The existence of an indecomposable minimal genus two Lefschetz fibration

TL;DR

This paper demonstrates the existence of an indecomposable minimal genus-2 Lefschetz fibration, providing a counterexample to a previously conjectured relationship between minimality and decomposability.

Contribution

It constructs an explicit example of an indecomposable minimal genus-2 Lefschetz fibration, disproving the conjecture that all minimal fibrations are decomposable.

Findings

Existence of an indecomposable minimal genus-2 Lefschetz fibration

Counterexample to the conjecture relating minimality and decomposability

Disproof of the converse of Usher's theorem

Abstract

It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists an indecomposable minimal genus-2 Lefschetz fibration.