turkmath.org

The theory of Lefschetz fibrations is a tool to understand the topology of symplectic 4 −manifolds via positive factorizations in mapping class group. The slope $\lambda_f$ of a Lefschetz fibration is a numerical invariant which is determined by the Euler characteristic and the signature. Hain conjectured that every relatively minimal genus-$g$ Lefschetz fibration $f: X \to \mathbb{S}^2$ satisfies the slope inequality $\lambda_f \geq 4 − 4/\lambda$. Recently, Naoyuki Monden constructed Lefschetz fibrations over the two-sphere which do not satisfy the slope inequality. In this talk, I will establish new examples of Lefschetz fibrations having slope less than the ones that Monden constructed.