The necessary condition on the fiber-sum decomposability of genus-2 Lefschetz fibrations

Abstract

The fiber-sum construction gives us many interesting examples of Lefschetz fibrations. Which Lefschetz fibrations can be decomposed as fiber-sums? Stipsicz obtained some results on the fiber-sum decomposition, which state about the relationship between the minimality and the fiber-sum decomposability of Lefschetz fibrations. He proved that every Lefschetz fibration with section of self-intersection number $-1$ cannot be decomposed as any nontrivial fiber-sum. In this paper, we show that the reverse of this theorem does not hold and we characterize genus-2 decomposable Lefschetz fibrations with $b_{2}^{+} = 1$.