Geography of symplectic 4-manifolds admitting Lefschetz fibrations and their indecomposability

Abstract

In this paper, we show that for a given finitely presented group 𝐺, there exist integers ℎ_𝐺 ≥ 0 and 𝑛_𝐺 ≥ 4 such that for all ℎ ≥ ℎ_𝐺 and 𝑛 ≥ 𝑛_𝐺, and for all 0 ≤ 𝑖 ≤ 2𝑛 − 2, there exists a genus-(2ℎ + 𝑛 − 1) Lefschetz fibration on a minimal symplectic 4-manifold with (𝜒, 𝑐₁²) = (𝑛, 𝑖) whose fundamental group is isomorphic to 𝐺. We also prove that such a fibration cannot be decomposed as a fiber sum for 1 ≤ 𝑖 ≤ 2𝑛 − 2 if ℎ > (5𝑛 − 3)/2. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.