeomorphism on the indecomposable summands.