Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds

TL;DR

The paper demonstrates that most symplectic 4-manifolds, after enough blow-ups, can have multiple distinct Lefschetz fibrations that are not related by known surgeries, revealing complex structures in their symplectic topology.

Contribution

It shows the existence of multiple nonisomorphic Lefschetz fibrations on certain symplectic 4-manifolds that cannot be connected through Luttinger surgeries or partial conjugations.

Findings

Existence of multiple nonisomorphic Lefschetz fibrations on symplectic 4-manifolds.

These fibrations cannot be related by Luttinger surgeries.

Monodromy factorizations with same characteristic numbers are not always related by partial conjugations.

Abstract

We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twists, answering a problem posed by Auorux.